Introduction and Brief Overview of Background Material. .
1. We may consider next what it was that Einstein was doing whilst Michelson, Hertz, Lorentz and Poincare were grappling with the foregoing* problems during the 1890s and early 1900s - as part of their 'traditional lines of enquiry'. These and other contemporary researchers were still seeking answers to the anomalies that science had recently thrown up in regard to the behaviour of light - in the usual time-honoured ways of physics, that is, of mechanics - albeit taken to the limit. Meanwhile, in 1894, Einstein was 15 and attending his Munich High school (Gymnasium) which he found rather rigid, and stifling of natural curiosity. He excelled at mathematics and while interested in physics, the subject there was based on rote learning of rather dated texts. Maxwell's ideas, for example, were not yet taught, never mind those of Hertz or Lorentz - whose 1892/95 theory (still developing) had not long been published. The Einstein family business failed that year and they decided to move to Milan but leave young Albert with friends in Munich, so he could complete his high school diploma there. However, after 6 months of disaffection at school, he was asked to leave - albeit with an excellent report on his mathematical abilities. This suited him as he had himself sought to leave on 'psycho-social' grounds (missing his family, etc). [* As described in the 'Background' section elsewhere.]
2. In Milan, he continued self study in mathematics and probably some physics - although what texts he used seems unknown. But they must have started him 'thinking' and influencing the direction of his future interests. On the basis of this and his good report from Munich, he applied in 1895 to the Zurich Polytechnic (which would teach in German) but failed the entrance exam, again doing very well in mathematics. The Director advised that he first study for a Swiss high school diploma - which he did in a sympathetic school in nearby Aarau where he was allowed to pursue further his own interests in physics and related lab experiments. He soon obtained his diploma and the Polytechnic then accepted him in 1896 without further examination. He was about 17.
3. By this point, he had decided that a future in mathematics required too much specialisation and preferred the more fundamental and general aspects of physics, later claiming that he realised he had a talent for manipulating the more general principles of science and 'scenting' out where any inconsistencies lay. He also maintained that his major capacity wasn't his intellect but having the stubbornness of a mule - to keep persisting on any such problem, for years if necessary. It was thus as a near 17 year old about to enter Zurich, having read certain accounts of the physics of the day, that he formulated in his mind his famous 'thought experiment' concerning what one would observe if one could travel with a beam of light - at its speed. [He must have mentioned this to someone at about that time in order that it was so fully accepted in later years.] In terms of the accepted physical principles of the time - namely Newtonian classical mechanics and the later Maxwellian electrodynamics (discussed in articles by Helmholtz and Mach which he had apparently read), answers to this question from these two points of view appeared mutually incompatible. Amazingly, this early question would remain the key motivator of his continued attention to this subject for several years and as such would contrast with the differently focused problems, paths and goals of Michelson and Lorentz, albeit in this same general area of the physics of light and motion.
4. That the young Einstein formulated such a question at all clearly implies he was indeed reading about current matters in physics by then and over the following years (ca 1896-99) - especially concerning the movement of light through the assumed ether - even though Maxwell's equations and Lorentz's ideas were still not taught yet at the Zurich Polytechnic. But it was just these aspects that led to certain anomalies. Would that wave of light of his thought experiment, originating behind him but which he observed by turning and facing them, remain as a still, frozen, dark wave as he and that wave travelled together at its immense speed, or would it still proceed into his eyes as normal light at its usual immense speed - ie relative to him ? But at what actual (vs perceived) speed would it have to go to do that? Newton's mechanics would predict one thing while Maxwell's electrodynamics would seem to predict another. Which was correct? He struggled with such thoughts for several years before finding the only valid solution (apparently) - in June 1905. Thus, the problem with the behaviour of light that Einstein would address was not quite the same as that addressed by Maxwell, Michelson, Lorentz and Poincare, as described earlier. Would their different suggested solutions to these slightly different problems nevertheless provide the answer to both (being based on the same required principles) and if so, which solution might prove the more valid overall ? [I've struggled myself with an idea of someone travelling past the Sun at or near the speed of light towards the Earth and being accompanied by a beam of light released just at the moment he passes the Sun. Do both take 8 minutes to reach the Earth and if so, how would the same beam of light appear to the traveller at his side? Does it depend on where the clock is situated ?]
5. It appears that there was at the time (ca 1900) some concern abroad about certain inconsistencies in the physics of light reported over the previous decade - following growing acceptance of Maxwell's ideas - as a stimulus to Einstein periodically to re-visit his early thought experiment. Similar inconsistencies in earlier actual experiments and explanations about light - concerning its velocity, frequency and direction (as by Fizeau, Doppler and Bradley, for example) of which he would gradually become more familiar while studying at Zurich - must have implied similar confused results to those of Michelson (of whose experiment, on the other hand, he would later say he was not sure he was then aware) - that is, in showing an unexpected constancy in light's speed and/or no clear evidence for the role of a still ether medium in its transmission. Was the motion of light affected by the motion of its source and/or of its assumed medium - as in air, a vacuum, through running water or from a moving Earth through a still ether, etc, etc? With no support found for a still ether, light's speed would possibly not be hindered thereby which may have accounted for its odd, unexpected constancy. One wonders if such findings, if not by Michelson then comparable findings by those earlier workers, may have led Einstein to an early suspicion about that suggested constancy and hence to a deeper analysis of Maxwell's electromagnetic equations. Would it mean that no matter how fast one travelled, light would/must always move away from its source (and such an observer!?) - at its one constant speed, as Maxwell's equations now (ca 1902) seemed to suggest - at least to Einstein? But, how could this be? That is, would light's speed remain constant at whatever speed any observer and his associated frame of reference was travelling (relative to whatever speed the frame of reference was moving on which the light's source was located)? It would seem to be counter-intuitive to traditional understanding.
6. On the other hand, light's speed was seemingly still expected to vary by most other workers whenever its source moved (despite Maxwell's equations). Even Maxwell himself, it seems, hadn't really taken on board the apparent universal reality and implications of its apparent constancy (in all situations). He had also maintained some belief in a role for an ether in light's transmission which itself may well affect its otherwise constant speed. [Note: One must be careful in analysing experiments in which the time taken for a body to move from a to b (even if that 'body' be a ray of light) might be confused with the velocity of that motion; ie is the distance travelled necessarily always the same ?]
7. While completing his studies, continuing to keep abreast of such contemporary ideas as best he could - as well as learning about earlier findings pertaining to this general area of concern in German publications, Einstein eventually gained non-academic employment just after the turn of the century. Nevertheless, he would continue to wrestle with essentially this same basic problem for several more years (1902-1905) - before finally resolving it satisfactorily before anyone else. [Some would say 'just before'.] We have enquired above how other researchers of the time were conceiving such problems in this same general area. Clearly, they had something to do with the propagation of light in its generally accepted medium or carrier - the ether. But, was it the motion or not of the latter or the constancy or otherwise of the former that concerned them most, if either?
8. To reiterate our earlier remarks in this regard, the problems in this general area seem to have arisen mainly out of Maxwell's conclusions regarding the character of light as it pertained to the contrary views of Fresnel and Stokes with respect to the motion of the generally assumed ether and its role as the medium for light. Maxwell seems to have believed that his new conception of light as an electromagnetic wave should still be propagated by means of an essentially still ether, if not now one of assumed elastic constitution. This had long been accepted. To this end, he suggested a means of testing this proposition although felt the effects concerned would likely be too subtle to be found by then current methods. However, this challenge was soon taken up by Albert Michelson - firstly in 1881 and later with Morley in 1887 (as described in detail earlier). Maxwell's suggestion entailed measuring the relative motion of the Earth through the assumed still ether and relied upon differences in the velocity of light sent through the ether from and in the direction of that moving Earth (compared to light travelling at 90 degrees to this direction) to establish this. Thus, paradoxically, he appeared to be going against his own (or, later, Einstein's) conclusion that light's speed was in fact a universal constant regardless.
9. After some period of ambivalence, Einstein (more than others) seems eventually (ca 1903) to have taken on board that the implication of Maxwell's equations was that light's speed was indeed an invariable constant and also (unlike Maxwell himself) that it didn't require an ether for its transmission (or, as Lorentz believed, as a source of physical or electromagnetic influence on the dimensions of bodies (and the time they took) passing through its assumed absolute stillness). Thus, various interpretations were advanced (eg by Lorentz), consistent with the mechanical model, to accounted for the awkward absence of any variation in light's speed under such conditions and the related support expected for the existence a still ether. Presumably therefore, the realisation and implications of any actual constancy of light's speed was frustrated and delayed. Moreover, neither Maxwell nor Michelson appeared to be seeking information that pertained directly to the question of the variability or otherwise of light's speed in any case; its assumed variability when originating from a moving vs stationary source and travelling through an ether 'wind' (despite the interpretaion given his equations by Einstein) was, it seems, generally assumed to be the case. Lorentz was of this same view. [Does the fact that the observers were also moving with the Earth affect this interpretation ?] Such expected variation in the speed of light was simply to be utilised as a convenient means to establish or confirm in particular the existence and effectively opposing motion of a still ether per se - thereby further confirming its role and indeed existence as the medium for (an electromagnetic form of) light and, as later hypothesised by Lorentz, as a previously unsuspected influence of the size (length) of material bodies passing through it and an associated effect on the passage of time. [One wonders what their conclusions would have been had their expectations been achieved ?]
10. There appears to be no reference in the relevant papers by either author about seeking to establish anything about the constancy of the speed of light itself (more the contrary), nor (initially) anything concerning a role that such an assumed still ether (as a medium), if so confirmed, may also have as a system of absolute rest - one that might pertain to concerns about...anything (as eg testing Newton's ideas about 'real' motion, real size or real time in mechanics, say) - at least not in explicit terms that were ever specified in the published reports of these two famous experiments (ie before ca 1904 when that absoluteness eventually became relevant to Lorentz with respect to electromagnetic effects of bodies moving through an assumed still ether). This implies that they were also not particularly interested in addressing anything about the validity of the existing principle of relativity (even though Micheslon's results could, I believe, be interpreted in terms of addressing/testing this important question). However, the still ether being assumed by Michelson would, as mentioned, later prove to be of interest to Lorentz - as an explanation of how light's unexpected constancy in those experiments could be accounted for (as just touched upon) in terms of a shortening of the apparatus used in its measurement, coupled with a less well explained variation in the associated times taken for that motion as any body passes through it. Apparently on the basis of reviewing Lorentz' papers of 1895 and 1898, Poincare in 1899 (and also in 1900 and 1902) had recalled the principle of relativity and had concluded that it applied only to the motion of matter and not to either light or the ether. He doesn't appear to have altered this view before 1904 or so but eventually felt that a new mechanics was probably needed to somehow also incorporate light's behaviour under its perview.
11. The hypothesised role of a truly still ether as part of Lorentz's later ideas (of ca 1895 and 1898) would appear to have been known by Einstein by about 1903 or so. When he eventually set out his own views on this area of concern (in his famous paper of 1905), he begins by referring to just this matter - in terms of the 'oversight' he felt was the case with respect to the symmetry that characterised the induction of a current. For the asymmetry generally assumed implied an acceptance or assumption of absolute motion vis a vis a truly fixed reference criterion by which means only could such motion be established. The appropriate reference points, he maintained, were however always equally relative, balanced and symmetrical, never one of them being absolute or somehow having priviledge, primacy or precedence (as more 'still') and thus the relationship asymmetrical. One wonders if others before him ever addressed this matter? Possibly Mach and/or Poincare? In any case, the implication was rather fundamental: there was no need for a still ether in some absolute sense - at least to provoke electromagnetic changes; the relative motions of the relevant elements could provoke each other (if such did in fact occur). Also, it meant that all motion was relative and symmetric and thus everywhere of equal significance on outcomes. No still ether or absolute motion based on same was required nor was there evidence for same. Another explanation was required for whereever that explanation was suggested (like mutual symmetry).
12. [Note: One might also point our here that Einstein was not trying to explain (as had Lorentz) an unexpected constancy in light's speed (in order to save the conclusion that would otherwise have been supported concerning the role of the ether medium in light's propagation), as found by Michelson with his particular experimental arrangements, as Einstein would actually have expected that constancy (had he been aware of that result) although claimed he was not then aware of Michelson's experiment (nor of Lorentz's or Fitzgerald's' immediate explantions). Rather, he sought to resolve his different question - namely, that if his interpretation of Maxwell's equations was right regarding the constancy of the speed of light, but Maxwell's ideas about an ether medium weren't, how could he explain certain* phenomena (which would later be more succinctly interpreted in terms of an incompatibility between that constancy and a principle of relativity that underlay Galileo's and Newton's mechanics)? The answer would however seem to entail some familiarity with and assistance from the ideas of Lorentz and Poincare (who were in frequent correspondence) which eventually emerged out of that earlier, if (to Einstein) essentially unknown, experiment by Michelson. Because of this and the fact that Einstein's solution to his question would however bear directly on Lorentz's answer to his different problem, the exposition of Einstein's approach and solution is often described in retrospect in terms of the contrasting Michelson-Lorentz-Poincare interpretation - as though Einstein followed on directly from their approach - essentially to solve their problem by his own methodology. Apparently, he didn't; he was pursuing his own agenda - but describing that on its own terms alone seems generally to have been avoided by many historians of science - as it proved useful in accounting for Michelson's and Lorentz's conclusions also.]
[* It would be useful to elaborate upon such phenomena here - ie beyond the basic elements pertaining to his 'thought experiment' with its 'conceivable' frozen waves - or not. That is, what was it that still remained inexplicable once one had accepted that light would always speed away no matter at what speed its source or any observer travelled - that is, before concerns about the principle of relativity informed his quest? Seemingly, it was eventually resolved (in part?) in terms of the invariable reliance of information concerning spatial size/length and time (ie on any distant body's velocity) on light signals. This is elaborated further below. The other part may have been the constraints placed on velocity by the 'ceiling' of maximum possible velocity (attainable only by light), although these could well be two ways of saying/accounting for the same thing.]
13. It should be possible therefore to delineate Einstein's solution of his distinct problem - independently from and without such frequent reference to those of Michelson, Lorentz et al - as the literature seems so typically to do. It is possible that Lorentz's later writings, with some of which Einstein no doubt did become familiar, made little or no direct reference to Michelson and the earlier, initial relevance of same to those later ideas of Lorentz. So Einstein may have been indirectly (and unknowingly) addressing Michelson's results nevertheless. And while it may have seemed probable that Einstein would be familiar with Lorentz's definitive paper of 1904 (as it bore so strongly on the very topics Einstein was soon to address), he had in fact allegedly not seen it before his own paper was written and sent to the publishers by late June 1905. And the 3 relevant papers referred to by Lorentz in that 1904 article (published in English journals) - by Rayleigh (1901), Trouton and Nobel (1902) and Brace (1903) - were very likely also unknown to Einstein. In these, negative results regarding the variability of the speed of light were again obtained (as they had been with Michelson) all of which Lorentz felt he could now explain in terms of his 'ether dependent contraction hypothesis' - as described more formally within his new (1904) electron theory.
14. We should note however that his new transformations (derived earlier) may have eventually provided Einstein with the quantitative framework or structure by means of which his own qualitatively different interpretations could be realised - by means of essentially the same equations. On the other hand, Einstein often referred to the transformations as the 'Lorentz transformations' (ie when writning about relativity some years later) when, in fact, he may well have been referring to those of identical form which in fact he had himself derived independently, simply referring to them now as 'Lorentz's' as a courtesy or as short-hand, they having been so accorded by Poincare by about 1904. This, I feel, has the disadvantage of associating those transformations too much with Lorentz alone and his ether-based concepts. We may note here that, as suggested above, neither Michelson nor Lorentz (in seeking to 'save' the former's experiment) were initially seeking evidence from those 1881/1887 experiments concerning the effects of a still ether (or anything else) on the dimensions of matter, nor on any variations in time. This seems to have been generally overlooked in later analyses, this focus only becoming central after the appearance of the Fitzgerald/Lorentz hypotheses by the early to mid-1890s.
[Note: One would like to know just when any implications of the original principle of relativity were brought to bear upon these matters during the 1890s. When did Lorentz firsst bring this principle into his discussions (as presumably brought to his attention by Poincare in their frequent correspondence - ca 1896-1904?).]
15. If light's speed had been accepted and recognised (after Michelson) as a true constant, it would - ie after that principle became a consideration - have provided the one and only known means by which it would (in theory) be possible to determine whether one was at rest or moving in a real, absolute sense - something which the principle of relativity for mechanics dictated should never be possible; all laws should operate identically in all uniformly moving frames (and in any absolutely still frame if such was ever available) such that there would be no way to distinguish such motion (or stillness) from that which could occur in any other frame. Thus, all observers (or their measuring instruments) should find all laws of nature identical wherever they were and whatever their differing respective (uniform) speeds relative to each other may be. If light could somehow over-ride this dictum (by not reflecting the speed of its own frame of reference as well as that inherent in itself - in relation to that of another frame - it would prove to be a worrying anomaly. One would apparently be able thereby to determine if one's own frame was the one moving or still in some absolute sense vis a vis some other frame. But was this implication ever considered, voiced or tested at that time (eg ca 1899)? As far as I can see, it wasn't then so considered in these terms - even though so much research was focused on this general area. [One wonders who in fact did first consider this aspect - Lorentz and/or Poincare, or was it only Einstein ? No, it was Poincare, I believe; but it was one of those crucial 'ingredients' which they couldn't quite fit into the final picture.]
16. Poincare suggested the relevance of the principle of relativity (as referred to by him, I believe) in 1899, 1900 and 1902 (see references in......) which was then 'picked up on' by Einstein who apparently 'saw' its true significance more than most (including Lorentz). Light should, he would later reason, also prove consistent with this same underlying principle. Its law should thus be unaffected by differently moving reference frames and, like all other 'mechanical' laws, that reality should be revealed in the true net values obtained after the application of one and the same form of appropriate transformations equations. It was the 'laws' per se which should remain unaffected (directly) if the principle of relativity must hold true, not the consequent velocities which albeit they (relatively indirectly) so determine.]
17. The same basic considerations were eventually discussed in terms of 'inertial systems', invariance and transformation equations, etc. This general area would no doubt have increasingly informed Einstein's thinking during 1904. [Note: Such references (as here) to the dictates of a principle of relativity are an example of the difficulties that arise when certain factors in the evolution of Einstein's thinking and theory are taken out of their proper (if often uncertain) chronology. When considered in retrospect, it is too easy to make assumptions about just when a given aspect, especially if seen later to be more fundamental than initially realised, was first considered and integrated into the sequence of a still emerging logic. Thus, it appears to me that we can't be certain just when the principle of relativity, say, began to inform or guide the focus of his (or anyone else's) resolution of 'the problem'; nor how fully. But as it gradually (or even suddenly) became an important element to consider (likely around 1903/4, one would assume that its early manifestations would be 'those uncertain phenomena' referred to above which would be more succinctly formulated and expressed later as 'the incompatibility between the constancy of light and this fundamental principle' - as it then stood. [I can now appreciate that they probably did exist - in terms at least of the above mentioned inertial systems, etc but, again, not necessarily as early as later reviewers seem to suggest. It seems quite possible that after his insight into the relevance of how Time should be measured that the particular problem that this helped resolve (ie an ether-less explanation with which he had been struggling for so long) could be later seen as a specific case of the more general problem of how to reveal that the two basic 'laws' were really compatible all along; that is, his 'breakthrough' didn't result in the first instance by considering the problem from that much more general principle approach alone.]
18. In any case, for seemingly other reasons, at least initially, Fitzgerald and Lorentz had taken up the challenge of Michelson's negative results in this regard - in order (effectively) to maintain the consistency of the mechanical model vis a vis Fresnel's view of the mainly still ether as the assumed medium for light. Maxwell and Michelson were (as mentioned above) seeking essentially to confirm and/or establish this, I believe. [All this has been described above but has relevance here also.] Such an orientation may have served also to rekindle interest in the idea of Newton's fixed reference system of an immobile, absolute space with a principle of relativity in which (without ever being mentioned seemingly) time and space would also continue to be quite reasonably assumed to be independent, absolute and invariable - as they had always (and seemingly had only ever) been so considered. All or even just some of this (including relevant inertial system considerations, invariance and valid transformation equations) was (eventually) riding on the ultimate validity or otherwise of Lorentz's efforts - which by about 1904 was felt by many (including Poincare) to have been virtually attained if, that is, one could just formulate a kind of 'new mechanics' to resolve one or two remaining difficulties (as regarding some logic for his explanation about time's seeming and required dilation) - for the 'old mechanics' had been stretched to (or even beyond?) its very limits.
19. It may have been such generalists who realised that it shouldn't have been possible to determine one's absolute motion by using a feature of mechanics (in which camp optics (as everything) was still assumed to fall) - ie the motion of light in its medium. To do so would require a special dispensation for that still ether medium. [See also discussion on this point by Einstein in our summary of his 1916 book below.] One has inserted the qualification 'eventually' in the above as it is still unclear if any such 'effective' concern (re the mechanical model and its associated (classical) principle of relativity) was at all to the fore by ca 1900-02, say - rather than that all researchers at the time were simply taking all that for granted and simply trying to explain certain anomalies in the time honoured ways of the only known model - mechanics - whose dominant position wasn't, I believe, yet being directly questioned. For some reason, it was often formulated in terms of whether or not the Earth or even the Solar system moved in absolute terms vis a vis a still ether (as an absolute reference system, as per Newton). If this question was relevant because it implied certain other relationships (re the mechanical principle of relativity), mention of any such implications seem to have been studiously avoided by Maxwell, Michelson and Lorentz (initially) in the most quoted papers of the day - ie before ca 1904. There concern was limited to the propagation of light through the ether.]
20. However, Poincare would eventually suggest (seemingly before Einstein) that there was probably no such thing as absolute rest or motion - as represented by a still ether, say (as Mach had also stated some years earlier) or seemingly by any other method, and hence that the principle of relativity should apply without any such special consideration to light or its medium - ie in all situations, whether mechanical or electrodynamic. As such, the latter must somehow 'fit in' to existing principles of mechanics and thus do so within its own constraints without invoking absolutivity. That is, that the aforementioned distinction of who is or is not really moving should not be determinable if there is no absolute fixed point or system of reference (as a still ether) even with the seeming constancy/independence of light's velocity. But, (in a sense 'paradoxically'), that only available principle (in Poincare's eyes) presumably still retained 'its' conceptions of invariable/constant/absolute time and space (ie possibly implicitly; I'm not aware that this was ever questioned overtly by these researchers or any alternative considered before Eimstein's 1905 'bombshell'). However, any conceptions of these that conceivably may have been considered instead as relative and/or varying would not reasonably emerge simply by noting any 'inconsistency' within this odd 'balance' or symmetry of these converses. A much more indirect and analytic evolution of this idea would, it seems, be required - arising out of the insistent demands of the implications of the unyielding constancy of the velocity of light vis a vis an apparently necessary, inevitable principle of relativity (seemingly with no still ether as a reference criterion to complicate such principles, short of it requiring some 'special case' interpretation). [NB The actual symmetry in current induction may also feed into this interpretation.] But the necessary 'malleability' of time and space (underlying velocity) that would allow this had yet to be appreciated or, if appreciated, to have its origin rationally accounted for. It was still 'just around the corner' - in Einstein's study.
21. Thus Poincare felt that the relativity principle that he advanced in 1902, 1904 and 1908 (and earlier) needed a more complete 'explanation' by which its application specifically to the matter of light's propagation could somehow be better reconciled - ie in that 'new' mechanics (but seemingly still 'of the ether') that he felt was needed - because Lorentz's mathematics still required awkward post hoc 'adjustments', especially with respect to Time which, with Space, were of course the two fundamental components of all calculations of the motion and velocity of any body, including light). In effect, Einstein did this such that the central implication of the principle of relativity - of never being able to distinguish one's absolute motion vis a vis others by means of any law of nature - mechanical or otherwise (ie by utilising light despite its constant speed) - was somehow maintained whereas in Lorentz's theory this was the case only if the awkwardness of light via its still ether was accepted. It seems that Einstein found Poincare's reference to the principle of relativity (around 1903/4?), and the need for it to have a 'new explanation', provided him with that more general principle of physics that he came to believe was necessary - to (?finally) resolve the problem as he saw it, rather than proceeding any further by the more usual hypothetico-deductive, trial and error methods of mechanical science, with which he was (even theoretically or with thought experiments) making little or no progress through 1904.
22. In one of the many chatroom dialogues concerning relativity one contributor comments on another's remarks when citing Lorentz (written 1914; published 1921) as "questioning Einstein's acceptance (in 1905) that his variable time (t') in a moving reference frame represented a true time - as represented by t of his 'local', stationary frame. He called the variables of the moving frame - ie x' and t' especially - as just "subsidiary quantities' introduced with the aid of a mathematical trick (and)..phenomena in the moving reference system could not be described in the same way as in the stationary system." [See later for details on the 'kinematic' analysis by Einstein in which these 'variables' are set out and analysed. We may note here also that this same criticism had been made with respect to Lorentz's own 'arithmeric tricks' when trying to account for his own time dilation aspects! Lorentz seems to have used the same symbols but reversed them as to the moving (Earth) and stationary (Ether) systems.] Now this quotation by Lorentz seems a little vague and abstract and may have been an initial view by him only; one would like more detail and specificity. Apparently, Lorentz's 1904 paper supports this initial stance in that his transformations were not, in his view, relating space-time measurements of the same event in the different inertial systems. Lorentz also wrote later (1921) that Poincare was indeed the one who had first formulated (or simply recalled?) the principle of relativity (not stating when or where), as well as providing the first 4-dimensional formulation of mechanics - before Minkowski.(But when and where was this first published?)
23. Thus, Lorentz apparently credited Poincare (who for some time also believed in a role for ether) as the true father of 'special relativity' and so rejected the idea of giving Einstein the Nobel prize on that basis. As Poincare was then dead, the committee apparently gave Einstein the prize instead for his important work on the photoelectric effect. Einstein had once agreed with Lorentz (in a letter 17 June 1916) that the special relativity theory does "..admit of an ether hypothesis..." although I believe he meant that while the ether theory wasn't inconsistent with relativity, the latter didn't require an ether and entailed fewer ad hoc hypotheses; as a result it was simpler and more credible. While Einstein was supposed not to have read Lorentz's 1904 paper before sending his own 1905 paper to the publishers, the contributor here quoted notes that the 1905 paper used the same notation as used in Lorentz's 1904 paper, and it begins the proof of the new transformations from the Galilean transformation - just as Lorentz had done earlier. But quite possibly such notation was then newly 'current'; why did Lorentz use it otherwise? Was it not based on Maxwell ? In any case, Einstein later agreed that he did know Lorentz's 1895 paper if not the later ones. And derivation of new transformations would quite naturally begin by modifying the existing ones, one would assume, as that would allow one to show why they had been inadequate and where they had to be modified.
24. [Did Poincare's 1902 reference to the principle of relativity imply that there were few if any earlier references to this principle before then - despite Lorentz and Hertz apparently also being concerned with related problems, if only implicitly ? But by what date? [I have an uncertain reference to Maxwell also referring to this long accepted principle as early as 1875 and Poincare in 1898 and 1900.] In any case, did Einstein see this general principle as possibly the best way to discover some explanation for Lorentz's space and time adjustments (which he suspected were somehow necessary) - but without an ether - eg after having his insight regarding Time (see below) and (?later) his seeming 'thought trial' of accepting, and 'holding fast to', the constancy of light's velocity - and seeing just 'where the chips (regarding time and space possibly) would fall in any improved transformations' ? [Note: A German researcher, Voldemar Voight (1850-1919), had already derived a form of the 'new' transformations before either Lorentz or Einstein - in 1887. They were with respect to a stationary frame sent into motion at velocity v along a coordinate x designed to better analyse the Doppler effect. It did not (as Voight himself later admitted) pertain to a general coordinate transformation as would apply in special relativity. He seems to have followed Maxwell's approach. See also later in this regard.]
25 A central 'problem' in physics by the early 1900s was thus essentially the fact that while the speed of light would (as Einstein believed) turn out to be constant, this was not generally appreciated nor 'taken on board' as a truly universal principle/reality by most other researchers at the time. For, implicitly, it didn't accord with the unquestioned model of Galileo's and Newton's mechanics and the implied principle of relativity (as it then stood, if generally taken for granted and hardly considered) - with its classical transformations and reasonable implication that, in a mechanical world, everything's speed relative to any defined external frame of reference was the total sum of its own speed plus (or minus) that of the associated frame on which it and its source may be moving. [Nor was the role of that principle of relativity yet appreciated - vis a vis this still uncertain light principle.] Such speeds of motion would be based upon the only conceptions of time and space (the components of speed) that had ever been considered - both being unquestioningly constant. A role for a still ether (of ultimately substantive/mechanical properties) seems to have been an assumed element in this stance as well. [Presumably, this is an example of the consequences of the aforementioned problem of being able to distinguish thereby one's motion, etc. One might point out here that all other bodies only move after the application of some force and that an existing moving reference frame (when viewed from elsewhere), constitutes an effective additional force - whereas light (alone) seems to move without any such impelling force; it is self-initiating and propagating and is apparently oblivious to any such additional (source-based) forces - of either kind. But, mote precisely, its velocity is already 'at the limit', and constant. Maybe there is an implicit 'force; impelling (or releasing?) it. What happens when a candle is lit ?
26. Awkward results of experiments with light due possibly to the unrealised constancy of its speed were thus explained instead by advancing various hypotheses concerning the motion or otherwise of light's assumed ether medium and, later, by the electro-dynamic effects on bodies moving through the atter's assumed absolute stillness. But if light's invariable constancy [I am aware that the term 'invariable' is redundant here, and elsewhere, but has been thus used for emphasis] had been recognised earlier, there may have been more focus directed to certain unstated assumptions underlying the elements of all motion (including the principle of relativity) and thereby resolve many of the confusing results obtained in the past. In particular, Poincare wouldn't be so dissatisfied with Lorentz's 'adjustments' of space and time (the very phenomena, ironically, to which such unconcious assumptions pertained) to allow his new equations to fit the facts. By dispensing with the ether that Lorentz said caused these effects, Einstein would (with the principle of relativity now resting on his shoulders as it were (if pointed out by Poincare) and, once generalized, guiding him) eventually question such assumptions [but only after his principle ot relativity, simultaneity and time insights?] and account for the consequent (and still necessary) 'adjustments' more thoroughly and acceptably, if possibly not as easy to comprehend.
27. Also in 1904, Poincare delivered a most prescient lecture in America in which he appears to have appreciated earlier than most that the principle of relativity must surely apply to all laws of nature and that because of the extreme accuracy of Michelson's experiment, its implications about the role of both the ether and the gradually accepted non-variability of light's speed indicated (as mentioned above) the (existing) principle must be somehow more fully 'explained' (but within a new mechanics) in a way that was more consonant with those findings (and being unable to determine who was really 'moving'). He felt that Lorentz was very close but that his contraction and local time hypotheses (designed in effect to overcome that 'problem') were too arbitrary (and also, contrary to Newton, restricted to speeds slower than light). Poincare apparently did recognise that a better analysis and interpretation of the role of time and 'simultaneity' of events in particular, would be helpful; particularly as Lorentz's 'explanation' in this regard was suspect - being merely a mathematical 'device' to fit the facts (oddly, the same criticism levelled by Lorentz regarding Einstein's analysis). How close he seems to have been in his publications on either side of 1905 - the year of Einstein's largely independent 'breakthrough' ('independent' but quite possibly benefitting from Poincare's recent ideas). But did Poincare relate such recognition to the effects, if any, of observations from differently-moving frames of reference ? Did he (then) appreciate that light's speed was quite possibly a (true) constant? And did he ever agree that the ether (still or otherwise) was not a sustainable concept ? [Yes - eventually to the latter point at least, but not to the others, I believe. What a brilliant mind he must have had! One can see how Lorentz may have felt that Poincare, possibly because of his earlier attempted analysis of time and simultaneity, had actually'discovered' 'special relativity' - just before Einstein. But this was not generally accepted by most later analysts.]
28 While Poincare hadn't by this stage formulated his ideas into a coherent final form, he did thus recognise the fundamental place in any such 'new mechanics' of (an inertia based) principle of relativity which (he eventually realised) required all laws of nature to function identically for 'stationary' and uniformly moving observers alike such that neither group could know if they were the ones truly moving or moving the faster or not. This was why he said a 'better explanation' was needed with respect to that principle (as if intuitively realizing that it required some re-interpretation in respect of the inherent components (time and space?) which the different velocities with which it was concerned - but couldn't quite 'see' how this could be done. Like everyone else, he apparently continued to assume that time and space were absolutes which never varied. [But he did address the matter of simultaneity: to what purpose exactly?]. It appears that he also continued to accept that light's speed needn't necessarily be unvarying, I believe. Or did he soon come to accept this constancy as a 'law' of nature - eg by the time of his 1906 paper on dynamics of the electron? In 1908, he again emphasised that the principle of relativity was a general law of nature and that there was no way to obtain evidence for anything except relative motion. Einstein had also concluded this by ca 1904 but Poincasre never mentions Einstein in any of his many papers and nor did Einstein ever refer to Poincare. Most odd. Egos?
29. ; If then one accepts that Michelson's findings had been as precise as one could obtain, they must accord with that principle (ie provide support for it) and to do that (in view of the lack of an expected variability in light's speed via its ether) some improved 'explanation' of the principle was needed in which (therefore) more realistically founded 'adjustments' in the intrinsic components of such motion or non-motion - (ie in our measures of time and distance?) - must somehow be incorporated. Those proferred by Lorentz seemed too inadequate and, to his credit, Poincare appears to have realised that (as mentioned above) a better conception of Time especially - analysed in terms of 'simultaneity' - may prove relevant - but seemingly not in terms of the same logic as Einstein - ie in terms of a symmetric balance as between differently moving observers, having no fixed reference point like a still ether, etc to afford either any primacy or precedence (or electromagnetic influences?) when having to rely on an inevitable 'lag' in receiving information concerning the time and distance variables of the velocity of any distant and differently moving body. Seemingly, Poincare hadn't appreciated such an explanation ? He was, after all, a mathematician, not a physicist, and yet his interests were very much in physics. As mentioned briefly above, this explanation by Einstein has yet to be elaborated in our present analysis and in fact does not appear to be made very explicit by him even in his 1905 paper which we shall be analysing below. (Even though it appears to have 'come to him' just hours before he wrote out his paper on that basis.) However, it is described in Einstein's biography by Albrecht Folsing (1993; translated 1997) which is more fully considered later; but this particular element may be usefully anticipated in conjunction with our prior analysis of the 1905 paper as it can, I believe, assist in interpreting the kinematic equations derived therein by Einstein when presenting his new theory; for he tends to avoid such reference to the perceptual activities of the observer or recording instruments that would necessarily be involved when actually measuring (empirically) the motion of bodies on a distant and differently moving reference system (even though these explanatory aspects themselves only 'came to him' in theory).
30. Newton's mechanics would predict that speeds beyond that of light should in theory be possible (presumably for actual bodies) whereas Lorentz saw that as an upper limit, after which problems of infinity of mass apparently obtrude. Poincare saw in this the idea that inertia in a moving body could probably only increase up to the speed of light (as mass increases to some limit and sufficient 'force'must not be available to further accelerate even something of such small mass as light). However, he still seems not to have considered that light's speed may have, as a consequence, been a true constant. While he apparently believed [on what basis?] that all laws of nature had to (?should) conform to this dictum, he didn't appear ready to accept (as asked about above) that one such law - in the electrodynamic sphere - would allow one to know who was moving if that law was (wrongly) taken as allowing light's speed through an ether to vary, and assumptions about time and space underlying motion (and the principle of relativity) were not fully recognised and more acceptably/rationally explained 'corrections' made thereto.
31. What was needed was someone who combined Lorentz's understanding of the physics of electrodynamics with Poincare's capacity to analyse and apply logically the most general principles of nature. Einstein alone (at that time) seems to have possessed just this combination to which was added a clearer appreciation of the apparent implications of Maxwell's equations - ie the ?likely constancy of the speed of light - coupled with an acceptance that there was no need for a still ether, nor any 'real, absolute motion' vis a vis some such 'fixed' reference system (as Lorentz felt necessary for his electrodynamic contraction hypothesis). However, it should again be mentioned that Poincare (1854-1912) would in 1906 publish his important paper on the dynamics of the electron, based on the latest ideas of electromagnetism (as advanced initially by Hertz and more recently Lorentz presumably), and apparently deduced essentially the same conclusions about the theory of special relativity as had Einstein - despite working quite independently of each other and, apparently, having not yet read Einstein's German paper. His ideas were, however, said to be restricted to a narrower compass than Einstein's - ie to electromagnetic phenomena only (as light) - and not to the electrodynamic implication for all moving bodies conceived as part of a single continuum - with its implication for variations in time and space measures over all magnitudes of the velocity of moving bodies. Nevertheless, such independent approaches which arrive at basically the same general conclusions greatly strengthens the probable ultimate validity and acceptance of same. Poincare was an outstanding French mathematician whose enormous contributions to science in that field apparently out-shone his more peripheral and sporadic activities in physics - as he so cleverly 'inched' his way nevertheless towards Einstein's slightly earlier-reported and more general conclusions.
32. It appears then that 'the problem' (the 'difficulties') as seen by most other contemporary physicists through the previous decade (1890s) (including Michelson's and Lorentz's views of it) was being approached by a different, if more traditional, route than it (or a similar problem) would be either by Einstein ('from elementary considerations (eventually) involving the motion of bodies, light's constancy, time, signalling and accepting no ether) or by Poincare (pertaining and 'limited to imponderable electromagnetic phenomena', rather than the motion of all bodies, ponderable and imponderable alike). Einstein would (eventually) see it primarily as a matter of making compatible those two seemingly incompatible basic realities/principles of physics (or nature) - viz: the constancy of light's speed - as indicated to him by Maxwell's equations, if not yet fully appreciated by most apparently (and certain related findings pertaining to measurements of light) - and the long established but generallly overlooked principle of relativity as it should apply to the motion of all bodies (and phenomena generally). At first, this later principle appeared to be incompatible with the former (if anyone before Einstein ever considered this matter; it appears not to have been a general consideration or concern). For how could anything not vary its speed relative to differently moving frames of reference giving it some boost of retardation? What it would come down to was that if the constancy of the speed of light was truly a universal, general law of nature, it must accord with the principle of relativity - at least IF it was accepted that all such laws must do so on the basis that all other features of nature point consistently to that conclusion. All moving bodies fell within a single conception and the laws governing them must not be affected by any differences in the velocities of the reference frames concerned - even those whose law may require it not to benefit (at all) from any added velocity (the very thing that all other bodies did allow). [Note: this latter 'shortcut' (in using the abstraction 'concerned' is utilised here, as elsewhere, to shorten the exposition, since this has been amplified many times elsewhere.]
33. This, ultimately, was what the theory proposed. As such, some way had, therefore, to be found that allowed that constancy (that 'unresponsiveness' to added (or subtracted) velocity) to function fully in accord with that principle - just as all other laws of motion must. This meant that a way had to be found by which the speed of light maintained its essential feature (its constancy) when measured from any frame of reference, however moving relative to any other, including any moving frame from which such light itself may originate. As with every other law of nature concerning motion, it would then be the case that, as the principle of relativity requires, no observer could conclude in terms of such light measurements whether any one reference frame was moving faster or slower than any other one, in any absolute sense, since such movement could have no affect on the motion of the bodies concerned. They should not be differentiable on that basis. This would maintain the validity of the propostion that there in no absolutely still reference system nor therefore, any abslute motion - only relative and symmetrically equal motion.
34. There was thus a need to adjust, if possible, some aspect of one or other of these otherwise apparent truths of nature in order that their actual compatibility (even if not yet recognized) became more apparent. They couldn't both be right as they stood. But to accept that light's measured speed alone may not be variable despite variations in the speed of its source or its observer/recorder and that the principle of relativity must somehow adapt to this so that light's speed alone couldn't be used to distinguish which of two reference frames was (truly) moving, or doing so the faster or slower, proved a hard nut to crack. [Whenever, that is, anyone finally addressed concern regarding this aspect of 'the problem'.] It seemed to go against all common sense (and against 'certain unspoken assumptions' regarding meassures of time and space underlying both that principle and all of science - as mentioned above). Possibly a principle of 'absolutivity', therefore, with its idea of absolute motion vis a vis a still ether, say, may have been the more valid construction after all (as discussed earlier) - one that also automatically assumed (as comprehensible, consistent concomitants) - the ultimate constancy and unquestioned absoluteness of our measures of time and space ? But that way was also frought with other anomalies and inconsistencies.
35. [I still 'feel' that Einstein didn't initially approach the problem (as he originally conceived it) 'head on' - from this laudable, 'general principle' point of view alone (toying with his two seemingly incompatible abstract playthings from the very start) - but that rather, as the pieces of the puzzle arranged initially in no particular logical order, these important elements gradually (or even suddenly) fell into a clearer cause and effect sequence along with the other elements of the problem area. After throwing out the ether and its absolutism, he would soon see that it could (only now) be more succinctly set out in that form, ie 'as though' he had approached it from that lofty abstract/generalist/efficient point of view from the very start. The essential logic behind it all may well have come down to this in the end - more succinctly than any other way. But prior to this, there was quite likely a more 'messy' stumbling upon some essential crux of the matter I feel. Later assuming the more organised approach would thus lend itself to a more generalised if abstract presentation of his theory in his eventual publication in 1905 (even if pared down to its most fundamental logic) - keeping to himself any much less general, but ultimate 'eureka' key to it all (eg re perception of 'time' being necessarily delayed (and thus 'dilated'), simultaneity or the 'glass ceiling-like' effects of a maximum possible velocity, etc) for quite some time after - if indeed he could ever re-capture the actual sequences later.
36. It is possibly useful to consider just what 'the problem' would be the case had Einstein not found a way to make the actual compatability of the two 'laws' of nature more apparent. For there was clearly such a problem recognised by both himself and other contemporary physicists - otherwise he wouldn't have been seeking the means to resolve it. And if it was so recognised, was this by virtue of recent empirical evidence or still by way of theoretical considerations ('thought experiments') of what apparently was the (disturbing/illogical) case which would be expected to be confirmed by such evidence once the techniques were perfected to confirm it?]
37. Certain related problem(s) had been recognised by most other workers - which had arisen out of Maxwell's findings (and certain earlier problems not fully addressed) - as needing attention in terms of their immediate mechanical difficulties. These required to be 'patched up' somehow and kept consistent within Newton's mechanical model and the constraints of the traditional principle of relativity. [These may have included those 3 English studies mentioned by Lorentz in 1904.] Einstein, on the other hand, would (eventually) approach them as above - in terms of a revised general principle (concerning the nature of motion - including that of light). Eventually (and especially after gaining a particular insight into the concept of time (which 'holding fast' to the application of a fixed velocity for light - as just another 'body' - had thrown up) as described below), this forced him to recognise, confront and question those unspoken assumptions - be they in the absolute or relative camps. Those unresolved remnants of absolutivity which had 'stuck' to the old principle of relativity were thus finally removed - because of the eventual appreciation and acceptance of the nature of light. And with a re-interpreted principle within this realm, relativity would finally shake off all other remnants of absolutivity. Thus, our original, earlier question as to whether investigations into imponderable phenomena (eg as light) may help resolve the fundamental dichotomy of the absolute/mechanical vs relative/electrodynamic world view (ie better than did studies of the ponderable bodies alone) would, it seems, be clearly answered in the affirmative - ultimately - and more besides.
Einstein's 1905 'Moving Bodies' Paper
38. Science normally progresses by a series of small steps provided by a sequence of researchers each addressing in turn problems and difficulties thrown up by earlier workers. Hypotheses are advanced and tested, results reported and subsequent hypotheses tested by later workers. Sometimes a major advance is made. Generally it is possible to trace the gradual development of most scientific ideas and conclusions by this means. Thus, after a century of incremental contributions by many well known workers in the sphere of light, Lorentz had provided, by about 1904, a fairly well formulated theory (based on Maxwell) concerning the electron and electromagnetic phenomena generally including, amongst other things, the behaviour of light and its assumed medium - the (still) ether. This background to his ideas would likely be traced with little difficulty through the introductory remarks and final conclusions in the scientific papers written by himself and all earlier workers in that progression between say 1830 and 1900. It likely continued in the 1906 paper by Poincare referred to above, and others by Lorentz that same year and in 1904. We may contrast this unobscured, stepwise progression with the style of reporting to be displayed in 1905 by Einstein in what his biographer, Ronald Clark, referred to as "...possibly the most important scientific paper written in the twentieth century..." namely, 'On the Electrodynamics of Moving Bodies'. But, it was, he said, a perfect example of a paper whose aim, as described by Hermann Bondi (a respected philosopher of science) was "to leave as disembodied and impersonal a piece of writing as anybody might be willing to read...(but one that was)...very likely to tell the reader almost nothing about how the result was actually found." He might well have added '...nor indeed exactly what the problem was that he was so addressing' - other than 'certain difficulties' which, in his paper, were even then not fully specified, nor was any result particularly obvious as a solution to those vague and uncertain 'difficulties'. And yet - 'the most important paper...' ??
39. Moreover, we may note that nowhere in this paper of such renown is his theory (of the electrodynamics of moving bodies) ever referred to as the 'theory of relativity', whether 'special' or not. It would appear that some time after its publication, Einstein must have realised that his new theory, based in part on the principle of relativity, was but a special case of a more general conception of the nature of motion - to which the name 'general theory of relativity' would later be given - and thus, after its publication, the earlier work became increasingly known as the 'special theory of relativity'. The motion of concern (of moving bodies) in the earlier paper is defined as constant or 'uniform' while in the later, more general theory, it can be thus or variable - as for example when accelerated as by gravity. But this important differentiation is not stressed in the earlier paper - although the motions concerned are properly described there (but on one brief occasion only) as 'uniform' - since inertia and the principle of relativity rely on same - but this feature is not particularly emphasised and seems almost taken as a given. [One wonders what name Poincare gave to his own similar theory of 1906 ??]
40. We would eventually learn also that, as a basis to his earlier paper, Einstein had been wrestling with a particular problem (relating to light) for about 10 years and that "...after abandoning many fruitless attempts, being visited by much conflict and confusion, 'at last it came to me...' that 'Time' was the key". For information about the motion of all bodies, including light was, he suddenly saw, necessarily transmitted only by reliable, but time consuming, (signals. This was just 5 or 6 weeks before he actually wrote up and submitted his paper in the summer of 1905. But the paper itself was almost cryptic in its presentation. It didn't mention that very recent insight regarding time and would contain no specific references to earlier studies by others, or a single footnote. There was just one brief acknowledgement - to help he had received from his friend M. Besso - 'on working on the problem here dealt with'. So, we can't easily analyse his conclusions in terms of their development from the ideas (as focused on some clearly specified problem) of his contemporaries or of those who came just before him, although he does refer initially if fleetingly to Maxwell. The problem 'door' that he decided to unlock with his newly discovered 'key' was thus a kind of side door - not the main entrance at the end of the more well trodden path by which others were, stepwise, approaching hoped-for answers inside - to some fairly well agreed and described problem. He must have realised that what he had discovered was that profound and fundamental and his confidence in it that certain that he felt no need to persuade the reader or scientific community of its bona fides - by setting out its detailed historical development; it would stand on its own succinct merits - forever.
41. We do have however three brief introductory paragraphs at the beginning of his paper - before he pursues the tight physical and mathematical development of his thesis at a more technical, if theoretical level. We can at least seek to analyse these paragraphs to see what they may reveal about the basis of his argument and just what was 'the problem here dealt with' and, hopefully, the answer he would suggest (his theory) as apparently the only way to resolve it. His paper wasn't the report of an experiment in physics the result of which is offered as evidence that a new hypothesis suggested therewith provides an answer to some prior (and clearly described) problem which is thereby supported. Rather, the paper is:
42. But somewhere within the body of the paper there must be, even if only implicit and theoretical, a 'result' or outcome (as referred to by Bondi) which supports his general thesis - one that is relevant to some specific problem area (the 'difficulties') in the sphere of physics that he is addressing. Once the following analysis of his paper is complete, we may see if this problem and its new answer have indeed been revealed more than we have suggested. [We have implied on several occasions above that this result concerns primarily his properly justified basis for previously unsuspected variations in the magnitudes of time and space when measured in others' (differently moving) frames of reference (and, symmetrically, every frame of reference is potentially some others' such one, seemingly, even our own, to those not of it).] Presumably it will also suggests both past and future phenomena which his theory (and no others) could properly explain and predict and be thereby further supported - so that his theory gradually becomes essentially a 'law' concerning the motion of all bodies. As this is inherent in all events and happenings in the universe (and in any case, apparently leads on to what has been described as probably the most important equation in science: E = m.c^{2} (the basis of nuclear power and 'the bomb'), it is obviously rather important and fundamental) !
43. Because Einstein chose to write in such a concise, abbreviated style (as part of the apparent 're-ordering' of his reasoning in more abstract, general terms as mentioned above), it may be helpful to first reproduce verbatim each of the three introductory paragraphs at least (in their English translations) and then try to 'interpret' them and their evolution and implications as best as one can Thus:
An Analysis of Einstein's 1905 paper 'On the Electrodynamics of Moving Bodies'. [Comments added in square brackets]:
44. The paper begins: "It is known that Maxwell's electrodynamics - as usually understood at the present time [ie ca 1895-1905] - when applied to moving bodies, leads to asymmetries which do not appear inherent in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. [Does this imply 'is in absolute motion'?] For if the magnet is in motion and the conductor at rest, there arises in the neighbourhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighbourhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise - assuming equality of relative motion in the two cases discussed - to electric currents of the same path and intensity as those produced by the electric forces in the former case."
45. [Now, when in 1820 Oersted discovered that when a current of electricity (ie electrons moving due to an electric force) began to flow in a conducting wire placed near a magnet at rest (ie a compass needle), the latter would 'instantly' move to point perpendicularly towards the wire. This was later concluded to be due to the creation around the wire by those moving charges of a circular magnetic field (its lines of force when viewed from above, say, appearing as effectively perpendicular to the direction of the wire which they encircled). Later, Faraday found a symmetric converse of this phenomenon when, rather than electric charges moving near a magnet, the magnet itself was moved near a conducting wire. This time an electric field was engendered around the moving magnet to which the electrons in the conductor responded - by moving as an electric current thus induced. The latter phenomenon was formalised within Faraday's 'law of induction' and the theory associated with it. It led to the developments of useful electrical machinery in which it may have been more practical to have a fast moving magnet encircled by fixed copper wires, say, rather than the converse (or vice versa?). But, in any case, in both Oersted's and Faraday's cases, the same outcomes would have arisen had the motions applied to the other elements in their respective situations. It was their symmetric relative motions only that produced these results. This was apparently long realised but tended to be forgotten in that Faraday's law was generally framed as though this symmetry wasn't actually the case. Einstein's first point in his paper is thus simply to point out this particular oversight or mis-representation in contemporary physics. Its relevance to anything in particular is not, at that point, discussed (although the implication of a role for an absolute rather than relative motion or stillness might be considered). [What did Hertz say on this aspect, if anything?] But he continues this topic in the next (his second) paragraph where its relevance may become clearer.] Thus:
46. "Examples of this sort [which indicate that there is no ?real, only relative, motion in either element], together with unsuccessful attempts to discover any [absolute,'real'] motion of the Earth relative to [an assumed absolutely stationary] 'light medium' [ie the ether], suggests that the phenomena of electrodynamics as well as mechanics possesses no properties corresponding to the idea of absolute rest [nor, therefore, of real, absolute motion]. [Note: Einstein places his 'asymmetry' example from the current induction situation (where no fixed component is needed or extant?) and the failure to confirm any fixed ether (as an absolutely still reference criterion to establish that the Earth's (or even the Solar system's) motion is absolute) as two examples of the same basic thing which, in effect, underlines the reality of the principle of relativity. Their comparable relevance to the latter may possibly need to be better defined. The principle is responsible (accounts) for it being impossible to decide which of two differntly moving reference bodies (systems) is more 'at rest' (or moving the slower, say) and which, relatively, the faster. Only their mutual relative motion can be validly described. And we may note, rather pardoxically, that the origins of the principle of relativity in Galileo's time was based on the then new idea that the Earth moved around the Sun and not the other way round. In fact, it is now appreciated that they likely both move relative to each other (symmetrically) as far as the mathematics of their motions are concerned.
47. He thus continues: "They [ie such examples that there is no real, absolute motion nor associated asymmetry or primacy of motion] between bodies suggest rather that, as has already been shown (my italics) to the first order of small quantities [ie v/c], 'the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good'. [That is, all laws of nature (regarding moving bodies measured on or from any frame of reference) may be treated as part of a single conception. The laws pertaining to moving bodies are unaffected by any differences in the motion of relevant reference frames.] We will raise this conjecture, says Einstein, to the status of a postulate (hereafter to be called the 'Principle of Relativity') and [in the very same sentence!] will also introduce another postulate, which is only apparently irreconcilable with the former, namely that light is always propagated in empty space with a definite [ie constant] velocity c which is independent of the state of motion of the emitting body [ie 'the light's source'; we may note that in subsequent descriptions of this postulate, the constancy of light was also as much referred to the motion of any observer (or method of measurement) which, counter-intuitively, also had no effect on the value of that constancy]. These two postulates suffice for the attainment of a simple and consistent theory of the electrodynamics of moving bodies - based on Maxwell's theory for stationary bodies". [We may re-state here also the fact that the relativity principle is often also described as what accounts for it being impossible to decide which of two differntly moving reference bodies (systems) is more 'at rest' (or moving the slower) and which, relatively, the faster.]
[Note: The word 'same' in the definition of the relativity principle presented by Einstein (which might appear a touch ambiguous) seems to imply that laws of electrodynamics which he feels must accord with the principle of relativity (as do those of mechanics) include no exceptions whose form might have been altered to fit any awkward facts that may seem to have emerged.]
48. [It is intriguing to speculate that the title of his paper could imply an allusion to Lorentz's contrary view that moving bodies truly 'shorten' when they move relative to a still ether - by virtue of an alleged electrodynamic effect which Lorentz (eventually) thought was so engendered. Einstein sought in his paper to show that such alleged effects were not required to explain the results of relevant experiments and that in fact they did not occur by that suggested means. Rather, a quite different form of electrodynamics applied to moving bodies the effects of which (regarding space and time) would be of a different kind to those suggested by Lorentz - not physical but perceptual.]
49. [His theory thus states, in effect, that 'there is a (primary) Principle - ie that 'of Relativity' which holds true for both mechanics and electrodynamics - even though we accept also (ie assert equally) a second Principle which holds in that latter sphere - that the velocity of light is a constant, whatever the velocity of its emitting source - this latter principle or 'law' not being (as it may initially appear*) inconsistent or irreconcilable with that first (relativity) principle.
[* It so wrongly 'appearing' because the speed of every other body to which this important first principle had previously always applied was assumed to be invariably a function not only of its own inherent motion (however instigated) but of the total relative speed of its source as well (ie added to it) whereas this was, uniquely, not the case (only) with the motion of 'bodies' that happen to be instigated (ie released and propagated) as light; this reality (which he would now have to find a way to allow) would have implications concerning the extent (proportion) which the prior speed of any moving source was actually additive with respect to the subsequent total velocity of all other relevant bodies so placed into motion. That proportion could vary between 0 and 100 %.]
50. There was thus the implication that the effect of the prior, on-going speed of a body's source on its final subsequent speed (as per its laws of motion) was something with which the principle of relativity always proved compatible and was associated (which would apparently not be the case however where, as with light completely, such a moving source did not affect subsequent speeds of all other bodies on a one to one, additive, basis as previously assumed, but rather and unexpectantly in a ratio (less than one) that was influenced by the proportion that those speeds were of that maximum possible velocity - ie that of light (whose speed would not be boosted at all) because it was the maximum possible speed below which the 'added' velocities all slower bodies' 'had to be'(**) proportionally accommodated); they couldn't sum to more than that of light. We might usefully add here that 'the problem' which was central to Einstein's concerns was summed up (rather too succinctly) in his phrase "...only apparently irreconcilable with..." or, in similar contexts, with the inclusion of such words as 'appearing' or 'apparently'. This 'problem' can be contrasted with that of concern to Fitzgerald and Lorentz after Michelson's negative result - which we may now better appreciate was not the same as that specifically addressed by Einstein.
[** 'had to be' is probably too proscriptive a phrase as it doesn't suggest a basis for that outcome - something to be further elaborated later.]
51. [Note: While the justification and bases for Einstein's two fundamental postulates will be addressed more fully below, we may add here that in his search for a way to resolve 'the problem' he eventually decided that as in other areas of physics where the laws of nature had been sought, an over-arching general principle was probably required to better guide one's search - with which various more specific relationships so encompassed would be required to prove consistent (ie be subsumed). His first postulate represents that general principle - one that had already existed but in a less generalized form - from the days of Galileo and Newton - and, as such, could (once so generalized) be considered the essential 'steering committee' (ie the 'gov'nor') with ultimate precedence or jurisdiction which nevertheless must still have its own explanation and justification; its relevance wasn't postulated 'out of the blue'. It had been referred to recently (1899 and 1902) by Poincare. Equally, there must also be a logical basis for how he came to advance and believe in his second postulate (as derived from Maxwell's equations apparently) which had implicitly served as a kind of 'thorn in the side' of mechanics (and the more restricted principle of relativity associated with the latter) but which the first postulate now 'demanded' (in Einstein's mind); that is, a way had to be found to accommodate it within its determining compass (ie under its over-arching umbrella). It too was a postulate not a deduction from more basic premises (as it was in the case of Lorentz who related any such constancy or independence of light's velocity to effects of the postulated ether which to him seemed the more fundamental and actual reality.]
52. Having stated his two essential postulates, Einstein then continues (apropos the last sentence of our preceding paragraph) : "The introduction of [essentially a third postulate] - regarding a 'luminiferous ether' - will prove to be superfluous (in our developing theory) inasmuch as the view to be developed here will not require 'an absolutely stationary space' provided with special properties [which could act on (affect) material objects moving through its assumed 'stillness'], nor assign a velocity-vector to a point of empty space in which electromagnetic processes take place." [Oddly, he doesn't mention directly that he sees no need to posit an ether medium for the propagation of light although naming it as he has (as a fixed or stationary entity) may imply that medium role also but if so, that doesn't itself imply that it must be an absolutely still medium, although this status may well have been assumed by most at the time). Maybe that was considered to be one of its 'special properties' (or even its primary one). This claimed superfluousness of the ether for his theory thereby reinforces his comment that his two postulates would alone suffice for the attainment of his theory; no 3rd, ether postulate, was needed - whether for either of its alleged properties - of 'stillness' or as a 'medium'.] We may recall that Michelson argued that the still ether provided a net resistence after the motion of the Earth (the light's moving 'source') gave his light waves an initial boost (vis a vis those sent perpendicularly) which I believe was expected to be manifested in a measurable altered ?speed of the light (or was it just the time light took if the distances were not equated ?) and that Lorentz explained the absence of this change to either that physical resistance of the ether (initially) or to an electromagnetic influence of its stationary status on the actual electronic/physical structure of the equipment (bodies) moving through it (subsequently) - resulting in an actual structural shortening of the measuring apparatus concerned (in some absolute, 'real', physical sense); 'local time' associated with the moving measuring equipment (ie on the moving Earth) was also somehow 'physically dilated' but no comparable (ether-based) explanation was initially provided for that equally important phenomenon associated with the velocity of moving bodies.]
53. [Einstein's example of the essential symmetry of current induction may be seen as somewhat analogous to but contrasting with the non-symmetry of Lorentz's explanation of how a still ether accounts for the contraction of bodies moving through it. If there was an ether that could have this effect, its motion vis a vis any bodies might only have to be relative not absolute (although this may never arise if, according to Einstein, there was in fact no ether; however, he doesn't quite say this but rather that he simply doesn't require it. That is, there is a simpler explanation and such simplicity generally takes precedence in scientific acceptability and validity.]
54. [We may note that Einstein implies that other examples are available where such relative motion has been similarly overlooked and absolute motion and a fixed point in space wrongly assumed (or possibly implied). However, we note that he doesn't specify these. [Recall my example of striking a match; either the match or that on which it is struck may appear to be the main element moving but, again mathematically, there is no difference; they are equivalent and essentially symmetric.] Also, it may be pointed out that although Copernicus and Galileo reversed the asymmetry as between which was the unmoving centre regarding the motion of the Earth and the Sun, it is now accepted that they too are characterised by an actual symmetry; neither are truly 'still' nor, ultimately, possess greater 'stillness' nor motion than the other; ditto vis a vis the galaxies.] They, with the ones he does describe, plus unsuccessful attempts at verifying any motion of the Earth relative to a fixed ether (as by Michelson presumably) suggests that both electrodynamics (which includes light) and mechanics (ie the motion of the Earth) operate perfectly well (ie predicatbly) without any involvement of a concept of absolute rest (or of a stationary ether of this sort) and thus are able to do so (in his new theory) within the dictates of a (new) Principle of Relativity to which this absence of absolute motion necessarily leads; ie 'new' and 'adjusted' in order to accommodate/guarantee the one 'body' whose speed doesn't meet the dictates of the original, mechanical principle of relativity (ie as varying according to the speed of its source). As Ronald Clark puts it in his biography of Einstein - such examples and attempts then suggested to him an "...inevitable consequence: namely, the destruction of the idea of [the reality or necessity of] absolute rest...". All motion was relative - as between any two or more moving bodies. (Poincare apparently arrived independently at a similar conclusion about the same time.) If there was an ether, it might serve as a medium, an energy source or even the seat of some other imponderable phenemena but without any absolutist' positional' attributes, stated Einstein. Galileo's thinking provided the basis for the relativity principle but, as noted, it needed generalizing - both to incorporate light's constancy and to remove any preference for any given frame of reference (as the Sun vs the Earth or the Earth and Sun vs any 'ether', etc.]
55. [This briefly expressed 'suggestion' - of there being no absolute motion - apparently leads logically and inevitably to his 'conjecture' - that all laws of nature operate identically in whatever reference system they do so when measured from any other such inertial system and thus accord with the appropriate principle of relativity. Every reference system is symmetric with every other one; none are 'favoured' and thus one can't say which is moving (the faster or slower). All laws (must) work the same therefore on all of them - as perceived from any other ones. Thus, the law of light's constant velocity must also be accommodated within this 'rule'; differently moving reference frames must have no effect on the electrodynamic law of that constancy just as they should have no effect on any other laws of nature, mechanical or otherwise. He suggests that this has in fact 'already been shown' (to a certain high degree of accuracy) although, rather naughtily, he again doesn't specify when or where it was thus 'shown'. [Is this line of logic different from that which justifies the demand that all laws of nature 'must' accord with the principle of relativity because its over-riding primacy is established in ways set out earlier, or as argued by Bondi?? Not quite, I believe; it simply means that all straight, uniformly-moving systems, whatever their differing speeds may be, will provide identical or physically equivalent environments for the operation therefore of all natural laws; their differing speeds, which are simply relative one to the other, (can) have no effect on the laws of nature. There is a consistency in nature - seemingly as it has successfully evolved in our universe and there are only relatively moving systems on which they all so evolved - on which inertia is equivalent and universal. I believe this may be what Bondi was getting at.]
56. [So, the asymmetry example is seen finally as pertaining to the importance of a mutual symmetry of relative motion (as Galileo first pointed out! re ship and dockside (although he didn't point out, I believe, that the dockside could equally be seen as the element that was as effectively 'moving' - relative to a still ship, as far as the mathematics (or nature) were concerned); ditto re the (observer on) the Berne tram and the (the observer) in the Berne square) - as opposed to any idea of an asymmetric, absolute motion of one or other element having any precedence or distinction (eg the Earth and/or the Tram) alone. The same thing applied to the magnet and the coil of wire. Einstein would thus refute both the action of a still ether in 'causing' contraction - ie as he asserted there was no (still) ether or any other absolute stillness (established by any other means) and thus of any absolute motion or effects of same. From this, it follows that all motion of any body is always and only relative to some specified reference point or frame which has equal precedence as being effectively in motion itself (whether seen as slower or faster) relative to that or any other body.
57. From this it can be shown that such mutual relative motion, no matter how fast as 'seen' from either point of view, has no absolute magnitude (it could effectively be zero (if this could ever be established) so that it becomes unsurprising that the one truly absolute speed of anything - that of light - always moves away from all other phenomena (seemingly moving but of unmeasurable magnitude in any absolute sense) - including its own electromagnetic sources - at its one and only (immense) speed. And if motion must be relative, then the inevitable concomitants of the velocity of any body's motion - ie space and/per time - should presumably be relative also (although I don't believe this was the route of the logic which led to that conclusion, even if it did so follow (although I have no basis for suggesting that). We may add here that the perception of both time and space on any moving frame/body from any other frame (seen from the former one equally as the one moving) is mutually affected by the inevitable 'lags' in the required measurement methods (using signals to convey the information). Only immediately local magnitudes would prove consistent. This is presumably expanded below although seems rarely cited when explanations of relativity are discussed.
58. Einstein's 'conjecture', already thus mysteriously supported, turns out to be one of the most profound hypotheses in science: namely, a 'Principle of Relativity' so generalised and defined as to apply to all laws of nature (including, therefore, to any constants of velocity which (perplexingly) may exist - ie in any sphere; although in only one sphere - electromagnetics - does such occur, eg as light (and other electromagnetic waves) - seemingly. It requires* all laws of nature, whether in the spheres of optics and electrodynamics, or of mechanics, to be equally valid, to apply fully, in all frames of reference (which, as it happens, are, in his view, all moving, often at different relative velocities) and in so doing do not provide any means by which any frame of reference could be differentiated from any others - as the faster, or slower or more or less 'stationary or mobile, etc. (in any absolute sense) through the different operation or results of any such law. Such a principle will not tolerate different sets of reference criteria when measuring the motions of bodies from whatever sphere of nature - be it optics/electrodynamics or mechanics. The unsuccessful attempts (to establish absolute motion of the Earth) that formed part of this foundation didn't initially include that of Michelson apparently - of which he later said he wasn't then sure he was even aware - despite that experiment being seen as the virtual sine quo non of this category of work and, as such, seduces most analyses of this topic as being the starting point of Einstein's focus; apparently, it wasn't.]
[* Bondi says the (generalised) principle of relativity so 'requires' because of the existence of a yet more fundamental principle (ie more fundamental than that of inertia on which the original principle of relativity was, it seems, securely based) - that of the internal consistency or 'unity of physics'; this needs further analysis on my part. Does it imply that there must be a consistency in our model of nature, with no awkward exceptions? I think I prefer a more 'mechanistic' explanation - as inertia seemed to provide to the rest of nature/mechanics - and as Einstein seems to indicate in his justification for the relativity principle's primacy. See.....]
59. [And while Michelson sought evidence supporting a still ether (and thus implicitly of the existence of absolute rest), there doesn't appear to have been a scintilla of any suggestion in either his papers or in Maxwell's prior comments about methods to so establish this, that such a still entity if found would have particular relevance....to anything other than to support Maxwell's ideas about the likely medium for his new conception of light waves (even if this was later typically described in converse terms - of seeking to show that the Earth really 'moved' - ie relative to an absolutely still ether, such proof of the reality of which appears to have been what was basically sought; but to what end?). Later, Lorentz would also seek (or assume) evidence concerning such absolutely stationary ether - as an explantion of why charged atomic particles within Michelson's interferometer (traveling on the Earth through that assumed ether) would contract into a smaller space and thereby account for an unexpected constancy in the measured velocity of light (that was, in Lorentz's view, apparently really variable therefore?) and its associated failure to verify the real, absolute motion of the Earth thereby. However, Einstein did not refer specifically to this failure. He would take exactly the opposite tack in his thesis - that there was no absolute rest (whether as a still ether or by any other means; his symmetry argument addresses this - ie relativity(!) and that light's speed was nevertheless acceptably constant, despite that principle's requirements and regardless of any motion or otherwise of its assumed medium. As such, it needn't be posited therefore - as a (?desperate) means of accounting otherwise for light's stubborn constancy.
60. It is thus intriguing that while Maxwell and Michelson were interested in the concept of a still ether - whose existence or not as based on such research would later be fundamental to the contrary theories advanced by Lorentz and Einstein - the former two themselves indicated at the time no apparent concern or interest in the potential relevance of such concepts to anything beyond the concerns of Fresnel, Stokes and Maxwell regarding the assumed necessary medium for light, latterly as an electromagnetic phenomenon. They weren't, one may assume, part of any imagined group of absolutists or mechanicists battling against those imbued with some contrary new philosophy (as written about in such terms only many years later); just contemporary scientists investigating current phenomena regarding light and its assumed medium within their usual mechanical model. The latter's existence might be supported or verified in terms of its motion being reflected in different velocities (or timings?) for light (ie as a convenient marker) without any concern that this might conceivably have regarding the fundamental velocity of light; even its assumed constancy otherwise. The existence of absolute motion based on the idea of a still ether (or of asymmetrical current induction) would appear to inhibit the possibility of (or need for?) a principle of relativity (entailing relative time and space) by which means the constancy of light's speed could prove compatible. Rather, it was used by Lorentz to allow an explanation of how that assumed still ether (allowing absolute motion) could be justified as masking the existence of the very opposite - ie the variability of light's speed, as thus utilised. [Or, again, was it simply the variability of the time taken for light to travel within the parameters of Michelson's or similar experiments and not its velocity?]
61. We may also mention here that both Lorentz and Michelson continued to promote their interpretations over the next 5 or so years (after 1905) - maintaining the existence, validity and relevance of a still, or a moving ether, respectively (but an ether nevertheless) - since when a long-established but generally discredited movement of 'anti-relativists' and 'pro-aetherists' have continued essentially this same orientation. Their arguments are usually much too abstruse and mathemetical for a layman as myself to take a confident view on. In the meantime, I will tend markedly to favour the views expressed by more recognised physicists, physics departments, books and journals.
62. [In any case, Einstein then introduces his second postulate (in the same sentence) - that the velocity of light is indeed a constant - which is independent of the speed of its source . He doesn't point out that this postulate was either implied or even explicitly stated within Maxwell's electromagnetic equations nor that certain of those 'examples' which indicate that the absolute motion of the Earth was not measurable (relative to a still ether, say), could also have been cited or interpreted as indicating that the velocity of light did not, as expected, vary - but was constant. Whenever and however he arrived at his conviction about it, he then points out that this light postulate is not really inconsistent or 'irreconcilable' with his new, all-encompassing, properly generalised principle of relativity (allowed by the absence in the universe of absolute motion) but only apparently so. For clearly, it would have to be consistent with it in so far as he has already stated in the first postulate that all (relevant) laws of electrodynamic and optics (as well as of mechanics - ie 'of nature') must (or simply are) consistent with this principle, in that the 'law' of the constancy of light's speed is indeed a part of such laws of nature (and, for reasons cited in his 1916 book), his theory states (requires?) that all general laws of nature must (or simply 'do') accord with that fundamental principle. Such 'reasons' must hold a position of primacy in Einstein's logic but they seem to be generally ignored by historians of science (or at least by 'lay' historians).
63. There can be no exceptions to being so compatible if one accepts, as Einstein has, that there is no absolute resting place in space since that means that there is no other principle except that of relativity available to account for the consistency and validity of all laws of nature concerned with motion regardless of the (inevitable) differing uniform motion of all possible frames of reference; there being no fixed frames of reference, everything must 'work' satisfactorily (whatever their uniform speed) in terms of the only relevant principle going - that of relativity - as it accounts for the total acceptability and equivalence of uniformly moving frames of reference (of often differing velocities) for all activities; that is, their complete equivalence for valid measurements for all laws and all observers. There must, therefore, be approriate transformation equations to verify the reality of that more comprehensive requirement just as there were for mechanics alone (ie in Galileo's more restricted transformations and relativity principle). One would assume that inertia remains fundamental in the foregoing logic.
64. In one sense, he doesn't have to provide a rationale as to why all laws of nature must, should or even do accord with his new principle of relativity (including those of electromagnetics)- since this is (?part of) his hypothesis or theory (the other part concerns his definitions of time and space differing from those of Newton as the basis (or outcome?) of that new principle) and if this allows predictions which are verified and nothing is found to disagree with these 'assertions' (definitions), its validity or rationale can be assumed - unless and until not supported - without stating the latter a priori, although he appears to have given one in terms of the logic which follows from there being no absolute rest in the universe. As mentioned, Bondi appears to give another (related?) one by citing his principle of 'the unity of physics'. I can myself see a certain inevitability of the all-encompassing principle of relativity in terms of the evolution of all laws of nature (that always work) within our universe in which there are only (uniformly) and differently-moving parts and their associated inertias. Otherwise, an infinity of different laws would have to have successfuly evolved - one set for every different moving environment. But this hasn't so evolved (nor we with it) and therefore we do have the one that proved the more sustainable. If there is a universe somewhere whose constituent parts are all mutually 'still', some other single set of laws could presumably have evolved (even if less imaginable).]
65. It was of course his resolution of the above 'apparentness' (cf 'obviousness') and its expression in such new transformation equations (rationally based) where the crux of his theory lay. That resolution in fact equates to his "..attainment of a simple and consistent theory - of the electrodynamics of [all] moving bodies'" [in which a rational basis for establishing that time and space are in fact relative, not absolute, is provided]. This theory was of course later termed a 'theory of (special) relativity'. It was presented as a theory rather than as an 'irrefutable new discovery or truism' simply because this is how science progresses; all such advances must be capable (in theory) of being falsified. It holds only until and if anything is ever shown to be inconsistent with it. So far, I believe, nothing has - now a century later. That he was convinced that there was a need for some kind of resolution between two apparently irreconcilable principles or truths in nature seems to have arisen after his lengthy anaysis of his thought experiment and comparable anomalies suggested by research (or 'theories' based on unlikely or unproved premises - as a fixed ether medium for light) as per Lorentz - with only one certain way out. [And also on his 'eureka' about time?]
66. [Note: one should probably explain further here why transformation equations are required - to account for the consistency of all laws of nature despite the manifest effects of adding or subtracting the velocities associated with the frames of reference concerned. That is, while laws of mechanics always appeared to prove consistent (and so accord with the then principle of relativity) even when any body's speed was measured as being greater or less than that expected for the magnitude of force applied to it, this was always reconciled by subtracting or adding the full amount of that frame's different velocity. With light, this adjustment did not result in the consistent (law-fulfilling) outcomes expected; thus new transformations were needed which would be consistent with the demands of the two principles and which therefore required adjustments in the basis of velocity (ie that time and space must somehow be relative/dependent). [We will see below that such adjustments will apply to various physical phenomena (in addition to space and time coordinates) - such as certain characteristics of light and the energy and mass of bodies - succinctly described as having to be 'transformed under changes of reference frame'.] One might consider whether the speed of light must be constant - if there is evidence that time and space are variable - or, must the latter be variable because the speed of light is constant? Which is the dependent variable and which the independent ? Which is a hypothesis (postulate) and which a proven 'fact' or reality ? Both ? In any case, the new transformations would show that not only the perception and measurement of space but of time also was dependent on (relative to) motion (as defined for all present cases) so that two reference systems in relative motion necessarily had differently perceived spatial and temporal intervals.]
67. Finally, it might be mentioned that it may be one thing to have a conviction, and a back-up of rational reasons for it, which takes prededence in one's thinking such that certain things must follow from it (as that both mechanics and electrodynamics must accord with a single principle of relativity and, to do so, time and space must prove variable (or relative)), but another thing to account for the exact amounts of such variabilities that allegedly come about in given circumstances - as this would imply more about the actual mechanisms which effect these variations, and so further establish their validity. Or, are they 'apparent' variations only ?? The answer must be in terms of just what it is to (or on) which these variables turn out to be relative (dependent). They are in fact relative to....'the difference, if any, in the velocity of the 'platform' on which any 'motionally-relevant' law of nature applies and that from which the effects of same are observed and measured'. They would arise due to some explicable qualitative 'mechanism' and quantitatively be relative to some particular quantitative function (and mechanism) of such velocity differences - concluded on the basis of the appropriate differential equations or analysis. One might reasonably assume that this function would be quite a straight forward one and to apply to any and all possible values of that measure of velocity - from, say, 1 mile per hour through 1000 miles per hour and on upto 500,000 miles per second, say, and poassibly even to an ?infinite velocity - ie the greater the velocity, the greater the effect on the variable magnitudes of time and space. This would presumably be the case if there was no restriction on the possible magnitudes of velocity. But there is such a restriction and hence the function must evolve (and prove compatible) within that constraint.
68. The restriction is that of the speed of light - at about 186,000 miles per second. And thus the effect of the velocity differences mentioned on the possible values of time and space (if they must so vary) must be determined as a function of some measure of both that difference and this upper limitation of possible velocity. That is, it has effectively to 'fit into' the available 'space' of possible velocities (and not some limitless, infinite range) and so the function must include reference to that upper limit - as sympbolised as c. That is, it must be a function of some ratio of the velocity concerned (v) to this limiting value (c) - ie some fuction of v/c. The exact value of this function was something which both Lorentz (for one particular calculation) and Einstein (for another) derived from their respective analyses, which will be considered further below. [With Lorentz, space (length) was dependent instead on the velocity at which any body moved relative to a still ether (which acted upon that body's electric function and sub-structure) but, again, he seems to have recognised that c represented an upper limit (how and when did he conclude this?) so his equations would necessarily also incorporate this same ratio v/c; time in his theory seems to have been dependent on other factors although with length thus determined then, for a given velocity, the time element would, it seems, have been thereby pre-determined quantitatively - if not qualitatively.] And in both cases, this same ratio would also be fundamental in the formation of their respective new transformation equations which, when applied, allow us to see (at least with respect to Einstein's theory) that all laws of nature are compatible (accord) with the principle of relativity. [But with Lorentz...they support (instead) the idea of a still ether affecting length of bodies moving through it. Thus, the extent of the variations in space and time were set by Lorentz for his transformations in such a way (by the function of v/c so derived) that the speed of light will (as it did) only appear constant - when it was really expected (by Michelson and Lorentz) to be variable - at least in their initial papers.]
69. The 1905 paper then continues with the third introductory paragraph: "The theory [ie of the electrodynamics of moving bodies] to be developed here is based - like all electrodynamics - on the kinematics of the rigid body, since the assertions of any such theory [advanced to answer the problem of....?] have to do with relationships between rigid bodies (systems of coordinates), clocks, and electromagnetic processes. Insufficient consideration of this [admittedly complex] circumstance lies at the root of the difficulties which the electrodynamics of moving bodies at present encounters." [my bracketed addition, italics and underlining]. That is, his theory will focus on (have to do with) those complex relationships - so analysed via kinematics. [One would like to have a list of the exact papers of ca 1900-1905, say, in which this topic encountered 'such difficulties' (and descriptions of the latter) at the root of which may thus be revealed that this circumstance was therein given insufficient consideration (which he is presumably now going to correct). [See Lorentz's 1904 paper for reference to some of these and in which there are likely references to earlier examples.] Such 'difficulties' must constitute 'the problem' to which he has alluded (without specificity) from time to time. Could it not have been spelt out more clearly? [One is tempted to conclude that it was Lorentz's seemingly 'non-rigid' bodies that Einstein was implying contrasted (in part) with his own approach to the same problem.]
70. [Note: Kinematics deals with the pure motion of bodies (even material 'points') in time and space without reference to the forces (as electromagnetic ones), energies or masses involved; that is, simply with the dynamic geometry, temporal and spatial, of bodies in motion - of whatever size and however 'moved' - in relation to their often differently moving frames of reference.] His analysis will thus deal with the relationships between the positions of such bodies (or points) within a system of 3-dimensional spatial coordinates - as established typically with the use of usually imagined perpendicular measuring rods - and of the time coordinates needed - using (again imagined) timing devices or clocks and electromagnetic processes (light). [The necessity for the latter might have been spelt out more explicitly here.] Exactly which of many kinds of potential relationships that are possible within such a melange of variables that Einstein will be analysing is not made clear at this point. However, he appears to be developing his theory in a similar way to that of Newton - who based his laws of mechanics/motion (vs those of electrodynamics) on a set of definitions of the basic concepts and relationships thereof involved in the mechanics/motion/velocity of bodies - over space and time - firstly as abstract geometric kinematics but later applied to the actual physics of same - entailing forces, energy levels and bodies of particular mass. Einstein follows this model also but does so with respect to the laws of motion of all 'bodies' (including light) and this requires that he add the additional category of electromagnetic processes into his final analysis of a single set of inter-relationships regarding all conceivable moving bodies. His model, his theory, is thus expanded from the limited sphere of mechanics to the broader and encompassing one of electrodynamics - of moving bodies within a single conception or continuum.
71. In both cases, it should be possible and relevant to show the application of the principle of relativity to the motion of all such bodies. But in order that 'everything behaves just the same' whether one's surroundings are moving uniformly or are stationary (as qualified earlier) - as required by that principle - for all laws of nature - will now entail a new consideration - not previously appreciated; for now a law of nature has been (belatedly) recognised that concerns motion itself - the 'bodies' pertaining to same of which must remain constant in both types of surroundings regardless of relative velocity differences. To accommodate this reality, the velocity of all other bodies, as viewed by those in the relatively stationary environment, must now be equally recognised (again belatedly) as actually composed of a variation in the perceived magnitudes of its two inevitable components - distance and time - and not of a constancy of same, as previously assumed. The greater the velocity of such other bodies (as perceived from the stationary or slower-moving perspective), the greater is the extent of this perceived variation. Seemingly, this recognition follows from the requirement that all moving bodies must so move within the constraints of a limited continuum of possible velocities - with a maximum at the 'far' end of that continuum.
72. [Describing the bodies and/or the coordinate system in which they may move as 'rigid' apparently obviates any subsequent explanations in terms of bodies physically 'contracting' or time inexplicably 'dilating' (as per Lorentz's hypotheses). It also makes consistent and predictable the measurements of such motion. It is only as viewed from a differently moving environment that the effect of the additional velocity of that other environment on activities therein can be appreciated. It is thus only from such a perspective that some allowance or adjustment in what is so perceived must be made in regard to the components of velocity of any and all moving bodies so observed - where those of light whose constancy of velocity (both actual and perceived as such) is responsible for and necessitates those adjusted perceptions of the distance and time elements of the velocity of all bodies moving within the differently moving environments so observed and measured. Thus, even light's 'body' should be subjectable to this same analysis despite it being at the extreme of such effects.] While all bodies moving at some velocity are thus conceived by Einstein as 'rigid' (as is the constant, non-adapting, velocity of light), it is the space and time components of that velocity of all moving bodies that in his conception turn out to be 'malleable' (and thus not, as Lorentz would maintain, the other way round.]
73. Newton began in his model by defining space and time as independent absolutes (see discussion elsewhere). I believe he set out his stall, his theory - of 'absolutivity' as it were - fairly unambiguously and without too much preamble. As such, these two fundamental aspects of motion were seen as not relative, dependent variables, varying according to any other independent factor(s), but remain unaffected in all circumstances - as two independent background elements to all 'the events' (motions/happenings) in the universe. As mentioned elsewhere also, this 'orientation' to scientific investgations soon became 'second nature' to all concerned for about 3 centuries; it wasn't a position to be defended; there was no opposition to it. Einstein, on the other hand, sets out his stall, his theory - 'of 'relativity' - rather more indirectly and less explicitly. He begins with an analysis and definition of simultaneity although we (eg those initial readers (and many laymen) of his article in 1905, say) are not really aware of what his actual goal may be in this particular regard and thus why felt he must he begins there. We may assume (based admittedly on our later knowledge of his ideas) that it will eventually pertain somehow to a new definition also 'of time' per se. We might also look out (subsequently) for any comparable analysis and definition of space - again as part of his theory. Presumably, they too will differ from those proffered by Newton - which seemed to work so well - until about 1900 at least. He could, for example, be laying the groundwork for an eventual definition of time (and space) not as independent absolutes but, on the contrary, as dependent, 'relative ones' (say) - as foreshadowed above. And as such, they could be applicable to both meanings of that term (and in the case of one of them, very close relations at that (ie as per Minkowski; see later re the new single concept of a four-dimensional 'space-time' continuum).
74. They would thus not be independent constants (which never vary according to some other independent factor), but rather dependent variables whose values do so depend; or rather now, as a single (if complex) dependent variable (space-time), dependent on factors described above. And, as with Newton, the validity of the ensuing theory and of the definitions of time and space on which they depend, will only be determined by the later availability or otherwise of supporting evidence (and not just on logical derivation or assertion). Both of these scientists thus went out on a limb, as it were, with their suggested interpretations (definitions) of time and space (as integral parts of their theories) and not necessarily as isolated concepts which they would necessarily 'swear by' as some kind of God-given truths to which they alone were privy. They were, rather, but parts of their overall theories and thus open to eventual questioning and testing. They would fall or not depending on their relevant predictive powers and internal consistencies. Moreover, while it may well structure his theory most effectively and logically to start his exposition by defining such fundamental concepts in the area of motion as time (via simultaneity) and space, we may recall that his startling 'discovery' of their relativity only materialised late in the (his 'long') day (as it were). But once so discovered, it must have become obvious to him that this was the essential foundation from which the theory could only now be more systematically constructed and formally (logically) explained (albeit given his earlier remarks about there being only relative motion - and no fixed ether, etc)
75. It may be useful to clarify, if we can, just what Einstein meant in the concluding sentence of this last (3rd) paragraph of his Introduction - that is, by the terms 'this circumstance' and 'the difficulties'. By 'this circumstance' he seems to be referring to the above mentioned relationships as they had been inadequately investigated within a large body of previous research in which, inter alia, the constancy of the velocity of light was not recognised as the actual basis of various anomalies so found. The variables which inter-relate in electrodynamics will do so in ways which go beyond that of mechanics {seemingly to considerations about the constancy or not of time and space(?); but then why would anyone have done so?? - before his happy tram journey) and sufficient consideration had thus been given to a kinematic analysis of same - specifically that the true constancy of the speed of light wasn't generally appreciated nor its implications so analysed (re time and space) as it didn't accord with the mechanical interpretation of the principle of relativity - so was not the focus of sufficient attention of that kind. To so correct that insufficiency would thus (it seems implied) provide the answer to 'the problem' - which, in turn, was implied in the phrase: 'the difficulties' - by which he seems to be referring to such as the negative results of (such as) Michelson's studies and the not quite adequate 'explanations' provided by Lorentz and/or Poincare and/or to those of Fizeau and others (Bradley and Doppler) which were equally ambiguous in their interpretations of matters which would be seen as ultimately germane to his present concerns and analysis.
76. However, he fails to exemplify the dictum proferred by a scientist heard recently on the BBC, who said that "The most important role for the scientist was 'To define clearly the Problem". I don't know to what extent he fulfilled that requirement in his two other important papers published that same year (in the same Journal), but in the one in which he presented what he described to his contemporay (M. Besso) as 'my great discovery' (or some such) - being the subject of the present account - Einstein appears for some reason to have been purposely vague in that regard. He certainly is very precise regarding the many 'definitions' on which his theory (ie 'the answer') is subsequently developed (each, in a sense, a mini-hypothesis) but was not so in respect of any description of the precise 'problem(s)' being so carefully addressed and thereby answered. However, the actual main problem may well be implied and derivable from that very, albeit complex, 'answer' (ie by working backwards)! At least, it should be. We shall see:
77. Thus, in the first three paragraphs of his paper, he at least did provide some clues about what 'the problem' was that he felt needed to be addressed and answered'. It 'had to do with' the need for a more accurate and logically consistent understanding of the principles which determine the motion of all bodies in nature - whether electromagnetic or otherwise. [This arose when one such 'body' proved awkward to handle in the 'old physics'.] Part of the answer was that they could apparently now all be explained within the one over-arching framework of electrodynamics - structured in terms of a generalised principal of relativity. The need for such a theory is revealed in Einstein's reference to the 'difficulties' which this topic was then encountering (presumably over the previous decade or so). They were thus manifested within various inconsistent results reported within this general sphere of research following on from those of Bradley, Fizeau and Doppler concerning light. In particular, the constancy of the velocity of light (as and when it became appreciated) appeared to conflict with the original principle of relativity but, before this was appreciated, there was equal confusion concerning the existence and motion or otherwise of light's assumed medium, the ether, and of its possible role as a fixed reference criterion - seemingly of importance to many as an explantion of other anomalies as mentioned earlier by Michelson; see paragraph 182). The work of Hertz, Michelson and especially Lorentz in the 1880s and '90s further revealed these 'difficulties' in electrodynamics. And the answer to this 'problem' was implied within his comment regarding the 'circumstance' (of the kinematics of certain relationships - as just mentioned) which required sufficient consideration (ie in order to resolve those difficulties/provide that answer). Examining any role of an ether had thus further confounded recognition of the real problem - seemingly - so this would not form a part of the present analysis.
78. Evidence that the Earth moved through a still ether - based on an appropriately varied velocity of light - would have saved them (eg Michelson and Lorentz) a lot of trouble. But this wasn't found and rather than point the finger at a stubborn constancy of that light's speed (which would have been incompatible with the current principle of relativity), they sought to account for the failure to establish that fixed ether by assuming that it existed nevertheless (despite that lack of evidence) and that it served not only as a substantive medium for light (consistent with the prevailing mechanical view) but also (by the 'real' motion implied) 'to contract the materials of the interferometer travelling at 30,000 kph relative to it (if not, as initially claimed, physically 'against' it. [Seemingly, Lorentz felt that the electromagnetic forces involved in that contraction of a body's particles were only generated when the inherent charges moving with same did so in absolute terms (ie 'relative' to an absolutely still ether in which they, as everything, were assumed to exist). Did this reflect the same 'misreading of Faraday's induction law as pointed out (as a most fundamental aspect of the basis of his theory) by Einstein? And if it did, were there other recent or contemporary ideas then 'abroad' to which Einstein's destruction of the idea of absolute rest also prove relevant? (As Hertz or...?) Or, was it only vis a vis Lorentz??
79. In any case, it is ironic that Einstein would say that the motion of the sub-atomic charges underlying Lorentz's contractions needn't actually be 'absolute'; their relative motion itself would have been sufficient - if indeed such contraction actually occurred thereby. But as he would deny that it did so, it was academic in any case.] And, by some alchemy, the 'local time' was also conversely (and conveniently) 'dilated' in Lorentz's equations while this was occurring. By appropriately adjusting the usual Galilean transformations (accordingly - in terms of both these hypothesised contractions and dilations) when applied to such measurements (later called 'Lorentzian transformations), everything appeared to come out alright in the end (when so re-interpreting Michelson). It was rather 'too neat' - even though, amazingly (or 'necessarily'?), the arithmetic was spot on.]
80. [While Einstein sets out his two postulates (based on his minimally supported assertion that (a) there is no absolute rest and (b) unstated findings as have 'already been shown', and (seemingly) on Maxwell's equations, respectively) and his equal assertion that there is no need for an ether - either as medium or as a location for such a stationary system which might be thought to be necessary to provoke spatial contraction or temporal dilation (for which, whether as ether-based or otherwise, there was no evidence and even some against), there is no indication that these elements in his reasoning transpired in any particular logical order or sequence. Once arrived at - by his admitted years of mental trial and error - he could set them out ('after the fact', as it were) in whatever form and order he felt would best (most logically and clearly) reveal the underlying truths (only thereby revealed) about the electrodynamics of moving bodies. That is, after he had 'figured it all out' - by who knows what means and in what order. We do know, however, that he did say at one point later that, after his long 10 year struggle "...suddenly it came to me...that 'time' (and its inevitable association with signal velocity) was the key.." - ie presumably to unlocking a (?side) door of the room which contained the answer to the problem (of finding a source for the variations in time and space that he saw (via Lorentz's theory and some of his own reasoning) very likely were necessary to resolve the incompatability - but to do so without recourse to a fixed ether).
81. As he reveals to us the 'kinematics' of moving bodies, the place and role of relative time (and seemingly of space as well) should become more apparent. However, it might have been better if he had first provided a clearer basis as to why he suspected that these particular factors might be the source of the difficulties he seeks to resolve in his theory. If we re-examine his last two sentences of paragraph 3, we see that he............(to be completed).
82. The original principle of relativity assumed that every body in nature should vary its velocity according to any variation in the velocity of relevant frames of reference (eg in that of their source). This was because all known laws of nature (according to the requirements of which all such bodies move in response to applied forces), while operating identically on all all such reference frames, did allow such bodies to move, in addition, according to the velocity of the frames on which those forces applied, when measured from a differently moving frame. But such laws were in fact those of mechanics only. Laws of electrodynamics, including that for the velocity of light, didn't allow the velocity of the frame emitting the light to be added to that of light itself (to determine its final overall velocity), since (unlike bodies moving in responses to the forces of mechanics), its law stated on the contrary, that its speed could not be added to (boosted), nor reduced, from its original velocity. This was a major part of the more general requirement that all (known) laws of nature should operate as their law requires in all reference frames. But accepting that light alone did not follow that of mechanical bodies in being boosted by their reference frame represented a huge problem for that original principle. Its terminology had to be re-phrased to encompass a greater generality - one that in fact had always if unknowingly applied; its effects at normal slow speeds being too slight to recognise.
83. It may well have been that it was only when Einstein took on board the validity of that constancy - ie first - that he (only then?) realised that something assumed as part of the original principle of relativity must therefore have to be adjusted to incorporate this later realised/accepted but precedent reality (and so manifest itself within new transformation equations which were compatible with these facts). While he may also have convinced himself early on that there was no absolute rest, and that therefore, the principle of relativity was the only means in terms of which the motion of all bodies could be explicable, he seems not to have concluded at about the same time that, with no absolute rest, another feature of nature must also (immediately) follow - namely that time and space can not be absolute themselves but must in fact vary. Rather, the latter conclusion likely only came to him (ie "suddenly...") when he realised that it was the incorrect assumptions of their absolute natures (underlying the original principle of relativity - which had gone some way to denying the need for a principle of absolutivity) that had to give in order to allow that principle, so adjusted, to prove compatible with the (already accepted?) constancy of the velocity of light. For they were the only elements left in the kinematics of the motion of all bodies that could so adjust - given that there was only relative motion and its associated principle; without a system of absolute rest, there was no 'absolutivity' principle to turn to - with its own set of laws. All laws functioned equally in all differently moving environments - the only kind there were; there was no stationary absolute environment that would require laws unique to itself. All laws had 'evolved' to prove mutually consistent within our universe of (only) moving parts. Uniform movement has no effect on them. Its where they 'grew up', as it were. They know nothing else. Hence the principle of relativity.
84. In order that the one law of nature which itself entails a 'constancy' of velocity of a particular category of 'happening' (changes/motion of physical systems) in nature (that is, despite any motion of its source or target, its always the same speed), the principle of relativity (which itself concerns velocity) must require a recognition that the two components of velocity - time and space - do (ie must) vary according to the relative velocities of the frame(s) of reference from or by which such happenings are measured. This 'in-built' reality is not apparent nor significant at the normal velocities of most everyday happenings, but becomes increasingly so as those speeds approach that of the one law of nature which entails this constancy - ie the velocity of light (in a vacuum) - by which, confusingly, such signal limitations (and thus our perceptions) are determined.
85. [It must have been while mentally manipulating the various factors of motion and their inter-relationships that it eventually came to him ...that 'time' was the key...'. This occurred after his imagined 'tram journey'. For even after concluding that some variability in time and space was beginning to appear to be the case, he needed a realistic basis to account for or exemplify this in nature - and his tram journey seems to have revealed this!] This must have been followed shortly by considering that the speed of anything, entailing as it does both time and distance (space), indicated that as such speeds approach that of light, so the extent of the alteration in the variable magnitudes of both these elements of velocity would have to alter accordingly - ie as such speeds became a greater proportion of that of light (although this seeming limitation may simply be a correlate of the actual cause of the limitation - namely the greater the lag in receipt of the information 'about' the time and space variables underlying the velocity measures concerned). They would no longer be seen (and wrongly assumed) as being absolute and constant. The original Galilean transformations (which Einstein was probably mentally manipulating when 'it suddenly came to him...' wouldn't accommodate the new reality of a constancy of any phenomena. [He likely had already accepted these.] Lorentz's transformations adapted to the anomalies of Michelson's results in ways that proved accurate but based on false conceptions of why such adjustments were necessary. He hadn't accepted that Michelson's result was due to light's speed being a constant. Einstein realised that such adjustments (in the transformation equations) could be accounted for in a more realistic (if very surprising) way. There would be no actual physical contraction or time dilation. What there would be was superficially very similar to these variations but rather difficult to explain. But at least these equations provided a kind of framework or check that his were of that same form. We may now continue with an analysis of the remainder of Einstein's paper where the basis for this conclusion, and the implications of light's speed being an upper limit of the velocity of anything, will hopefully be revealed to us]:
[Note: An additional resume of the 1st three paragraphs of the 1905 paper is firstly placed here - to be integrated into the foregoing - in the event that its different focus may help clarify certain aspects:
A Second Resume of the 1st 3 papagraphs of the 1905 Paper.
86. 'This second resume and analysis of the early part of Einstein's paper 'On the Electrodynamics of Moving Bodies' is intended to better determine its line of reasoning - as the latter seems to be based on certain aspects of mechanics that are not fully described in that part of the paper in any detail. Thus he begins (as we've noted above) by citing two examples from the domain of electrodynamics which of itself is quite reasonable since this is, after all, the essential topic/title of his paper. But both its development and the title of the paper might be better understood if considered initially within the wider or even contrasting context of the mechanics of moving bodies, as evolved earlier by Galileo and Newton, since it emerges out of this sphere. [A review of Maxwell's theory of the electrodynamics of stationary bodies would also prove useful here.]
87. Thus, Galileo first explained how activities on the (to him) moving Earth (or, equally, on a smoothly-moving ship) could and did proceed just as the laws of mechanics would predict, without such difficulties (as 'things flying all over the place'), despite his (and Copernicus') conviction that the Earth actually moved, around the Sun, rather than it being the other way around. Before this, the Church had always maintained that the Earth was the unmoving centre of the Universe. So, anything on the Earth forced into some movement would begin the resulting motion effectively from a truly 'still' condition and then move by an amount that exactly reflected the extent of the force so moving it, no more and no less; there would be no extra motion due to any movement by the Earth itself. But if the Earth actually did move, relative to the Sun, an explanation was needed as to why things upon it would nevertheless move in very predictable ways, according to the laws of mechanics, and not be disturbed by the Earth's motion. This explanation was provided by Galileo's concept of Inertia. Because of inertia, the prior status of all bodies moving steadily with the essentially uniformly-moving Earth had no effect on the laws of mechanics applied to them. For all bodies to which any force was applied would begin their response from exactly the same neutral 'starting point' - that is, one of steady, unpreturbed motion but effectively zero acceleration - whatever the relative velocity of their particular frames of reference. Newton later incorporated this same principle within his famous 'Principia Mathematics' concerning his important laws of motion.
88. This application or expression of Inertia in the domain of mechanics was eventually referred to as the Principle of Relativity which states essentially that any difference in the smooth motion (velocity) of large bodies (as planets, trains, planes etc) between themselves has no effect on the mechanics of smaller bodies that may transpire on any of them, providing only that such pre-existing, relative motion is straight and uniform. While this principle is often quoted in the abstract as the rationale of why certain conclusions in mechanics (or, later, electrodynamics) may be arrived at, it is important (I believe) to be able to analyse and keep in mind the actual basis for such a conclusion - ie in terms of the foregoing mechanisms or rationale underlying that principle, rather than typically mentioning only the principle per se. And while the Earth is not absolutely still but moves, its actual motion isn't absolute either; it does move but that movement is relative - to the Sun - not absolute. For it to be absolute, there would have to be some body to which such movement could be compared or referred that was itself absolutely 'at rest'. But all known bodies in the universe, including even the Sun, are in motion relative to something else. The motion of a ship or plane is therefore also not absolute with respect to such as a nearby dockside, say, or the Earth itself. In all such cases in the domain of mechanics, motion is always relative to something else; and either element in such comparisons may be seen as the one 'moving' more (or faster) or less (or slower) than the other - since the mathematics are identical whether the sign is positive or negatve. Thus, all bodies moving in relation to each other possess no properties that correspond to absolute motion or rest and all a have equal precedence as far as which is conceived as being the more relevant viewpoint ('observer's platform').
89. On the other hand, the two examples from the domain of electrodynamics (vs mechanics) which Einstein cites at the actual beginning of his paper concern the motion of certain bodies in nature (namely, electrons) that, in his view, has been wrongly accepted as being affected only by what may be termed the apparent absolute motion of one of the elements concerned (or at least something comparable to that). Thus, firstly, it was generally believed that the electrons in a conductor could only be provoked into motion (as an electric current) by the force produced when a magnet moved in the vicinity of a stationary (electron-containing) conductor. The converse was generally assumed not to result in their motion. The former case was thus tantamount to assuming that a kind of absolute motion by the magnet, with respect to a 'still' conductor, was the necessary arrangement to provoke this electrodynamic action. However, Einstein points out that this motion of the electrons actually depends only on the mutually relative motion of either or both these respective elements - the magnet and the conductor - as first noted by Faraday. Either may be viewed as the moving element - but with the effects depending only on their relative motion, one to the other, with neither ever being absolute or taking precedence. When generalized to all of physics (or nature), this will have profound significance and implications - as expresed belwow through the principle of relativity.
90. Secondly, he points out that attempts to establish that the Earth (like the magnet in a way) moves in some absolute sense - in this case not with respect to the Sun but with respect to an absolutely stationary ether that was alleged to surround the Earth - have been unsuccessful. (This must refer in particular to Michelson's experiments of the 1880s. The failed attempt to establish this relied on obtaining a difference in the velocity of light (propagated by means of such an ether) measured with and then perpendicular to the direction of the Earth's motion.) Without specifying either Michelson's specific experiments, nor Lorentz's questionable attempts to explain such negative results, Einstein then alludes to other failed examples of these sorts which, he says, "...suggests that...the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good." In other words, just as all the laws of mechanics obey the principle of relativity (for the detailed reasons elaborated earlier), so the above and similar examples (and other rationales which he only later cites) suggest that all laws of electrodynamics also should do so (without being adjusted for any special cases; ie the same laws). In both domains, maintains Einstein, there is only relative motion and relative rest - never any absolute forms of either. [Note that in my earlier 'insertion' after paragraph 38, I have interpreted Einstein's second example as representing the mechanical sphere; this may need further thought.]
91. Being thus merely 'suggested', he initially calls this statement a 'conjecture' but, reflecting his confidence in it seemingly, it is soon expressed in his paper as a more formalised 'postulate'. As such, it is referred to as 'the principle of relativity' but, as will become apparent, it will be of a more generalised form than the similarly named principle of Galileo or Newton - which (as it turned out) applied only to mechanics (and not entirely accurately either); this altered version incorporates electrodynamics into which mechanics will be subsumed. (Note however that one should, as above, be able nevertheless to account for why this principle should apply to all laws of nature (including that for light) in terms of the actual 'mechanism' by which this principle functions, applies or comes about and not rely simply on some abstract reasoning or analogy as to why it should (somehow) be universally applicable - as a kind of abstraction.) As with mechanics, so with electrodynamics, bodies moving in relation to each other should possess no properties that correspond to absolute motion or rest. [Unless light itself (and its very small 'bodies' (quantum packets)) is an exception ? No, it is merely 'at a limit' and the absence of any absolutism in nature and the role of inertia apparently applies equally here as well. ]
92. In any case, without making it explicit or clear, there seems to have been an implication in his conjecture about the more general applicability of this principle that without such failed examples to cite (and those 'other rationales' that he would later quote), there must have been some basis on which such laws of electrodynamics and/or optics were, before he pointed out such examples and rationales, more generally asssumed not to have come under the purview of the traditional principle of relativity - at least as it was interpreted at that time. This basis is implied in his second postulate which he ?necessarily presents almost simultaneously with the first one (ie in the very same sentence). This is that "the velocity of light (in a vacuum) is a constant not varied by the velocity of its emitting source" (nor, as later indicated (where?), by that of its observer or measuring instrument). To the extent that this hypothesised feature of light was generally accepted before 1905, it could be understood why it was necessary to present some evidence (and related hypothesis) that suggested that, despite this unusual (indeed unique) characteristic of light (and other electromagnetic waves), such electrodynamic phenomena should somehow also fall within the ambit of a principle of relativity (necessarily of a more general application) and do so for the reasons cited above. Given that this 'law; (of light's constant speed) was simply one of the total laws of nature to which the now generalized principle of relativity applied (by definition), the addition of the 2nd postulate might be considered redundant - as it was already implied within the main postulate - but it must have been felt that its unusual nature required special focus (as it had so conflicted with the traditional principle of relativity) and thus its equal presence and attention - to explain how its uniqueness can now be incorporated into the perview of a broader principle of relativity, as defined above, and elaborated further below:
93. Thus the term 'somehow' is appropriate above because this characteristic of light may well, on initial consideration, appear to be incompatible with the (traditional) principle of relativity - as that principle had applied to mechanics. In fact, Einstein's theory of the electrodynamics of moving bodies (later to be called the theory of special relativity), consists essentially in providing a rationale for showing that his two postulates are in fact not incompatible; that light should also be expected to accord with this now more generalised (and adapted) principle. That is, that the constancy of the velocity of light whether measured within the same system of reference from whence it was sourced or measured from a different system, moving at a different velocity, will always remain and be measured as its law requires - at its one constant speed even after considering the extra velocity to which its source may have been subjected. This is what the new principle of relativity requires of all its laws - that they perform exactly as they should (as they state) whatever the difference in the speeds of the frames of reference concerned. His theory would thus seek to show how the application of the relevant transformation equations while allowing fitting increases or decreases in the net values of the motion of all 'normal' bodies due to the speeds of their immediate frames of reference (vis a vis some 'neutral' viewpoint - as the traditional principle of relativity expected), would also allow the complete lack of any such increase or decrease in the speed of light regardless of the speed of its immediate reference frame - as the law of the constancy of the speed of light required - thus fulfilling the dictum of the new principle of relativity with which all laws of nature must accord/be compatible with.
94. While this actual compatibility (in the case of light) is exactly what Einstein's postulate concerning his new general principle of relativity would explain and predict - that is, that this electrodynamic law about light's constant velocity (like any law of nature, including those of mechanics), and the measurement of same, should not be affected by any difference in the velocities of its own source as compare to that of a differently-moving system from where it may be observed and measured, there was a difficulty. The seeming incompatibility referred to (vis a vis the old principle of relativity) came about because the consistency of all laws of mechanics, whatever the differing speeds of the reference systems concerned in their occurence and measured outcomes (as represented by the principle of relativity), is only manifested directly when measured within their 'own' reference systems (as on a fast-moving train or on an airplane, say). When occuring on the train but measured from the train station say, through which it is moving at some (smooth) speed, any velocity of a body caused to move on the train will, in relation to the station, amount to its own newly achieved motion, as measured relative to the inside of the train (ie just as measured before), but plus now that due to the speed of the train itself (in relation to the station). This reality causes the original principle of relativity no problem however since its focus is really on the the actual net speed of the body concerned within the train alone (as meets the law of its mechanics, after some force is applied to it). This is what the original principle required and generally correctly predicts. To verify this, it is simply necessary to subtract - by way of the appropriate transformation equation - that latter velocity from the totality of the velocity of the body as measure from outside.
95. A difficulty arises when such a subtraction (as provided by Galileo's transformations) does not maintain the law's requirement - as in the case of the electrodynamic law for light! For this particular law requires that light's velocity is not affected by the velocity of the faster emitting source - which would thus make it incompatible with Galileo's version of the principle of relativity, at least with its usual transformations which effectively seek the 'net' velocity of a moving body after adjusting for the full velocity of the emitting source - by subtracting that latter speed. And yet, Einstein was convinced that that principle, in its essence at least (ie as described in its detail above), was basically valid. But it should apply to all laws of motion, including that pertaining to light (despite its unique character - of not, in fact, gaining any boost (say) from a faster moving source (ie its immediate frame of reference). It is for this reason that Einstein felt that a more generalised and adapted principle of relativity was probably required, one which would apply to all laws in nature, both mechanical and electrodynamic - and thus with necessarily adjusted transformations. Such adjustments would therefore require an explanatory rationale as to the basis of their new form.
96. A new formula was thus necessary by which means the net velocity of all bodies in nature, ie after subtracting (some proportion - as all, some or none - of) that due to say a faster reference or emmitting system, was still as their various laws dictated. For light, this would require the subtraction of none - ie zero percent - of the velocity represented by the faster (emitting) system, but for all other bodies a subtraction of somewhere between just above zero and (as formerly found and most typically) 100 percent (or thereabouts) of the velocity of the relevant (faster) system concerned.* Between these two extremes, the percentage of the faster system's velocity that would be subtracted (from the total speed measured incuding that due to the force acting on any body therein) would depend on the extent of the totality of such faster velocities (ie of the body and its system) and, crucially, this would be provided by new transformation equations in which the values of time and space would necessarily become variable (which, effectively, 'allows' the velocity of light to remain constant, as its law requires, and allows the dimensions of moving bodies to remain rigid not physically contractable and the clock time measures for their motion to also remain physically unmalleable; apparently, only the perception of these so alters although there is no other means of establishing any other 'view' of time and space than those perceptions. They are our reality.
[* Such adjustments means that the net velocities remaining after the subtraction of often less than 100 percent of the faster reference system's added velocity would be slightly different than the values calculated previously using the Galilean transformations.]p> 97. It is fortunate that Einstein's theory doesn't require a concept of absolute rest (as allegedly provided by a still ether which was hypothesised by Fitzgerald and Lorentz to cause a contraction in the material structure of bodies moving in such an absolute sense through it) as it would resolve its one difficulty by another means - namely by adjusting the original principle of relativity (to become his 2nd postulate) so that the apparent incompatability with it (ie the difficulty) which accepting his 2nd postulate about light's constant velocity appeared to indicate, would be resolved. That adjustment entailed instead revised conceptions (?perceptions) about the previously assumed invariablity (non-relativity) of Time and Space underlying the velocity of the bodies concerned.
99. 'Note: It would appear that the Constancy of the velocity of light (ie c) - and the eventual realisation of its universal validity by Einstein - was the prime reality which ultimately insisted upon an altered conception of nature, but that the consequent recognition that a more validly-based principle of relativity was (subsequently therefore) also required indicated that this latter (revised) principle (based on a new conception of time and space) - once discovered - would prove to be an equally necessary contribution before the final conception could be fully and finally realised. Once that principle was so adapted (generalised), it could later be maintained that it was the over-arching determinant all the time. [Again, however, the basis for this must be explicable in terms of the mechanism through which that principle operates!] Once it was so appreciated, the Constancy of the velocity of light (c) had either to be a single exception in nature and thus an exception to the existing principle of relativity or a way had to be found to 'modify' and generalise (the actual workings of!) that principle (in response) so that it could incorporate (be compatible with) that equally insistent Constancy (and of all other laws of nature) - and so would not be an exception in that regard, and not the other way around, and thus be recognised as falling under the umberella of that more general principle of relativity.
100. But the former was not really an option, since without any absolute resting place in space, there was nowhere in which laws of nature could be (or had ever been) distinguished in their operation on that (moving vs 'still) basis (ie and so identify such a unique environment) which left as an alternative means some law which alone could allow such a differentiation. All our laws of nature have evolved in relation to our variously (if relatively) moving environments on which they prove consistent, reliable and valid (almost by definition); if they hadn't, we wouldn't be here; nor they! [But inertia?] ). But while the reality of the Constancy of the velocity of light had always been the case (ie one such evolved law), even if only recently recognised as such, there had never been any evidence pertaining to light's speed which could be cited as indicating that any particular frame of reference (environment) could be distinguished in terms of being 'faster, slower, more stationary or more mobile (or whatever) than any other - ie on that basis.
101. It was concluded therefore that, as far as one could tell, the principle of relativity must in reality be so based (ie on 'relative', variable or 'malleable' magnitudes of time and space) as to be compatible with all laws of nature within a single conception of motion, including that concerned with the velocity of light (and must of course have been so based all along). This was required in order that the differing speeds of reference systems 'made no difference' (which is the principle of relativity) - for all laws. The reason why this must be the case may also need explaining. Thus, it was as if light's speed couldn't be the 'malleable' element but had to remain constant or rigid then time and space had to give up being seen as unyieldingly constant and have their 'malleable' roles recognised. This strikes me as providing (a little) more in the way of explanation than simply asserting, without explanation, that 'all laws of nature 'must' accord with the principle of relativity' per se (even if they do) - as a basis for justifying...anything.
102. However, another justification of similar form is provided by Hermann Bondi when he says, in effect, light's speed can't be an exception to the principle of relativity because of something he calls 'the principle of the unity of physics'. That is, he says: 'The Principle of the unity of physics requires that systems that cannot be distinguished by internal dynamic experiments (ie in mechanics) should be indistiguishable by any internal experiments'. We are thus driven, says Bondi, with virtually no means of escape, to Einstein's more general Principle of Relativity.' The 'force' of that drive was (?prior) realisation of the Constancy of light's velocity. Well, if ever asked, I can quote this but must admit, I'd prefer rather more concrete detail or an actual example of what he means by that generalised abstraction concerning the apparent primacy of something called 'the principle of unity'. We may also ask ourselves whether Einstein felt that it would be better to present his theory as though it unfolded in his mind in some particular logical order even if he 'happened upon' certain insights in some different order, worked it all out from that perspective and only later 're-arranged' it as now presented. Will we ever know?
103. Further discussion on this point is given below in which the demands of there being only relative motion and only moving parts to our universe (no still platforms) resulted in the evolution of laws of nature (ie those that survived and work consistently here) adapted to this reality. They all must work equally in all moving environments such that there is a consistency (a unity) of their operation from whatever perspective/platform/frame of reference they are perceived. Otherwise, they may well not be here; nor we to consider it. [But, the idea of 'perception' brings observers (like us) into the equation and surely the laws must work equally whether we happened to have evolved or not? True, but within each differently-moving environment all laws of motion operate identically in any case; it is just that they must appear to do so to any (who have evolved) viewing same from other platforms. Why!? Because they're 'our' laws...or? Yes, partly; we construct concepts which prove consistent with our sensory and perceptive apparatus as our only way of knowing/experiencing our ever-moving universe and its apparent 'laws'. Thus, we have concepts of space and time (now combined into one) by which means we can make sense of our perceptions of differing motions of bodies of either differing masses or acted upon by differing magnitudes of force, or both from our constantly moving perspectives. There are only moving perspectives and smooth, uniform motion - whatever its relative speeds - does not affect our laws - even that one entailing motion itself - and it apparently does so by virtue of adjustments in our perceptions of time, distance and mass. But it was soon appreciated by Einstein that one must be able to account equally for all (?uniformly) accelerated (vs unchanging) motions as well - as would be represented within his later general theory of relativity.
104. [Note: The theory of special relativity can be defined in a fairly general and succinct manner as a theory based on the idea that all laws of nature should be the same for all observers whatever their differing (uniform) speeds and viewpoints. But this tends to subsume that most unique law of nature - that pertaining to the constancy of the speed of light - within some anonomous totality of such laws and so masks almost entirely the crucial role of having to find the only way by which the one law of nature pertaining to a constant speed of motion (of anything) could be made compatible with that principle of relativity - concerned as it is with the effective non-effect of differently-moving reference systems. It would seem too simple to suggest that Einstein merely reasoned that his assumption that the laws of nature do not depend on one's motion meant that the speed of light too must therefore be found to be the same by different observers whatever their differing speeds - ie simply because it too was just another 'law of nature'. Such conclusions arrived at solely on the basis of such grand general principles and reasoning seems almost too easy. Rather, he would seem to have concluded that all laws of nature, including that relating to the constancy of the speed of light, may well not depend on one's speed - but only after he figured out how that latter constancy could somehow be reconciled with an appropriate principle of relativity - ie by conceiving all motion (of bodies) as falling within a single conception of motion from zero mph up to a maximum possible velocity of 186,000 mpsecond (which is the speed at which light 'happens ' to move. Once he had resolved that (eg 'Is there an answer...etc'?), he could see that observers moving at different speeds could report that light's speed did remain the same for both of them - but only by disagreeing on the time and distance each believed the light travelled. This important latter aspect would seem to have been concluded only after much analysis and confusion. Measurements of distance and time thus depended on one's speed relative to some relevant reference frame. Seemingly, measures of mass would also so depend (see below).]
105. And, having worked this out, Einstein (or others?) could then state the more general truth that all laws remain the same whatever one's speed - and call that 'the (special) theory of relativity': all motion is relative to a given frame of reference and any such motion (of bodies) implies time, distance and mass - which are thus all relative themselves to the relevant reference system. Everything remains internally consistent therein but may be perceived as having different values when observed from a system moving at a (relatively) different speed (due ultimately to the inevitable lag in receipt of the relevant information). The lengths of feet and seconds, say, to measure some standard motion of any body, including those of light, will appear quite normal to those in one moving environment but will appear different (ie longer or shorter) to those observing same from their own, differently moving (ie faster or slower) environment, while their own measures for these same motions will appear to themselves just the same as those used in the other environment appear to those there and vice versa. Their perceived magnitudes are thus relative to the relative speeds of each other's frames of reference. At slow speeds (little different from each others), these magnitudes will appear almost identical but if their respective speeds are somehow vastly different, they (lengths of time and distance) may appear quite significantly different in the others' locality despite measuring the same (rigid) things. Is it all similar to the fact that a house or tree you're standing beside looks quite tall but ordinary, yet those someway down the street look much smaller - yet we quite accept this without question or surprise?]
106. Seemingly, light would not only not require ether's assumed property of total stillness (absolute rest), it wouldn't require its alleged property as a luminiferous medium either; It must simply propagates itself somehow (neither aided nor hindered by any such 'ethereal substance') despite its assumed wave-like character. Thus the inclusion of the term 'luminiferous' is a little diverting from his actual concern here. For such a medium is primarily not required in Einstein's perspective on it, as a repository for a concept of absolute rest - which would follow from the primacy he gave to his denial that such an entity existed. There was thus, in his conception, only relative motion. [But, he also appears to make no reference to the third role (ie beyond that of either a medium or an absolutely stationaty space/system) for an ether - ie as the very crucial factor in Lorentz's interpretation - as that which is responsible for the contraction of a body's space - due to that still ether's effect on electromagnetic spatial arrangements within any body passing through it. But, in any case, that seems to be due to the ether's state of absolute rest which (somehow) acts on the truly (absolutely) moving electron's front-back dimensions.]
107. We may contrast Einstein's brief reference here (and not earlier) to this concept of the ether and the relevance its stated non-existence (or non-necessity) had on his theory with that seemingly implied by Hawking - as a more definite raison d'etre for elaborating the theory in the particular form and order he did.] And while this 'stillness' aspect was unrequired by him, something like an ether might still exist as 'a wave medium' (a kind of 'field') - although Einstein's reading of Maxwell's theory may have indicated to him that there was no need for this other (medium) role for an ether either - something that Maxwell himself appears not to have accepted before his early death. However, Einstein seems to confound these two roles of ether - without making any clear differentiation in terms of their relevance to his own theory. My later conclusion that the main development of his reasoning might best place this assertion and premise at or near the beginning of his logic (even if it may have been placed there only in retrospect - by anyone trying to set out a more logical format of general principles) - is certainly not supported in any obvious way when he inserts this aspect at the point he did. We may reasonably assume that he was simply saying essentially that his two postulates alone were sufficient for his theory - with no need additionally for an ether concept, whether as a medium or as a source of absolute rest. He also denies the need for any 'velocity-vector to be assigned to a point in this otherwise imagined still space for the proper functioning there of electromagnetic processes. [I'm not yet sure to what this latter aspect refers although it seems to allude to the idea that there is no (or no need for any) absolute (non-relative) positional or directional attribute to space.]
Einstein's Kinematics and Electrodynamics of Moving Bodies
108. Following his three Introductory paragraphs, Einstein presents his actual theory in two major Parts - focused on Kinematics and Electrodynamics, respectively. [His paper may equally well have been titled 'On the Kinematics and Electrodynamics of Moving Bodies' in that his kinematics prove to be so fundamental to his theory.] He begins his elaboration of the basis of his theory rather ambiguously in Part I - by immediately defining in Section 1 the concept of 'Simultaneity'. He provides no basis for choosing to focus firstly on this particular topic at the very outset of his analysis although it may reasonably be appreciated that this is a slightly indirect means of focusing on the relativity of Time - which topic we might at least suspect to be rather fundamental in that it was the subject of his eventual 'eureka' moment. (But on what basis might we suspect this ?) We can only point out that he states in summation at the end of his 3 introductory paragraphs that the theory of the electrodynamics of moving bodies to be developed arises from his 2 postulates alone, with no need for a concept of absolute rest (as some believed was associated with an alleged still ether) - as the utility of that approach would apparently be replaced by that of a new (generalised) form of the principle of relativity (postulate 1) (but again, how do we know this?) in which (as, admittedly, we are not yet privy in his paper) the long accepted concepts of invariable time and space will necessarily be replaced with their actual (or perceptual?) variabilities. (His earlier assertion that the incompatability as between that postulate and his 2nd one regarding light was in fact only 'apparent' might also point to some necessary adjustment in the measurement of the underlying time and space - as the basis of resolving that suggested misunderstanding. But, again, why?
109. In any case, such a new theory will also be based (as are all electrodynamics, says Einstein) on a kinematic analysis of the relationships between (the velocities) of all moving bodies, including electromagnetic ones, and the clocks, measuring rods and systems of coordinates by which (only) such velocities and relationships can be validly determined. Kinematics thus deals with the geometry and perceptions of moving bodies - of whatever size and however moved - by whatever forces and energy. The inclusion of clocks and measuring rods in this theoretical analysis certainly again hints at his likely focus on the components of measures of velocity - namely, both time and distance (space) - in which the idea of their suspect invariablity might again possibly be suggested; so we would seem to be partly privy, at least, to his intentions by this point. An analysis of such relationships will thus, he states, be require (at a more sufficient level than has been the case previously) in order to resolve 'the difficulties' which this subject was then still encountering - ie in making consistent and valid predictions concerning the motion (ie velocity) of all such bodies. Again, as with the 1904 paper by Lorentz, this is a theoretical analysis which, while it may prove consistent with explaining certain results concerning the motion of light, it will ultimately require objective evidence that this new interpretation about light, time and space are confirmed. And even then (as with Lorentz's views) it should be the case that any such new factors involved can't be explained as well by some other interpretation. Thus, for example, might there still be some kind of 'ether' extant - as electric, magnetic or gravitational 'fields' - which has a role - as moving with the Earth, say ?
110. Thus, as mentioned, Einstein begins his analysis by considering the subject of simultaneity. This is an indirect means of addressing the related and fundamental matter of the relativity of Time. It seems it was through his 'sudden' realization that 'Time' was a crucial consideration in the problem with which he had been wrestling in 1903-05 (and even earlier), and specifically that it was not possible to establish the same time at two distant points other than by light signals, that he arrived at his crucial insights into the fact that time and space must vary according to the speeds of the observers (or instruments?) measuring moving bodies in each other's (moving) frames of reference, and the more so the greater those speed differences - up to a limit (ie the speed of light). We may note here that Poincare at least had already made reference to this (simultaneity) aspect of analysing time, although in exactly what context, I'm unaware. After he has thus deduced his laws of the Kinematics of moving bodies (as part of his overall theory of same), Einstein develops in Part II the Electrodynamic aspects (with more theory?) in which he can apply the laws so deduced in terms of actual bodies of certain mass subjected to given forces. This would lead on to deductions concerning his famous equation of E = m.c
[Note; It may be useful to interject the various succinct if rather general and abstract paragraphs (as and where relevant) of Einstein's 1921 Nature article on Relativity amongst the different sections of our analyses of the two theories - eg in a different coloured ink to so identify same - to provide a kind of road map as so considered retrospectively by Einstein - when he may well have seen for the first time just where he had been in his mental travels and how it may now be represented more succinctly - if rather abstractly - in terms then (post-1918, say) of more general concepts.]
[This is described in the article within 16 numbered Sections]:
1. Definition of Simultaneity.
111. Because it is a kinematic analysis, Einstein begins by defining a (?relatively) stationary three dimensional system of coordinates (x, y, z) - as one initially in which 'Newton's mechanics hold good'. A point (as also, a body) can be defined therein - in terms of x, y, z - which is 'at rest' in this system - its position there being measured by rigid standards of measurement (ie measuring rods or 'rulers'). If this point moves - to a new position within this defined context - it must do so over time and thus at some velocity (as relatively slow, fast or whatever). Now, we must also be clear, says Einstein, just what we mean by 'time' in such a context. Here we see how he is probably leading us to an eventual definition of this latter basic concept - but via this initial analysis of simultaneity - within the classical coordinate system. We may note that, at this point at least, he doesn't also say we must be careful as to exactly what we mean by a 'position in space' (or the 'distance' between two such positions) - although he has possibly already ensured clarity in this regard by defining same carefully - with cartesian spatial coordinates and standard measurements. Presumably these prove just as robust when measuring the point's movement over space (ie 'distance'). The use of a standard time measuring device isn't as objective seemingly (eg cartesian 'temporal coordinates' are presumably needed) as are rigid rulers for distance, in that he points out that when using a clock, we are in effect assuming (making a judgement) about a simultaneity between what it (the position of its hands - often both at the beginning of a body's (here 'point's) movement and, after a time interval, its position at the completion of that movement) shows to our eyes and the actual occurrence of the event (a movement of the point) being timed (or even the actual time?). (This of course entails two such judgements - at the time it begins to move and when it arrives at its new spatial position.)
112. Thus, we can now better appreciate why he approaches his analysis of time per se from the point of view of defining simultaneity. [Possibly some parallel argument could be developed concerning distance across space entailing assumptions about a 'simulspateity' (or some such) between what a ruler shows and the (?actual movement/distance) of the point over an interval of space being measured (ie not actually merely 'judged') but, as suggested, this is probably already provided - by the cartesian coordinates?] We might also differentiate the simultaneity as between an event (as the motion of a body or even a point in space) and the position of the hands of a clock (both to be perceived by an observer either there or at some distance) and the (lack of) simultaneity as between (one of) the event(s) being the position of the clock's hands themselves, or a point on same (and the other some such motion of a body), and our perception of both at some distance.
113. Einstein then asserts that such judgements of time - while accurate for events which occur 'near to where the clock is' - 'are no longer satisfactory...when they occur remote from the clock'. [Should he not have added that this was even more the case if such remote events occur in a frame that is also moving at a different speed to that of the local events?] No rationale is provided at this point as to the basis of this important assertion. [We may point out however that it was allegedly while returning from a visit with his colleague Michel Besso on the tram one evening that he noticed the clock on the town hall in Berne and considered how one could establish that it was that same time at some other distant place (as after a journey on that tram taken at near the speed of light). The next morning, he returned to Besso and excitedly reported that he had completely solved their problem (which they had been discussing over some months before). He later said his joy was unbounded for weeks. [Possibly this is why he focused initially on 'time'; maybe this remoteness doesn't apply with respect to the spatial variables?] He then gives an example of a way in which this claimed inadequacy (where such remoteness of the clock may apply) may be overcome (after giving initially a similar one - ie by using light signals - without explaining why this might be seen as a reasonable way of overcoming the inadequacy which he asserted earlier). He provides this initial method after saying that 'of course, we might content ourselves by simply doing this...(such and such)...in using such light signals' (as though this was the more obvious method) but then informs us that this method would in fact be inadequate - 'because it has the disadvantage that it is not independent of the 'standpoint of the observer' with the clock - as we know from experience'. [Presumably he felt that judgements about time and distance may be quite accurate when the observer and clock and ruler are near the event concerned which is thus measured directly re its time and distance but is much less satisfactory when the values to be measured (judged) occurs remote from these measuring devices (and the 'standpoint of the observer').]
114. He then describes his much more practical (and presumably adequate) method of establishing the accuracy of our timings and thus better guarantees the valid simultaneity (or synchrony) by which such timings can apparently only be validly 'judged' - that is between an actual event and a time reading for it - even where the clock (and observer?) is at a (?fixed and ?known) distance remote from the event (one assumes). This method also uses light signals to which methodology we have in a sense been introduced in the prior if inadequate method - without its rationale being properly explained; it is as though we have been thus 'softened up' to its suitability for this purpose without its relevance, necessity, rationale or suitability being really explained. [Ditto in regard to the accuracy of measuring the distance involved in the remote event.] The events concerned, whether near or remote, appear to still be within the one stationary system.
115. The timing for the more adequate method is then described thus: Two observers are at markedly separated points A and B in space (eg A could be at x1, y1, z1 and B at x2, y2, z2 some distance away) and each has an identical clock with which they can both accurately time events at or very near their respective points. But neither can accurately time events at the other's location (without certain assumptions, which are not revealed) nor therefore make mutual comparisons. The two respective times are defined as A time (tA) and B time (tB); no single 'common Time' (for A and B) is yet defined however - as Newton would assume with his definition that absolute time was the same everywhere. Of course, thus far, we have no reason to believe that the two times are not identical nor, therefore, the common time for both. We may accept that they may or may not be the same however. The only way that we can define the common time for A and B, says Einstein, is if we establish, by definition, that the time for light to travel from A to B equals that for it to travel from B to A. This assumes (ie by definition) that both the speed of light and the distance between the two point remains constant. [Does Time not exist in the universe other than in terms of our methods of measuring it? What if the Sun went out and there was no light?] If now we let a beam of light leave point A at its A time and travel to B, it will be reflected from B at the latter's now B time back to A, where it will arrive at a new A time - t'A. In accordance with our definition, the two clocks will be synchronised if:
As such, if either clock synchronises with a 3rd clock at C, then all 3 will be synchronised. All such clocks are assumed to be stationary - relative to one another and are thereby synchronous according to our definition. TIME (ie of an event) itself is so defined in these terms - as: that which is given simultaneously with an event by a stationary clock located at the place of the event when that clock is synchronous with (?another) specified stationary clock (located...?where). [For some reason, Einstein doesn't cover the two aspects here questioned.) He then continues with a further assumption (which he says is in agreement 'with experience' - again not revealed here) - namely, that the quantity
116. That is , that the round trip distance 2AB divided by the time light takes to travel from A to B and back to A is equal to the speed of light in empty space and accepted here as being the universal constant c (as, he has concluded, is shown in Maxwell's equations). Such Time as shown by stationary clocks so synchronised in the stationary system is called 'the Time in the stationary system'. Such a definition thus objectifies what would otherwise be (possibly erroneous) 'judgements' of the simultaneity of clock readings and actual events anywhere in a stationary system. Seemingly, however, the time (if 'seeable') on either distant clock (as viewed from near the other one) would differ slightly from that viewed on the near one by an amount due to the 'lag' in the time taken for the light conveying the information of those clock-hand positions from the one distant location to the other. So, they may have been confirmed to be simultaneous, as described, but would not be seen to be so from either distant location. [One wonders if it might have been possible to define SPACE by like means - only by using what would be assumed to be an already accurately defined Time by which to do so - just as Time has been so defined (albeit later) - by using what one assumes was an already acceptably accurate measures (not judgements) of the distance (space) involved; a bit of a chicken and egg situation? Are we pulling ourselves up by our own bootstraps here?]
117. In any case, his frequent qualification as to the (?internally) stationary status of his described systems (we may have simply assumed that they were so) would seem to imply that he will shortly be contrasting these with 'moving systems' - to be described subsequently - in each case relative to...?). In such moving systems we might expect that the distance between the two systems would now not be known (and would also be changing continuously at a rate possibly also unknown) and hence the clocks couldn't be accurately synchronized and, crucially, one couldn't calculate the time lag involved in receiving the information about the time or distance intervals comprising any distamt event to be measured. Whatever readings were obtained would have to represent the only available valid magnitudes. This would be exactly the same for anyone viewing the event from the other stance. Is this in fact what he reports below ?] Thus, in Section 2, Einstein then addresses the matter of the relativity of both Time and Space - and presumably in situations where the systems concerned are not (just) stationary, but move. ie:
2. On the Relativity of Lengths and Times.
118. This section seems to contain the kernel of his theory and is, he states, based upon the two key principles - of relativity and the constancy of the speed of light. He thus starts by defining each of these carefully. The Principle of Relativity has a number of slightly different definitions in the scientific literature (including more than one by Einstein). On this (early) occasion he defines it thus: 'The laws by which the states of physical systems undergo change are not affected, whether these changes (generally the movement of bodies) be referred to one or the other of two systems of coordinates in uniform translatory motion'. In other words, different environments moving at different uniform motions have no discernible effects upon the operation of all laws of nature operating within or on either, as viewed from either (they themselves being effectively oblivious as to which is moving or moving the faster ot slower); Such laws are oblivious to such motion and work the same everywhere, within their own reference systems, regardless. The Principle of the Constancy of the Speed of Light (one of those laws) is then defined as: 'Any ray of light moves within a stationary system of coordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving source (therein)' (and, it might have been added, whether it is measured from either). As such, the speed of light (c) must always = the distance of light's path (as defined...where?) divided by the duration of the time interval (as defined above) taken by such light. In other words, a pulse of light released within a moving source must travel at the same measured speed within that source's system as it will if measured from another system moving at a different speed. The speed of the first system (the source), if relativly faster than that of the viewing system, would hence provide no additional speed to the light as measured from that latter system - ie using as units of time and space those which prove consistent with the value c - ie equaling the distance the light travels divided by the time taken, in each case where the magnitude of the units concerned for both measures are those as perceived as valid from the (more stationary) viewing platform.
119. He then presents the logic with which he will demonstrate the consequentrelativity of Space and Time - by which means or outcome the foregoing example whereby no extra speed can be added to that of light (c) (by means of an apparently faster reference system when viewed from a slower one) may be accounted for. Such relativity thus allows the sought after compatibility (where light's law is indeed not affected by any attempted boost in its speed) to be appreciated. Thus, whereas Newton began by defining Space and Time as absolutes - without too much justification or background - and built up his laws of motion and his model generally from that basis (associated with concepts of mass and force), Einstein (after providing some preliminary definitions about simultaneity and stationary systems of coordinates which will prove necessary in his subsequent developments) begins instead by defining or presenting as axioms the aforementioned two apparently (but not actually) incompatible principles of nature - which, individually, can be shown to be valid. But applying these equally (as they had always stood) to the usual measures of motion - ie based on unvarying (absolute) measures of time and space - led to difficulties (as reported by other researchers over the previous 20 years or more). This eventually pointed to a need for those latter measures to so adjust that such difficulties would be overcome when the two principles would be seen to be compatible and so find that the speed of light was indeed unaffected by the speed of its source. The magnitudes of the elements of that speed would have thereby appropriately adjusted' to this end. The principle of relativity would be seen to hold true, with the law of light's speed never found to travel at a speed different from its accepted constant value whatever the velocity of its source.
[We may interject here what ....... Tao described as the 1st step in his 5 step sequence in the derivation (by Einstein) of his famous equation of E = m.c^{2} (described below in paragraph .....); that is, "...by using the two postulates of special relativity, determine how space and time coordinates 'transform under changes of reference frame'...". That is, derive firstly, those new transformations which will effect exactly that - based on his (Einstein's) own premises (just as Lorentz did - based on his own (if invalid) ones) - even if Einstein would honour Lorentz by typically referring in future to these as 'Lorentz transformations' - as they were quantitatively identical and derived slightly earlier.
In detailing this first step towards that famous equation (which thus arises out of the special (not the general) theory of relativity, Tao utilizes mathematical logic of a higher form (in order that all 5 steps can proceed in continuity using a consistent methodology. We shall seek to follow Einstein's own form of reasoning - both here as it arises from his 1905 paper and (later) as given in his 1920 version for the layman - at a less technical level. We intend nevertheless to point out Tao's similar descriptions of steps 2 to 5 - where the phrase 'transform under changes of reference frame' (ie by applying the correct transformation equations based on v/c as described below). 120. The adjustment of such magnitudes of time and space (which conveniently both allowed that compatibility and overcame those difficulties) proved to depend on the proposition that (the perceptions of) Space and Time must vary (be relative) according to the speed of the immediate environment in which it was being measured when observed from a (relatively) slower or faster neutral environment (due to an unavoidable delay in receipt of the relevant perceptual information conveyed by light signals). Their relativity turns out to be the only way that such an undeniable compatibility can be realized. His theory thus says in effect that these two principles are in fact compatible (by virtue of this surprising but eventually verified re-conception) and shows us the means by which they may be seen to be so - ie by repealing Newton's absolutes and replacing them with their relative equivalents (which of course had been the unappreciated case all along). By this means, the need for a revised interpretion of one of the principles (that of relativity) was appreciated thus allowing its actual compatabliity with the light principle to be the case. On this basis, Einstein could then proceed to develop his laws of (electrodynamic) motion with their apparently inevitable implications for (the perception of) time and distance, and for mass and energy (to be described later).
121. [Any analysis of the evolution of Einstein's resolution of the problem he is addressing in his famous paper would thus require an explanation of (a) exactly which problem it is he is so addressing (as mentioned above) and (b) the logic of his decision that the application of the two principles as defined, not individually but in combination, could provide a resolution of that problem (and have other significant implications). My own view is that he probably first realised that the Constancy of light's velocity was a fact that had been generally overlooked by most researchers of the later 19th century and that when he analysed what the application of that constancy to various relevant research findings led to then (only secondly) realised that the Principle of relativity would appear to be thereby compromised. As this was apparently not acceptable in contemporary physics (and/or was not consistent with the logic on which he felt that principle was justified), he concluded that something within the set of factors involved in motion generally (as 'overseen', as it were, by the principle of relativity, would have to (be seen to be the case) - something that quite likely related to the factors which Fitzgerald and Lorentz had been manipulating within the relevant Transformation Equations - ie measures of Time and Space - when trying to fit awkward data into the usual mechanical model with its principle of relativity - which assumed constant/absolute time and space. If the previously unrecognised relative values for these (which were revealed or made apparent by holding fast to the value of c in the analysis) resolved the problem, one would of course have then to find a reasonable means by which such values - so revealed) - could come about. What was 'the mechanism'? The valid application of the principle of relativity to the law of light's speed is a statement about what must be the case according to such fundamental principles of nature (apaprently) but this doesn't itself account for how such an outcome actually comes about - only why it should (or even must). These are two different things, one must assume.
122. While both factors comprising velocity were seemingly involved, it seems it was the concept of Time in particular through which Einstein first had his eureka! moment - that is, that its (?perceived) value/magnitude was not absolute and constant but that it somehow varied according to the speed of the immediate environment in which it was being measured when observed from a (relatively) slower or faster environment. [And he somehow came to this realization when considering difficulties in establishing simultaneity - as provoked initially by his imagined tram trip from that clock in Berne's town square - and subsequently (on further reflection) by thought experiments concerning such as lightening strikes as described below) Poincare's article touching on simultaneity may also have been an influence. The same implications on the measured magnitude of Space (distance/length) must have followed soon after; That is, they must both be relative (in the relevant frames of reference) and so vary in value within the new transformation equations by which means these (velocity-relevant) outcomes are verified/confirmed such that the laws concerned fulfil thereby the dictates of the principle of relativity. But by what mechanism could he account for thses seeming findings ?] We may contrast Einstein's rationale in concluding that these two factors must adjust (vary) when one (necessarily) holds to the demands of the constancy of the speed of light (or indeed simply accepts that this is always the case), with Lorentz's rationale that by starting with an acceptance that time and distance vary when bodies move through the assumed ether (as he and Fitzgerald both postulated) and that it is those factors which account for a kind of forced constancy of the speed of light which masks the actual variability of same that Michelson was apparently expecting otherwise (and that of certain other expected electromagnetic effects).]
123. The ultimate acceptance of the validity of that (?theoretical) conclusion regarding the compatibility required, and of the recognition of the relativity of Space and Time on which this was based, (as part of the dictates of the newly perceived principle of relativity), would depend ultimately upon the eventual verification of any predictions such a theory may present. That is, they don't necessarily have to be shown to be logically deduced at the outset although such a prima facia rational would no doubt help in the serious consideration of their likelihood and later testing. Apparently, Einstein's paper did not provoke any immediate interest and was only gradually appreciated - by about 1908 or so. Its predictions were of course fully verified but only later. But his purpose in establishing the relativity of time and space would appear to have been equally important as a means of explaining the reasons why many previous findings ('the difficulties') were so awkward to account for. It would seem to follow that if he could 'prove' that time and space do indeed vary then he could equally assert that the two principles are valid and compatible. But what is egg and what is chicken here?
124. He continued his demonstration (ie theoretical proof) of the relativity of Space and Time (which his theory says must be the case given the existence, validity and actual compatibility of the two postulates described thus:
'Let there be a stationary Rod of length l which lies along the x axis of a stationary system of coordinates (the rod's length measured by a ruler that is also stationary in the same system). This may be called the 'length of the stationary Rod' (ie in the stationary system). Let the Rod then move with velocity v along the x axis of the stationary system in the direction of increasing values of x and then have its length re-measured by two methods: (a) by the ruler held by an observer also traveling with the rod. This may be called 'the length of the moving Rod in the stationary system'; And (b) by the use of two stationary synchronised clocks. In this latter method, the observer ascertains at what spatial points (on axis x) in the stationary system the two ends of the (?moving) Rod are located at a definite time. The distance (space) between them, as measured by the ruler, is a length which may be designated as 'the length of the (moving) Rod in the stationary system'. According to the Principle of Relativity, the length of the stationary Rod (l) should equal the length as measured by method (A) when moving in that stationary system.
125. Einstein then asks 'how do these (equal) lengths compare with the length of the moving Rod as ascertained in (b) by the stationary clocks? Does the latter still equal l ? [Note: for some reason he adds that in method (b) 'we shall determine the length on the basis of our two (now compatible/unified) principles (or does this qualification apply to both methods?).] He replies that 'current (ie Newtonian) kinematics tacitly assumes that the lengths as determined by operations A and B should be precisely equal'. In other words, he adds, a moving rigid body at the time (epoch) t may in geometric respects be perfectly represented by the same body at rest in a definite position. But, on the basis of applying both laws/principles (which we have accepted/defined as being valid), we will find (ie to be explained later seemingly) that these two lengths will in fact not be (?measured/perceived as being) equal. One must note that at this point, Einstein is again simply asserting that this will be so found. He doesn't yet proceed to show this to occur in a practical demonstration or how it must be the case by virtue of some logical argument. It has yet to be proven to us - in theory or practice.]
126. Thus, he continues, we place clocks at the 2 ends of the moving Rod (at A and B) which have been synchronised with clocks of the stationary system (as observed by stationary observers) and so show the true time of the stationary system. But we have moving observers also - observing those two synchronised clocks located at the ends of the moving Rod. We then let a ray of light leave from end A of the moving Rod at time A (as noted in the stationary system) and travel to end B where it is reflected at time B to return to end A, at time A'. All this is as described before except that now the points A and B (the same distance apart) are Moving. Einstein continues: 'Taking into consideration the Constancy of light's velocity (oten ignored in the past), we 'will' find that:
where rAB denotes the length of the moving Rod measured as in the stationary system. That is, (tB - tA) will be greater than (tA' - tB) whereas, if light was assumed to vary according to the speed of its source (being truly additive and subtractive), these two times should (I believe) have been the same - as found in the earlier example. Observers moving with the moving Rod would thus report that the two clocks were not synchronous, while observers in the stationary system would report that they were! The conclusion is thus that we cannot attach any absolute significance to the concept of simultaneity (as Newton would assume) in that two events when viewed from a system of coordinates (ie from a given environment) appear to be simultaneous, appear not to be so when observed from a system which is in motion relative to the one where they do so appear. Time would thus seem to be dependent upon the motion of one's viewing platform and so not be invariably constant whatever may be such motion as always thought to be previously. But is the time as perceived from either platform more correct or valid than from the other? Apparently not. Both are equally valid. Both systems appear to be stationary to those in them (if closed off from other environments and the motion is uniform). And what about Space? [Note: I should relate these ideas to the brief mention above re the symmetry of the relativity of motion - as per the Berne tram example.] Has Einstein thus 'asserted' (earlier) that two apparently equal lengths are not so and (here) that two apparently equal time intervals are also not so? If so, he would presumably next prove, explain or clarify such assertions. [And the conclusions above may well be the outcome of the lag explanation even if this is not referred to here directly.]
127. In Section 3, he does appear to address these latter issues, amongst others, and is entitled:
3. 'Theory of the Transformation of Coordinates (of Space) and of Time) from a Stationary System to another System in Uniform Motion of Translation Relative to the Former.'.
In deriving these required* transformations, he again asks us to imagine two systems of coordinates situated along the same x axis - namely, a stationary one (K) and one that will move along that same axis (k or K'). We may again conveniently imagine these as the stationary railway bed and tracks and the moveable train, respectively. Each system is provided with an identical measuring rule and a number of identical clocks. System (k) is then placed into uniform motion along and parallel with the x axis of the stationary system, as are its rule and clocks and any observers. At any definite time (t) of the stationary system (K), there will thus correspond a definite and parallel position of the axes of the moving system (k). [* Required, and fundamental, as they provide the 'test' by which the validity of the relevant (generalised) principle of relativity can be established and confirmed as the means by which the 'difficulties' (re the problem about light's constant speed) previously confronting this aspect of physics can finally be resolved and overcome.]
128. [Note that 'Translation' appears to be a precise technical term in the present context implying that everything concerned moves together (at a different speed?) in one direction to a new position along just one axis (here X) without rotating.] That he describes this section in terms of a 'theory' seems to imply that, again, he's going to set out his thinking as though his conclusions follow inevitably from his premises but without, at this point, being able to completely justify or verify them with actual evidence; it is still a theory following from given if reasonable premises. From the time of Galileo (ca 1600) upto that of Lorentz (ca 1900), it was accepted that one could describe the distance, time and speed pertaining to the motion of any object in its own immediate environment from the point of view either of that environment or, by the use of appropriate and quite simple transformation equations, from that of any uniformly moving environment outside the former one. Thus a ball thrown down the corridor of a train moving at 100 mph would appear to move at, say, 30 mph within the train itself but actually travel at 100+30 = 130 mph - as seen by those observing from the station outside - through which the train runs.
129. Such a simple addition of velocities is one such transformation equation. Others could be derived to show the difference in distances travelled as viewed from each perspective, etc. Normally, however, no discernible differences in the times taken for the different views of the throw would be expected; the greater distance travelled by the ball as seen from the station would be accounted for in regard to the shorter time taken by its extra speed. If the ball was thrown in the opposite direction, the equation would entail a subtraction - of 100-30 = 70 mph as the apparent resultant speed. If what was 'thrown' was a rather 'magic' ball (or a small bundle of 'slow' light beams) which, miraculously, always travelled at just 30 mph (ie its speed being a constant), the transformation equation would have to show that its speed as viewed from both within the train and from the station somehow always remained at just 30 mph. Clearly, some aspect of the transformation equation would have to be different to allow for this unexpected 'reality' of there now being no additive or subtractive factors in the calculations. The same would apply to any comparisons made in respect of the (other) two components of a body's velocity - distance and time. But should that 'adjustment' not have to have (already) applied also to the motions of all other bodies - necessitating some re-calculations to discover the actual but over-looked results of the additions described above?
130. The above example (which we may call 'A') could be described equally from the point of view of the throwing 'event' occurring instead on the stationary platform and now being observed and measured from the moving train. In this case ('B'), the platform could appear (to those on the train) to be travelling effectively in the opposite direction to the train and, for the analogous situation, the ball would be thrown in the opposite direction. But the same transformation equations would equally apply. Also, 'the event' need not necessarily be conceived as the motion of an object over time and space, but as occurring at a single 'point' (as simply turning over in the thrower's hand, say) in one or other environment; the equations would simply calculate that the ball was travelling at 0 mph in respect of its surrounding carriage but still at 100 mph (the train's velocity) in relation to observers in the station - in example 'A' - and vice versa in 'B'. The transformation equations would be to that extent simpler - whether for some specified distance travelled (by the train alone), the velocity of same or the time it so travelled. It may be pointed out that throughout the development of his theory, Einstein doesn't always make it clear just what 'the event' of concern is or what is the body if any that is moving, and to where. Is it a material point or a rod or anything or...? And does time refer to a moment in time (?now; ?then) or the duration of some 'time-requiring' motion - of a body moving from a to b? And finally, is the observer at the side or behind the distant event ? If at some angle at the side one could presumably apply the appropriate trigonometry and still utilize the correct transformations re times, distances and velocities concerned.
131. In this next demonstration, Einstein sets out to derive just such transformation equations (and in doing so, as dictated by his two principles, will presumably find that the values of the time and distance elements involved will no longer remain absolute and constant (as formerly assumed) but will now have to be relative to/dependent on - the differing speeds of the two systems - and display thereby their theorized variability. [Note that this outcome is as expected by virtue of the logical consequences of accepting his two premises/postulates, including the outcome when holding fast to the unchanging value of c; presumably, this somehow equates to the effect of the inevitable lag in receipt of the information about the times and distances involved in the motion of the body of concern - as measured from (eg) the slower moving reference system. The equations applied would presumably incorporate exactly those quantitative effects - being (the same?) function of the ratio of v to c - which can vary between v/c = 0/c = 0 to v/c = 1 (ie where v = c); v can never exceed c, by definition.] Seemingly also, this derivation provides the same variable values for time and space as Lorentz had calculated to account for Michelson's result (for the Earth's motion of 30k per second through an assumed still ether). Did that result, we may ask, provide Einstein with any clues about the way to calculate/derive the correctly premised equations that validly apply in his own interpretation ? 132. We may recall here that in 1887, Voight had derived very similar transformation equations when working on a different problem in mechanics. His were: x' = x - vt; y' = y/(1/sq rt 1-v2/c2); z' = z/(1/sq rt 1-v2/c2 and t' = t - vx/c2. If the right side of these are multiplied by v, they apparently prove the equivalent of the 'new' transformations of Lorentz, and of Einstein - which may explain why (apparently) Voight's transformations implied something concerning the relativity of simultaneity and thus the dilation of time. In 1908, Minkowski said that the latter's transformations were indeed 1st 'examined' by Voight; Lorentz (1909) soon responded that had he known of Voight's earlier transformations, he could have taken them into his own electrodynamic theory directly rather than having to derive his own. I'm not aware of whether Einstein knew of this early version of the transformations (published where?) but they are surprisingly similar to his own - as derived for his theory of special relativity (and to Lorentz's - derived for an explanation of Michelson's ether-assumed experiment and hence for his own electrodynamic and electron theory arising thereby). Lorentz was later in correspondence with Voight (ca 1910-12?) when they apparently discussed Michelson's 1887 experiments.
[The material in the paragraph beginning 'We now imagine space to be measured....etc' should be covered from about here. Also, I want to be certain that I have covered the implications of his next paragraph (starting 'To any system of values x,y,z,t ...and ...'our task is now to find...! I believe the material in paragraphs below (133 and on) were as per my earlier version and may now have to be replaced by the above - which should then continue from 'We now imagine...,etc (unless I should re-think para 112 in terms now of K and k (ie K') etc) ? In any case, it is the following form of reasoning which we may wish to compare with that of Tao in order to better understand (ultimately) the full derivation of E = m.c^{2}. 133. As mentioned above, Einstein begins with two systems of coordinates (reference systems) in stationary space (which I believe I previously termed systems M (moving) and S (stationary) (or K and k as above) whose x axes coincide (and seem to be considered as a single shared axis), while the other two, y and z of each system, are independent but parallel. Each system has identical measuring rulers and clocks associated with it. While Einstein doesn't specify which system is right or left of the other, we may take system M to be to the right of S initially (as observed before the reader). (If it was otherwise, the kinematics and arithmetic may simply entail more subtractions than additions but the final results would presumably be mathematically identical.) System M is thus seen as moving to the right - away from the stationary system S. One might observe them, instead, from behind.
134. He then states that: 'for any event that occurs in S - at a point position (called here x(s), y(s), z(s) at time t(s) as measured therein by its own rulers and clocks - which conveniently could be at its origin where all 4 coordinates would = 0), there will 'belong' a comparable set of values which pertain to the same event but as perceived from system M (to which Einstein gives rather obscure Greek symbols; for our purposes, they may be called position x(m), y(m), z(m) and time T(m) (or just T - for 'Tau' or time in the moving system). [Note: We could equally assume that 'the event' occurs in M (as in our 'a' example of the moving train above) and the position of same considered similarly in terms of coordinates of M for which a comparable set of coordinates 'belong' in system S (eg 'the station'). However, his derivation is based on an arrangement that equates more to our example 'b' so we shall utilise that one.] He then says 'we wish to discover the system of equations which 'connect' these two sets of values. He thus uses the terms 'belong' and 'connect' without too much explanation as to meanings - in regard to the relationships and significance implied.
135. We thus assume that both terms indicate that we are seeking the set of coordinate values in one system by or in terms of which an event in the other system may be validly described, predicted and measured, one being a particular function of the other. That is, as in the examples above, transformation equations are to be derived which allow one to describe the elements of motion from either point of view (S or M) - and, crucially, which are compatible with or follow from the two stated (mutually-compatible) principles, and so confirm that compatibility. It is a way of saying (and showing) that the motion of the moving system has no net effect on the outcome of any law of motion operating on any body therein when viewed from a differently moving (eg relatively stationary) system (or vice versa) whatever the difference in their velocities; that is, that the proper, generalised principle of relativity is thus shown/confirmed to validly apply as it should - whatever the velocities, body and law concerned. The motion (velocity) of the body's background reference system (as system M) has no effect on its law or outcome since only the appropriate proportion of its speed (depending on its law) is added and subtracted from the total (combined) velocity). The net remainder will thus be the same as was found in the stationary system. Knowing the latter should therefore allow one to calcualte the same (unchanged) net value in the other (moving) one if the appropriate equations are applied - since these effectively incorporate the necessary adjusted values for time and space (ie velocity) from that perspective.
136. To maintain some contact with our concrete examples, we may thus imagine the two systems M and S to be comparable to the moving train and the station, respectively, as described in example 'b' above. We may imagine in addition that the events of concern occur at the same height on the vertical axis (y) - say at eye level - and at the same depth on the horizontal axis (z) in the two systems - the z axis extending into the background away from the station and railway line (ie perpendicular to them), the events occurring at a point mid-way between the rails, say. The other horizontal axis (x) could be an embankment running parallel with and just beyond the tracks. In system M, the value of x(m) on its part of the x axis would be at its origin initially (ie = 0) and the position of a reflecting mirror (introduced below) some constant distance beyond that. In terms of system S, these values of x(m) and that of the mirror (x(m)+ ?) (in system M) will move from the origin of S at the rate of vt. While the two sets of coordinate values would appear to be directly measurable with appropriate measuring devices as eg in the case of the ball, so that the relevant transformation equation would, as always assumed in the past, be based on our usual values of distance, time and thus speed (v), the consideration of a 'body' (as light) - which travels at a constant speed (c) as measured from both systems (ie is observed so to do from either system) will require a modification of those values in the new and more valid (generalised) transformation equations now to be derived. And that constant speed will have consequences for the perceived speeds (v) of any slower and (potentially-)variably moving bodies and/or of the origins (and entire systems) of moving frames of reference as well.
137. Einstein begins his analysis by saying that the equations to be found (by which distance, time and thus velocity will be correctly transformed) will be linear (as time and space are deemed to be homogeneous). But then (for no reason that is immediately apparent) he begins the actual derivation of the new transformation equations by defining a value on the x axis called x' which is equal to x(s) - vt. Now we have noted that the x axis can apply equally to either system; that is, it is common to both - extending in a sense from system S into and through system M as the latter moves increasingly further to the right along it. The value of v pertains to system M while that of the position x(s) and t(s)) pertain to system S. The value of the newly introduced term x' is thus a function of values in both systems. He described earlier an event as occurring at a point in system S - ie at x(s), y(s), and z(s). As it uses the x axis symbol of system S (ie x(s), and not the rather obscure one suggested by Einstein for system M (which we call here x(m)), the derived value x' would itself thus appear to represent essentially a positional value on the x axis in the former system (S). Initially (when t(s) = 0), this position would be nearer the origin of S (than whatever value x(s) itself represented) - unless x(s) was taken as being at its origin (ie x(s)= 0) in which case x'(s) would become increasingly negative as t(s) increased - and system M consequently moved to the right at velocity v. If the origins of systems S and M initially coincided so that x(s) = x(m) when both t(s) and t(m) would then = x'(s) = x(s) - vt. As M moved to the right at vt, so the value of x(s) in terms of system M's coordinates would continue to = x(s) - vt = x'(s). Thus, the value of x(m) that represents in M an event at x(s) in S would equal x(s) - vt; that is, would equal x'(s). The position of the original value of x(s) in terms of M's coordinates would also be increasingly to the left of system M's origin - ie would have negative values. [The foregong provides some ideas about the derived value x'(s) which Einstein defined as above (as = x(s) - vt). But what its real meaning or relevance may be in regard to the derivation of the transformation equations, I am as yet unaware. One would hope that it would reflect the quantities inherent in the inevitable lag in receipt of the information regarding the time and distance (length) of the motion of the event (of some body or point) of concern.]
[Note: See how Tao treats these present aspects in his steps 1 and 2.]
138. In any case, he then points out that any point at rest in system M - as, for example, its origin (ie 'at rest' relative to that moving system but actually 'moving' with it) would also have a set of positional values describable in terms of a set of coordinates of system S - ie as x(s), y(s) and z(s), say. While y(s) and z(s) would remain constant in value as M moves to the right, x(s) of this set would have to increase in value continuously at the rate of vt in order to represent the point in S where the point 'at rest in moving M' is located. But all components of the set of x'(s), y(s) and z(s), on the other hand, would remain constant and fixed in terms of system S, irrespective of passing time t(s); ie they would be independent of time - since x'(s) = x(s) - vt and so always exactly balance the distance that the origin of M moves to the right. Thus, whatever is the value of t(s) as system M moves increasingly to the right over time, the position of x'(s), y(s), z(s) would remain stationary within S. Of course, initially, when t = 0, x'(s) will = x(s) and the situation would not be differentiated from that which would be described for x(s), y(s) and z(s) (with t = 0). At that moment, the event in S could be described in terms of the coordinates of its own system S by either (ie both) of the foregoing sets, ie equally - as they would coincide. But over subsequent times of t(s), these two sets would increasingly differ.
139. One may then ask: 'What, respectively, do they each represent' - especially in regard to the set of coordinates of system M (that 'belong' to one or other of those two of S - ie where 'the event' occurred) that we wish to determine? We should point out that the 'point at rest in system M' which Einstein says we can describe in terms of its position in S, is apparently not where 'the event' he mentioned earlier occurs (at least as far as one understands). However, as far as calculating the equivalent points (ie that mutually 'belong' to each other) in the two systems is concerned and the transformation equations of relevance thereto, it is probably not necessary to describe an actual event. The respective 'points' alone seem to represent the 'events' (which are either stationary or move in their respective systems). As such, it may be the case that 'the point at rest in system M' - as discussed by Einstein above - may be considered in that light. In any case, any conclusions reached regarding an event in system S (as represented in system M) would be identical to those reached in the converse situation. They're symmetrical. Thus, the station could appear to be moving to those on the train (and their own environment the stationary one) given sufficient blinds, one peep hole and a super smooth, quiet train, etc.
140. [We must also presume that Einstein's purpose could not be achieved by calculating his equations in terms of x(s), y(s), z(s) and t(s) alone but that the role of the value x'(s) (kept 'stationary) is somehow necessary. The former set of coordinates may seem the more straight forward and understandable. By 'forcing' these measures to accord with the two principles, one could imagine (having previously 'read around' the subject) deriving a set of equations in which some factor or function incorporated the demands of the constancy of light's speed (c) in ratio with that of the moving system such that a variation and relativity of time and distance would have to emerge in order that the restricted velocity of the moving system accords with the requirements of the new principle re c. However, for whatever reason, Einstein proceeds along his seemingly necessary if more complex path - ie by utilising this stationary/constant position on axis X - ie x'(s) - rather than the ever-increasing x(s) - at least in respect of the (non-event?) point in M that he has just described; (although to what end?). Possibly this derivation provides the quantities involved in the 'lag' aspects mentioned above as much or more than those pertaining more directly to the conclusions based just on the logic of applying/holding to the two postulates per se ?]
141. In any case, he then defines the time (T - as 'Tau' in Greek)) in system M (also called T(m) here) in terms (ie as a function) of the latter 3 coordinates of system S - ie x'(s), y(s) and z(s) - but now including that of time t(s) in that system as well. That is, the time T(m) in M will be a function of the time t(s) in S - somehow calculated in terms of x'(s), y(s) and z(s. Clearly this is a different position to the point x(s), y(s), z(s) and t(s). As the clocks are synchronised in the stationary system in which both x and x' exist, one wonders how the time t(s) in S (of which T is to be shown as a function) could be affected by this one way or the other and thus how it would affect time T in system M. One will shortly be introduced to the inclusion of a reflective mirror (see below) some constant distance from a light source at the origin of system M. The position on x of both the light source at M's origin and that of the mirror at some value of x(m) (in terms of system S) - would continually increase as M moves at velocity v over time t(s), whereas that of x'(s) would, I believe, remain stationary. We apparently must interpret all these positions and movements in terms of Einstein's initial description of an event occurring in system S - at x(s), y(s), z(s) - for which we seek to find the set of coordinates in system M that 'belong' to or 'connect' with same. That is, in terms of our train-station example ('B'), the 'event' at a point in the station occurs at this known position and time and we wish to calculate the position and time of that event (or point?) in terms of the coordinates of the train. This may (as suggested above) entail thinking of the station as (apparently) 'moving' (in the opposite direction) relative to what appears to those on the (otherwise moving) train as their own 'stationary' system. In either case, any point on the x axis of system S would be described as a point on that same axis with respect to system M as one that was increasingly left of that system over time - at the rate of vt.
142. The values of T (as To, T1, T2; see below) in its own moving system M would be as shown on its own moving synchronised clocks. But we wish to calculate their values as seen from system S which we were earlier led to believe would differ from t(s) of system S (or as measured as T within system M itself). [But our original example of those on the station observing the thrown ball on the moving train calculated its speed at 100 mph plus 30 mph = 130 mph. That is, 'the event' was in system M and we calculated its speed in terms of system S's parameters. Einstein focuses on the other direction - describing 'the event' as occuring at a point in system S and thus (one assumes) calculating its equivalent from the point of view of (ie as seen from) system M. But he now says we wish to calculate one of the parameters (time) 'as seen from system S' (as the Station)?! Possibly this light signalling element in his derivation does not represent the moving body event per se (for which transformation equations and relative values of time and space are required or are entailed), but some necessary adjunct needed to calculate (derive) these?]
143. He thus continues: 'Let a ray of light be emitted from the (moving) origin of M at the time To (in M) and have it travel to (stationary) x'(s) along the x axis (presumably in system M) where it is reflected (at time T1) back towards the origin of M (travelling towards it) - where it arrives at time T2. The time at T1 (the middle point on its journey) is then shown by Einstein to be one half of the total time for the return trip. ie:
[Note: the distance from M's moving origin to x'(s) of system S (and effectively stationary re system M) and from it back to that origin (moving towards it) would thus differ. The distance from the origin of system S to x'(s) (and back) would however remain constant, I believe (as x' = vt) - but how that may or may not prove relevant, I'm uncertain. But the 'moving toward' element accounts for the difference in time (or simultaneity) measured/observed/recorded and this equates to our 'lag' concerns !]
144. Despite the remarks made above, transformation calculations should still apply if the moving body concerned was such as a small 'packet' of light albeit moving at a very great speed - or at least would normally in the past have been expected to be the same - by anyone who assumed that the speed of light was not a constant (as the speed of everything else in the universe was similarly taken not to be). One could thus follow Einstein's reasoning and mathematics in which he focuses initially not on the speed of the projectile concerned (ie light) nor on the distance travelled, but on the times taken - as in the equation above re To, T1 and T2. But, by ignoring the constancy of the speed of light, such that the values of c+v and/or c-v would retain their full arithmetic values throughout, the value of T (in system M) would prove to be the same as that of t of system S - as seen from either system. If transformation equations were calculated on this basis, any factor or function applied to the values of time and distance (ie typically called Beta) would presumably equal 1 and thus have no effect. This can be contrasted with the case where the Constancy of light's speed is properly assumed. In this case, it appears that Einstein proceeds on the prior understanding that time in a moving system (as seen from a stationary one) is not equal to that in the stationary system. That is, he isn't going to discover this by doing the calculations; he somehow already 'knows' or suspects this. He also 'knows' that c is indeed a constant so that slower and shorter values of time and distance in M, respectively (as measured from S), will have to allow for this and be expected.
145. That is, where in the false example it was assumed (wrongly) that a real value of c+v would apply and influence the outcome accordingly, in the proper calculation (or experiment) this sum (c+v), as well as (c-v) where applicable, must somehow always equal c - no more and no less - so that the values of v can only be some proportion (from 100% down to 0%) of their originals according to the proportion thay are of the velocity limit of c. For the principle of relativity to continue to hold universally true, time and distance in system M as viewed from system S have to adjust to accommodate that actual inviolable constancy of light's speed (where only 0% of the moving system's velocity can be added to or subtracted from it). Whatever the complexities of his mathematics in showing the derivation of his equations, the 'bottom line' is that the 'sums' entailing any addition (or subtraction) of the velocity of either frame of reference vis a vis the other one to that of the velocity of light must in a sense be neutralised by virtue of an 'adjustment' in the relevant values of perceivable time and space which constiture that velocity. This perceptual adjustment is required by the principle of relativity so that observers in either environment would be unable to differentiate their own state of motion by any difference in that particular law of nature (concerning the constancy of the velocity of light), as they would be so unable by any difference in any other (mechanical) law of nature affecting motion.
146. Thus he continues by saying that this ('half-way') time at T1 can also - "by inserting 'the 'arguments' of the function T' and 'applying the principle of the Constancy of the velocity of light' in the stationary system", be expressed (necessarily in terms now of both time and spatial coordinates (at time = 0 or ?) as:
[That is, the purely 'temporal' statement regarding time in M alone (with its origin's spatial coordinates understood) is converted thereby into one in which both temporal and spatial aspects of system S and the velocity (v) of system M and that of light (c), are now also incorporated. He is thus analysing Time in system M in terms of all relevant measurements involved in the motion of some body (ie here light) from both systems (one also 'in (relative) motion') - thus implying the basis of the transformation equations needed to calculate time T in terms of time t - of which the former is (partly) a function - ie because of the 'demands' of the constancy of light's speed. [Note: One would like to know how to verbalise the above equation. That is, would it be such as: " Time in system M (of an event occurring at a point x(s)=x'(s), y(s)=0, z(s)=0 and at time = t(s)+x'(s)/c-v) equals one half of the sum of the Time at...etc)" - with the coordinates so described being of the relevant system(s).]
[Again, check Tao's version (still in steps 1/2.) at about this stage.]
147. Presumably, he could equally have analysed Space (distance) in system M in terms of similar measurements from both systems and so calculate distance (from origin to x', say) in that moving system in terms of the relevant Distance in system S - it again (or also) varying due to those same 'demands' and the velocity measures to which this applies affected accordingly (with time). In any case, this 'resolution' of his equation would appear to be very fundamental in the overall derivation of his theory. By the phrase 'the arguments of the function T' he appears to mean that he will be calculating time T in system M as a function of - time t in system S - and do so by 'inserting' into his simple equation for times (T1, etc) alone those necessary other parameters of the motion (of a body) - ie distance (origin to x' and back?) and velocity - of system M (ie v) and of the body concerned - light (c). These are seen to have been incorporated within the 4th (time) coordinates of both T1 and T2 - as t+(x'/c-v) and t+[(x'/c-v)+(x'/c+v)], respectively. That is, these represent the extent to which the perception of time (t) (alone?) is altered in system M (as viewed from S) to give its values as T. [Is Distance so perceived not also altered thus? Seemingly 'yes' - as the eventual transformation equations will of course apply equally to that parameter (along the x axis) as well.]
148. Had he sought the function by which Distance in M varied relative to that in S there should be equivalent parameters so incorporated, one imagines. In either case, one would be seeking to discover/calculate just what function T is of t and such other parameters (and/or what function Distance in M (as seen from S) is of some equivalent distance parameter in S and its other relevant parameters. That is, how much - in terms of t, v, c, and distance on x do the values of system M's time (T), space or velocity (as seen from S) have to be altered to 'allow for' the other element mentioned above by Einstein to be considered in expressing any such initial equation (re T1 or some Distance equivalent) in the more comprehensive form needed - viz: the Constancy of light's speed (c)? ('Needed' - to derive the transformation equations.) By seeking the answer to 'how much' - one is in effect seeking to calculate some constant (eg B (beta) by which time t and Distance are appropriately dilated, shortened, or whatever - ie to 'allow for' the constancy of light's speed at any given velocity of a moving system. When something is a function of something else, then, unless that function = 1 presumably, there would be more than one factor of which it is a function. Thus, if something (x) is a function of y (ie which equates to, say, .75 of y (vs 1)), then it must be .25 a function of something else (say z, or z and q or whatever). Thus, while T may well be a function of t, it is also partly determined by (is a function of) such as the velocity of system M, the Distances involved, and probably of c as well(?) - ie parameters that comprise that function beta. [Note: As suggested already, one could proceed as above but without imposing the constraint of that 'other element' to be considered - when expressing the equation in its more comprehensive form. This would imply that the magnitude of M's velocity would be added to that of the velocity of the body (ie light) whose motion is being so assessed. How would that affect the values of the 4th coordinates (ie of time) used in the more comprehensive equation - and its derivatives? The differences with those used above may help define just what those latter entail. This is touched on further below.]
149. Einstein then continues his derivation of this crucial function (beta) (which we may recall is essentially the answer to his initial question 'what is the relationship (and associated equation) between x and t in S and the equivalents (x' and t') in M so that by knowing one set, we can calculate the others) by means of a calculus step - thus: 'If x'(s) be chosen infinitesimally small (does this mean the distance origin to x'(s) approaches zero or ?), the foregoing equation can (he notes) then be expressed as:
[Which must say approximately the same thing - but as applied at the level of a ?point or whatever is implied via the apparent efficiency of calculus. [Sadly, I don't follow this particular step. But it seems possible that his earlier example of deriving the equation W = v+w/1+v.w/c.c is relevant to these more detailed steps.] He continues by 'explaining that: 'As light is always propagated (as an expanding sphere proceeding from the origin of M) along all three axes equally - at the velocity (when viewed from system S) of the square root of (c2 - v2) (ie = c - v, with negatives removed?). [But does it not always travel at a constant c ?] In any case, we will then find that dT/dy and dT/dz both equal 0 - whereas dT/dx' would have a value which is less than 0 by an amount (v/c2-v2)(dT/dt). Referring back to the important assumption that T is a linear function, it then follows (says Einstein) that:
That is, the time T in the Moving system is some unknown function (a) of the time t in the Stationary system - less a (normally tiny) fraction of the speed of light; that is, time T in the Moving system is (typically) very slightly less (ie 'slower' or 'more dilated') than it is seen to be by those in the Stationary system - at least if we take the (previously unknown) function (a) to = 1 which in fact it is later shown conveniently to be. ('Typically' in the sense that at 'normal' velocities, the relevant fraction and thus the difference would be exceedingly small; but if and where v was very large (ie fast), the time T in the system perceived as Moving that fast (by those in the Stationary system) would apparently appear even more dilated (ie slowed) as compared to their own time t. In the above equation, it is assumed also that at the origin of system M, T = 0 when t = 0. This rather important latter feature is described by Einstein in the phrase "...and where for brevity it is assumed that..." ie that the foregoing values of t and T are as thus given.
150. [As we know, measurements pertaining to the elements of motion (ie distance per time equals velocity) must be made with respect to a defined frame of reference, and do so in regard to both elements - space and time. They can't be made in relation to a vacuum - for either. Thus, if there is no point in space that is fixed and absolutely at rest, then there is no point in time that is absolute 'zero' either - only relative space and time are available; all reference points (of both) are thus 'moving' and an appropriate/relevant comparative/reference set must be specified. [One wonders if the basis for the reference point specified for time in Einstein's explanation/derivation of relativity may be partially concealed in his use of the term 'briefly' - and the subsequent comment about time in system M at its origin being assumed to be equal to zero - ie 'when t = 0' - ie without further explanation.] Thus, at the origin of M, x(m), y(m), z(m) and T are all valued at zero (ie 0,0,0,0) as are x, y, z, and t in S - as these will apply to his important equation T1 = To + T2. [We may recall that Einstein did say that he struggled for years ...'until suddenly, it came to me...that 'Time' was the key...'. Was the contents of this 'eureka' moment concealed at all within this term 'briefly', or was this but one aspect of a more extensive insight and quite irrelevant to these matters?]
151. It is, in any case, with the foregoing equation - ie:
that it becomes possible to calculate the system of values in system M (equivalent to x, y, z, t of system S) and called here x(m), y(m), z(m) and T(m) (or just T) in terms of which an event* occurring at x, y, z - at time t - in system S, can be determined. [Note that one could equally derive the coordinate values in system M from those in S. To use Einstein's term - the two sets of values (in S and M) 'belong' to each other.) and so do so by means of the transformation equation thus derived. Such equations incorporate/express the fact that light is (also) propagated at its constant value c in system M (ie just as it is in S) - this being the requirement demanded by the two principles of concern. It can gain nothing in speed from the relative extra speed of its source in M. We thus find that, for a ray of light emitted at the time T = 0 in the direction of increasing values of x(m) (in system M) that:
152. A system of transformation equations, so derived, are utilised when one wishes to measures any aspect of the motion of a body (or of light) occuring in, say, a moving system for which there are already local measurements available but are also wanted from the point of view of another, slower or seemingly 'stationary' system (or vice versa). The available measurements are to be referred instead to the latter reference frame such that the speed of light is (as mentioned above) not shown to benefit in its speed whatsoever by virtue of the attempted additional speed of its moving source (ie in the moving system) and that of anything else (any body or particle) only by some proportion of that system's speed - this added proportion decreasing as the velocity of that system (as seen from a stationary system) increases until, at the speed of light, it (the body concerned) would benefit not at all and so would travel at the same speed as it would in system S. The speed of that frame and anything else moving with(in) it will be proportionally influenced (by virtue of the variations now in the perceived magnitudes of time and space as determinbed by the inevitable time lags in receiving the relevant informnation about them) such that the principle of relativity would continue to hold true. [This, I know, has been analysed elsewhere.]
153. Thus, by using the appropriate transformation equations (which recognise that the speed of light is the same in both systems - gaining no speed advantage from the Moving system - yet according with the principle of relativity), it becomes possible to describe the elements of motion for any event that may occur in one system as perceived from (ie referred to) a differently moving system - and do so in a way that allows the two principles of concern to prove mutually compatible. The velocities concerned must adjust according to the v to c ratio and not accept the values of v alone. The crucial 'function' by which the normal (Newtonian or Galilean) transformation equations are revised thus turns out to be based on a ratio between the relative velocity (v) (of either system to the other) and the velocity of light (c). This was derived previously by Lorentz (for his electrodynamic studies) but on a different basis with suspect premises. Einstein derived them independently - in terms of valid premises. [The simple equation of W = v+w/1 +(v.w/c.c) seems to me to sum it up more succinctly! We may again check Tao's step 1. (near its end) about here and compare.] [* Note: An 'event' would normally entail the motion of a body through space, over time, although one can imagine an event occurring at a 'point' (or particle) if it simply 'turned over' (on its own spot, as it were) or just suddenly 'appeared' or 'exploded' at some point. As such, an event could happen at a given point (x, y, z or wherever) and so avoid having to complicate its transformation by considering a body's motion - from a to b, over time and space etc. But an emmision of light rays, at least, does represent the motion from a to b or whatever of a kind of body and the moving point of an origin may represent another - as applies in the present analysis seemingly.]
154. We continue by considering the equation which gives us the value of x(m) in the Moving system M - ie x(m) = ac(t - v/c2 - v2 times x'). [Note: as elsewhere, the symbol for squaring is provided by the number 2 (not superscripted) throughout.] The ray of light moves, relative to its origin in sytem M (ie at the latter's origin), with a velocity of c - v - when measured in the stationary system S. [Again this seeming inconsistency in light's otherwise Constant speed! Is he saying that 'the ray moves' - ie 'the velocity of light (c) is...' = (c - v)?? How can this be? Can 4 ever = 4 - 2 ?] However, he continues by showing (on what basis?) that the time t in system S may be shown as:
[I hope to see just what this equation, however derived, means and how it is the case.] If then we insert this value of t (the time in system S) into the equation for x(m), we obtain
In an analogous manner we can find that for rays moving along the other two axes:
Thus
By substituting 0 for x' (its value when t = y/sq rt (c2 - v2) [why?], we obtain a qualified set of the transformation equations we are seeking - ie:
where B (beta) = 1/sq rt(1 - v2/c2)
(in each case qualified by (being multiplied by) an unknown function of v, as calculated later). In addition, an additive constant would apparently be required on the right side of each equation - if no assumptions are made as to the initial position of system M and as to the zero point of T in that system. Thus, we have calculated the crucial value of the function as sought. [It is interesting to consider how the foregoing mathematical derivation would proceed if one assumed throughout that the speed of light (c) was additive with the velocity of system M (as was always assumed in the past) such that the eventual value of beta would equal 1 and thus that time T (as seen from S) would be found always to equal that in S itself (ie t) and that relevant distance values would also remain the same as viewed fron either system. By this means, it should be possible to see exactly where the effects of the proper value of c on the derivation ultimately occur and thus 'how it all works'!]
155. It is now necessary, says Einstein, to prove that any ray of light as measured in system M is only propagated at velocity c (as often indicated above) if this is its speed in system S. This would provide the proof that the two principles are indeed compatible - as has been asserted (and would require that time and space necessarily adapt accordingly - from absolutes to 'relatives', although this is not the immediate focus of this proof). Thus - at the time t = T = 0, when the origin of the coordinates of the two systems coincide, let a spherical wave of light be emitted therefrom and be propagated with velocity c in stationary system S. When a point (x, y, z) is just attained by this wave, then:
Transforming this equation with the aid of our recently derived equations of transformation, we attain after a simple calculation:
The wave (ie the light) under consideration is therefore no less a spherical wave (of light) with a velocity equal to c when viewed in or from either system. This shows that the two principles are indeed compatible. That is, by applying the adjustments in the values of time and space inherent in the transformation equations, it is possible to reveal the constancy of light's speed as observed from a stationary environment regardless of it travelling within and sourced from an even faster moving environment. The two principles are compatible because of those suggested 'adjustments' which, we must emphasise, have already (at least in theory) always been thus. Their validity as such, in turn, can only be established by the empirical realisation of relevant predictions in which such adjusted values have been confirmed. [See parag 133 below where another spherical form is considered similarly but this time as a rigid body, not a wave of light.]
156. The transformation equations derived above included the unknown function of velocity. This function turns out to equal unity and hence the final form of the equations is as shown above in blue. It apparently relates to the fact that the direction (sign) of the velocity of the moving system does not affect the dimensions of any Rod thereof set parallel to the y or z axes (ie perpendicular to x - in which direction is the movement concerned). But the kinematics of this proof is even more complex that that of the foregoing and we shall accept it in terms of its conclusions per se.
[We insert here (as part of section 3) some further comments regarding the foregoing transformation equations and how their derivation seems to have fitted into the order of Einstein's thinking ca 1903-05. We may refer to Tao's sequence of the derivation of same also.] 157. If Einstein began by seeking a way to incorporate the motion of all bodies within (meet the demands of) one, more general principle of relativity, he would realise (as just described above) that he had to find (derive) those transformation equations which would allow the speed of light as released on a moving system (frame, vehicle, train) but perceived/recorded/ measured from a more stationary one, to not indicate an apparent reduction in its speed after applying the traditional transformations. For they would remove the full speed of the moving system from the actual final speed of the light as measured from the stationary system (a speed which necessarily must be that at which light always travels - ie at c (only) - so giving an invalid/faulty result (ie too slow by that full amount of the speed of the moving system). For light, therefore, the principle would not have been confirmed thereby and the transformtion equations applied would appear to be invalid for that purpose. If he wished to confirm the validity of the general principle of relativity, he would have to derive an equation which would only remove 'an appropriate amount of the velocity of the moving system' - ie that amount over and above the speed expected for the body concerned (according to that body's relevant law of motion). In the case of light, this would of course equal None whatsoever (ie 0 mph extra) since light only travels as per its law at its constant speed - this being also the maximum possible speed (c). This would apply exactly the same for any other body which (conceivably) might be able to travel that fast, for any additional speed is not possible.
158. If, however, such a body travelled at, say, one tenth of this maximum possible speed, then the transformation equation would, again, have to remove an amount (proportion) of the total speed (of the body plus the moving system) which, as required by the principle of relativity, would show the body concerned to have a net velocity equal to that which its particular law required (for a given force and mass). Thus, effectively, the proportion removed would be from the attempted 'boost' in velocity provided by the moving system and not from the body concerned per se (or would it be from the total speed measured ?) In the case of light (and in this latter case as well), these amounts of the moving systems' speeds that must be removed are proportionally reduced not directly because of a realisation that nothing can exceed the speed of light (or rather the speed at which light (happens to) travel) as that itself is not so much an explanation as an assertion, but more directly (in the sense that it seeks to account for this result) because the information regarding the velocity of the body concerned (whether light or any other), added to that of the moving system, (?usually) entails a 'time lag' (and a comparable distance effect) which together underlay such velocity perceptions - as the body concerned and its moving system moves, for example, away from the observer or measuring device. The faster it moves away, the greater the temporal delay (and associated spatial distortion) when perceiving the relevant information regarding its velocity - ie the distance per time comprising that velocity on which any judgement concerning that body's motion in one reference system from another moving, say, more slowly is inevitably based.
159. When an event (a moving body) takes place on a distant reference system (M) moving away from a system (S) from which the event is to be measured (ie with respect to its velocity (distance per time), the (1) Time taken for the body to move - eg from a to b therein - as noted by those on system S - but by means of a clock in system M near the event - will appear Slowed - to an extent that depends on the distance the clock is from the observer on S and on the velocity (v) by which the two systems are moving apart. Thus, at say distance x' = ...... and v = 1000 mph, the time taken may appear to be say 10.3 minutes rather than 10 minutes exactly, as may be expected if it took place instead very near to where the observer and clock are on S. That longer (slower) time is accounted for by a delay in the time it takes for the information from the clock's time on M to reach the observer on S, while the (2) Distance travelled by the body between a and b as measured by a local measuring rod there will appear Shorter to the observer on S to an extent that depends on the same two variables - the distance (x) and the velocity (v) concerned. In the present example, the distance may appear to be say 4 miles rather than the 4.2 miles it would be (say) if measured near the observer on S. This shorter distance is apparently also accounted for by the fact that the light signal information has a longer distance to travel to the observer. In this case,...(to be completed; note: this explanation would presumably follow a similar logic to that described below (parag 133) in regard to the shortening of the spherical wave).
160. In the case of a slower moving body boosted by a fast-moving system, the perceived velocity of that body so boosted would include a considerable proportion (say 99.x%) of the moving system's nominal velocity added to it, while for a very fast moving body such as light released on a moving system, the recorded measurement of its velocity so 'boosted' would have very little or indeed no (0) % of the moving system's nominal velocity (v) added to it. Thus, in the former case, if a body other than light was forced into motion on a moving vehicle travelling at, say, 1000 mph and its speed as measured from a stationary system was 1500 mph, the application of the traditional transformation equation would subtract 1000 mph from that body's total (1500 mph) speed and so leave a result of 500 mph as the net speed of the body concerned (say a fired bullet) within the vehicle. One might reasonably conclude that the transformation applied was a valid one (if that speed was indeed as expected for the bullet) and thereby verify the associated principle of relativity for such a moving body (ie that the speed of the moving vehicle had no effect on the law governing the expected net speed of that body subjected to a given force).
161. However, as suggested above, it appears that this total speed (of 1500 mph) would in fact not be what was found; rather, it would be something like, say, 1498 or '99 mph and therefore a subtraction of 1000 mph would not result in the expected (and correct) value of 500 mph for the bullet but (as in the case of the light particle) a value that was too low. And so, again, a new, more appropriate transformation would be required - one that, as it turns out, subtracts only a particular proportion of the velocity of the moving vehicle (and thus of the combined bullet plus vehicle speed). This may be, say, 99.3 % of the latter figure (rather than 100 %), so that the total is reduced not to an invalid value of less than the expected net speed of the bullet (500 mph) but, by subtracting only 999 mph (say), the actual total would be reduced to that correct latter value exactly. The proportion subtracted by means of the new transformations is determined on the basis of Einstein's theory and is a particular function* of the proportion that the body's speed (v) is of the maximum possible speed (c) (as this apparently correlates precisely with the effect of the inevitable delay in the receipt of the information regarding the velocity of the body as viewed from the stationary system. In the case of light its velocity (v) being = to (c), only 0 % of the moving vehicle's speed would be available to add to that of the light and hence this amount only should be subtracted (via the appropriate transformation equation) from the recorded velocity of light. [* The correct function is as derived by Einstein and already shown above.]
162. The information (in the form of light signals) regarding (?both time and distance measures for) the slower moving body would travel (back) to the perceiver/measuring device concerning a velocity composed partly of that due to the body itself moving within the moving system (depending on the force propelling it therein) and partly of that due to (say 98% of) the velocity of that moving system, while the information (again in the form of light signals) regarding (the same measures for) the very fast moving light particle would travel (back) to the perceiver/measuring device concerning the maximum possible speed for any body - leaving no scope (0 %) for any added speed due to the velocity of its moving source (whatever its nominal speed). For the laws underlying these bodies' velocities to be verified as not being influenced by any such added speeds of the moving systems concerned (that is, for the principle of relativity, which demands this, to be verified), the same transformation equation should apply validly in all such cases [see Einstein's derivation of his new equations] such that the proportion of the respective moving systems' velocities that are (eg) subtracted (in the cases specified here, namely 98% and 0%) should confirm those bodies' net velocities to be exactly as required by the principle of relativity for their respective, if quite contrasting, laws. [We must note that all velocities discussed here are of course composed of distance per time measures and hence attention is drawn to the variations in these particular variables that, in turn, account for the variations in velocity noted. We may enquire whether Einstein decided to examine what we mean by time and space in such measures after noting such velocity concerns or whether this followed some other precursor (such as....). See parag 111 above also.]
163. Section 4 - Physical Meaning of the Equations Obtained in Respect of Moving Rigid Bodies and Moving Clocks
[Hopefully, this Section will show that the necessarily delayed perceptions involved, being symmetrical, somehow turn out in the end to be (?effectively) real, physical changes in time and space...or...will they still be 'just' perceptual, dependent on the stance of the perceiver/viewer/measurer and the (?inevitable) lag involved in transmitting information between reference systems moving at different velocities ? Thus, what happened to the previously used terms such as 'appears', 'as viewed', 'envisaged', judges', 'would report', 'would be seen as', etc, etc ? Certainly, the equivalence of the perceptions as noted from the 'other' system (be it viewed as the moving or the stationary one, or vice versa (they being of symmetric significance), argues strongly for the 'effective reality' of the associated perceptions; they can't both be 'just appearances' but rather the only available 'realities' - at least from those respective and equally precedented points of view/perceptual stances, etc. Its just that one can't (yet) see how this apparent reality is logically 'forged' out of the examples given nor indeed how it translates into the objective realities that the theory, so based, clearly successfully predicts when it is (?was) based so much on acceptance of such qualifications as 'appears', is viewed as', 'will report', etc, etc. Presumably, in those later verification studies, the findings which thus prove to be confirmatory must themselves also be necessarily 'as viewed', 'seen', 'reported', 'measured', etc because the same restrictions would apply as in the examples utilised to develop the theory. But, do the reports of such studies include such qualifications ? [For example, does the clock at the Equator really tick slower than one at the poles or does it just appear to do so - even if necessarily so ?]
164. Moreover, it appears to be the case that in his 1905 paper, Einstein focuses on the inevitable outcomes of applying his two postulates (as in sections 2 and 3 above) as though that logic alone accounts for and justifies same without any mention of the actual 'lag mechanism' by which this apparently occurs. Why not ?? One might also ask here (for now) whether there may be any basis for explaining the necessary variations in time and space (that must be the case if one wishes to accept that the constancy of thr speed of light must accord with the principle of relativity) not by the so called 'lag phenomenon' desctibed above but instead by the fact that if the speed c is a limit due to an infinite increase in any body's mass with such velocity, would there not be a proportional increase in such mass for slower moving bodies so that for a body of given size subject to a given force, any increase in its velocity attempted by a faster moving reference system would have to cope with a slightly greater mass ? If so, would that not account equally for any reduction in the combined velocity, and do so by means of a more recognized physical mechanism ? Or...?]
165. This present section (4) may thus be relevant to this query in that Einstein addresses here the matter of the ?apparent physical effects on the Space (length) of rigid moving bodies and on Time (rate of ticking of rigid moving clocks) during that movement (as measured by such moving clocks) - in both cases as observed/recorded/measured from a slower-moving or relatively 'non-moving' ('stationary') perspective. Such effects would, presumably, be the manifestation of the relativity of these components of motion as concluded via the arguments presented in the first 3 sections discussed above. The body to be thus examined for this purpose is a rigid sphere of radius R (ie 'seen' as a sphere 'when examined' at rest) which is now moving with system M (and thus at rest relative to it) with their common velocity = v (relative to system S), with its centre coinciding with the origin of system M. In terms of the coordinates - x(m), y(m), and z(m) (of the 3 axes of system M) - with time there (T) not mentioned), the equation of the (physical) surface of the sphere (its shape?) is then given as:
In terms of the coordinates of system S - ie x, y, z - when time t in that system = 0 - that equation becomes:
A rigid body with the form of a sphere when measured in a state of rest (eg by those moving with it in system M), has when viewed/measured in a state of motion (by those in a stationary system), the latter form - of an ellipsoid (of revolution) with the (?apparent) magnitudes of its 3 spatial (length; width; height) dimensions X, Y and Z becoming:
166. Thus, whereas the spatial dimensions Y and Z of the sphere (or of any rigid body of whatever shape) do not appear to be modified by the motion, the X dimension does 'appear shortened' (ie 'has' the form of an ellisoid) - in the ratio of 1 : sq rt(1 - v2/c2). The greater the value of v, therefore, the greater the (?apparent) shortening. [Note: these appear to be unaccounted 'assertions' only, at this point.] So, when v increases to the value of c, all moving objects, as viewed from a 'stationary' system, would (on the same unexplained logic) thus be seen to have shrunk (with respect to the dimension of its motion) to zero and would (?be seen to) become a two dimensional plane. The speed of light at c is taken to be an infinite limit of speed for anything so that v can never (?equal or) be greater than c. Being symmetric, the same (?apparent) results would hold for bodies at rest in the 'stationary' system when viewed from a uniformly Moving one of whatever velocity. With the effects on time also applying (see below), the combined effects would presumably translate into a slower velocity of the now ellipsoidal sphere. But would this slower, differently appearing sphere be due to the perceptual lag effects on time (and a comparable effect of the length dimension) or to such as the increase in mass of the sphere ? [But, it may well be the case that any increase in a body's mass (as the velocity of its moving reference system increases) not benefitting by the total amount of that increase (for a given force) would not apply in the case of a sphere (if of ?little weight) - where interest is focused on its shape mainly ??. Note here that the question as to how perceiving the body on M from S is affected by the time lag (ie the 'comparable effect' of motion on Space referred to above and elsewhere) is seemingly answered above; that is, in the phrase "..would thus be seen to have shrunk...", etc. One must study this further with respect to the associated 'physical' implications of both the shape and velocity of such a body.]
{Note: Is Tao's step 1. (or even 2.) relevant here ?]
167. With respect to the (?physical) effect of motion on Time, we may imagine one of the clocks qualified to mark the time t when at rest in system S, and the time T when at rest relative to system M, to be located at the origin of system M and adjusted to mark the time T there. When that moving clock is viewed from system S, what rate of time passage does it display ? Einstein continues: 'Between the quantities x, t, and T, which refer to the position of the clock (?), 'it is evident that': **
Therefore, T = t sq rt(1 - v2/c2) = t - (1 - sq rt(1 - v2/c2))t from which it follows that the time T marked by the clock (as viewed from the stationary system) is (?seen to be) slowed by 1 - sq rt(1 - v2/c2) seconds per second - ie by 1/2 v2/c2 (where, he points out, magnitudes of fourth and higher order are neglected). The greater the value of v, the greater would be the (?apparent) slowing (dilation) of time - as well (as shown above) the greater the degree of (?apparent) shortening of bodies seen as so moving. A shorter body (or is it a shorter distance of its movement?) moving at a slower time would result in an effect on its velocity - namely.......?.... [Note: Has Einstein's Kinematics provided sufficient 'practical physics and geometry in his analysis or is it weighted mainly in terms of the mathenmtics of such kinematics? That is, do we know or is it relevant from what direction and angle the observer and/or measuring instruments are facing the events of concern ? Are they parallel to them, perpendicular or...what?? In any case, it must follow that such a body as described if timed by a clock as described would be seen to be slowed by virtue of these effects on its moving reference system (M) - or is this not relevant in these examples ?]
[** This wording is a paraphrase of Einstein's (who actually says 'we have, evidently, x = vt...' which, in English, could suggest 'apparently' which is not, I believe, what he intends here; rather, it likely signifies that the outcome is quite evident from the precursors). Thus, the equality is evident in that...while system S is stationary, the position of x, y, z therein is Not (necessarily) - at least with respect to its 'equivalence/connection/belonging to' the position in M of an event which occurs in M - itself moving at velocity v along the X axis (for a distance x?) . Thus, the position on that axis in S of the event in M is represented by x (of system S) = vt. The event thus has two sets of positional values: (i) within M - as x(m), y(m), z(m) - (at time T) and (ii) within S (as x, y, z and t. Hence, calculations entailing the extent of motion in terms of position x, can be substituted in the equations with vt.]
168. [A short time after Einstein's 1905 paper was published, his former physics professor at Zurich (Minkowski) (ca 1908?) suggested that the 3 dimensions of space and the one of time (in the paper's equations) could be conceived as a single 4-dimensional concept of 'space-time'; as they always occur together and are interdependent. It occurred to me that this concept (and those of its component 'parts') is/are definable in terms of motion only. They have been so conceived (by man) in order to help us understand our universe and what 'happens' (ie 'when bodies move/events happen') in it. One could thus imagine time to be simply: 'that which allows two or more events to occur at exactly the same place in space, and space to be that which allows two or more events to occur at exactly the same time'. In each case, such motional events, which manifestly do or can occur (as far as we can perceive), would be impossible without the construction of such concepts (as differing velocities of such motion also require?) - or now, apparently, without that single, mutually-interdependent concept of a dimension of space-time, each element being necessarily defined in terms of the other (?tautologically).
169. In this description, any event/happening must entail motion of a body. Two succeeding events will typically be separated by various other events (sometimes a few and sometimes many) and these provide us with a sense of time passing - which it continues to do throughout our successive lifetimes. Without motion/events, there would seem to be no need for time although a totally 'frozen' universe (or part of same; black hole?) devoid of any motion of anything would presumably always require space in which/where such bodies exist (occupy space) - except possibly at the instant of the big crunch or moments before the big bang - if all matter disappeared into (almost) infinite energy such that no space or time (or space-time) existed (or was needed)?? But then, even an instant implies time and does energy (that final single 'superstring' as we proceed back to the big bang) not occupy at least a miniscule 'dot' of space? Otherwise...oblivion? Or, just 'God'? Why and how is there any energy, matter or force in existence?? Without them, nothing would be...and...would it really matter ? To whom ? The 'non-existence' (of anything) would of course never be known about! It almost seems as if that would be the most expected, most probable status...of...nothing..and for eternity....... (which of course could be a very long 'time'..or a very short 'time'. But...'what 'time'' ?]
170. To return: The foregoing effect of velocity on (?perceived) Time leads, says Einstein, to a 'peculiar consequence' [or, does he mean 'a peculiar apparent consequence' ?]: Time (?apparently) runs more slowly in (non-pendulum) clocks at the Equator than at the Poles! He explains this by showing that such a clock (A) synchronised with another of identical type (B) at the start of a journey which travels some distance, either in a straight or curved line at uniform velocity v, will no longer be synchronous with the stationary one but will lag behind it (as shown earlier) by an amount = 1/2 v2/c2 times t (ie the amount of time it so travels). If clock B is at a (relatively stationary) Pole and A on the Equator, after 24 hours the latter (having travelled about 24,000 miles will thus be (not appear apparently) very slightly slower than that of B, which has remained relatively stationary (albeit both are travelling much faster through space with the Earth around the Sun also). At the speed of light, does Time (appear to) stand still therefore - and be of infinite duration? If so, nothing could happen/transpire presumably. And at absolute stillness, would Time (appear to be) infinitely fast - allowing 'everything' to happen - instantly and simultaneously? Who knows? In any case, does the clock run more slowly at the Equator because of the lag in receipt of the hand position information and if so, by whom or what observer ?
171. Section 5. On the Composition of Velocities
Einstein shows here the impossibility of combining velocities to sum to more than the velocity of light (at c). The relevant proofs that two velocities cannot sum to more than c entail further algebraic manipulations of the variables of velocity (ie distance and time) - in terms of the restrictions on same inherent in his new transformation equations. We may accept that adding together ratios of (shortened) distance to (increased) time (ie in eg high velocities) may well prove to be less than previously believed - when it was not appreciated that such variables were not apparently so altered. [We may note here that the (incorrect) Galilean transformations may be derived from the Lorentzian ones, but only by allowing the speed of light (c) to become infinitely fast which is not possible; if it were, one could have instantaneous information from a distance and thus unvarying measures of time and space - as Newton assumed.] We assume that the proportional effects of v/c affect only the moving system itself and not the body moving within it - so leaving the body's net velocity unchanged as required by the principle of relativity. [Note: Tao's steps 2 to 5 aren't relevant until nearer the end of this entire section on Einstein (ie at about patagraphs .. to ....]
Additional Analyses and Review of Part I on Kinematics. [This descussion is intended to further our understanding of Part I; it is based primarily on....'further considerations'.]
172. As we've noted, Einstein refers in his 1905 paper on the electrodynamics of moving bodies to "...the difficulties which this subject was then encountering...". He then addressed those difficulties (which, inconveniently, he doesn't directly identify) by a careful analysis of the motion of all bodies in nature, over time and space, but now treated as parts of a single category of such motion. This analysis, referred to in physics as 'kinematics', thus focuses on the means of measuring the distances covered and the times taken (ie their velocities) by all bodies in motion and does so in terms of the necessary frames (bodies) of reference (or system of coordinates) and associated rulers and clocks required for such an analysis. Einstein felt that 'insufficient consideration (analysis) of the relationships amongst these core elements of kinematics lay at the root of the difficulties alluded to'. His analysis of these relationships would thus seek thereby to resolve those difficulties, and its associated exposition in his paper would effectively comprise his 'theory of special relativity' (if not then named as such by him); it was, rather, effectively considered as a 'theory of the electrodynamics of moving bodies'.
173. He begins by noting examples in the scientific literature which strongly suggest that the motion of all such bodies (whether in the former categories of mechanics or electrodynamics) can only be measured in relation to some other agreed 'body' (system) - of reference (itself inevitably also moving relatively) since there is no evidence for any reference body that is unmoving (ie still or 'at rest') in any absolute sense - against which the consequent absolute motion of anything might be conveniently assessed. Thus, only relative motion is measurable - with some such reference body always implied (if not always apparent). The examples quoted - one from mechanics and one from electrodynamics (there are doubtless others) suggested to Einstein that 'the same laws and equations of motion that pertain to and are valid for bodies traditionally considered within the sphere of mechanics will also prove valid for those bodies that pertain to electrodynamics (as light) and do so validly in both cases for all frames of reference' (ie whatever their different but uniform velocities may be). That is, they can be treated as part of one consistent sphere of nature (moving bodies) all of which accord with this newly conjectured and more general 'principle of relativity' (as he has so named it) - which he refers to as the 1st postulate of his theory. [We may point out here that while this principle was first formulated by Galileo and later incorporated by Newton within his Principia, they hadn't so named it and that it had been largely overlooked, if taken for granted, during all of the 19th century until Poincare mentioned it once of twice, with that terminolgy, around 1900-1902. It had, however, always referred to the motion of classical mechanics, not electrodynamics.]
174. For this principle to hold true over a single conception of moving bodies, including electrodynamics, it must therefore (as now defined) accommodate the laws of all moving bodies within the same one equation or relationship - including therefore that regarding the constancy of the speed of light (taken here to be effectively a series of discrete 'bodies' - whether as particles or as discrete 'packets' (quanta) of light waves. While this is implicit therefore within his 1st postulate, he must have realised that the restriction implicit within the constancy element of this particular law of motion (or velocity), in contrast to all other such laws concerning moving bodies, would provide a greater challenge to the former's requirements and acceptance since with those more typical laws (of mechanics), extensive potential variability of speed was always inherent (according to the speed of any reference system which was their source) whereas with light, its law denied this very potential flexibility of speed - as perceived on a variably moving frames of reference from a differently moving one. He therefore specifies or points out this particular law as a separate 2nd postulate (a kind if sub-postulate) which must somehow also 'hold' just as all the others do, despite its less typical nature. Its inclusion as a separate basic postulate of his theory is also justified by the fact that it concerns the very source of 'the difficulties' referred to since it alone didn't accord (wasn't compatible) with the existing or traditional principle of relativity with its tacitly assumed absolute values of time and space and its associated transformation equations (as the laws of mechanics appeared to be consistent with) even if this limitation wasn't appreciated by most physicists at that time. By holding to the value of c in whatever circumstances, his theory would adjust the velocity factors underlying that latter principle (as his 1st postulate) so that this seeming incompatibility with the 2nd postulate would be resolved.
175. Those laws of mechanics (which allowed the motion of the bodies concerned to vary according not only to the strength of the forces applied to them - relative to their associated reference system - but to be further boosted or retarded (fully, not just proportionally) according to the exact extra or reduced speed provided by that local body of reference on/in which they were so impelled - when the body concerned and that moving reference system were perceived and measured from some other 'neutral' position (reference system) moving at a different speed - were those laws to which the principle of relativity (as formerly understood) appeared always to apply validly. This stated that such laws performed on their bodies exactly the same, with the same outcomes, no matter at what relative speed their local frames of reference may themselves move. Any such boost in a body's apparent speed (due otherwise only to some force acting on it within/relative to its own refernce system ) could be accounted form exactly, it was always assumed, by applying the appropriate transformation equation (as a subtraction of any such extra speed) to thus reveal the actual net speed of that body relative to its own reference system to be due only to the force applied to it within that system - as the relevant law of its motion would expect/require.
176. Such relevant tranformation equations should serve the fundamental function of manifesting the operation (and confirming the validity) of the application of the traditional principle of relativity. This principle (as formerly understood) essentially reflected the fact that any such extra speed (provided it was uniform) did not affect the validity of the underlying law of motion (due effectively to the operation of the inertial influences so applying). In the case of light however, its underlying law required that any extra speed attempted (as measured from the neutral system concerned) would not be successful in adding (or subtracting) any such speed to or from the total net speed measured (since its law required that this was as fast as anything could move) nor, therefore, for any to be appropriately subtracted or added (say) from it subsequently by an existing transformation equation - to determine the actual (true) net speed of the light which any such 'booster' may have so attempted to alter. What was required was a different principle of relativity which, as would transpire, would recognise that its time and space measures should never have been assumed ro be absolute (but still requiring the law concerned to be unaffected by any attempted boosts or reductions in the speeds of the relevant frames of reference), one whose validity would be confirmed by a new transformation equation which could prove consistent with the single full range of moving bodies (from those whose speeds could be increased (or decreased) almost without limit - to those, as light, whose speeds could not be altered at all. A rather tall order.
177. The added speed of virtually unlimited extent (in the sphere of mechanics) had always been adequately 'handled' (that is, accounted for to leave just the net effects of the forces and masses concerned) by the application of the former transformation equations of Galileo and Newton which were an integral part of the operation/application of the original principle of relativity (and the apparent validity of which their successful application had appeared always to have confirmed; such confirmation being their essential raison d'etre). They would remove the extra speed provided by any such booster frame and reveal the true net speed of the body concerned (as far as could be measured) that its particular law required (after the application of a given force) whatever the speed of such a 'booster' (and therby confirm the validity of the traditional principle of relativity). Such traditional equations would however not validly handle or prove consistent with regard to the motion of any body (such as light) which did not show this unlimited variation (eg increase) in its speed when viewed from slowed system, nor indeed any variation in same, due to differently moving reference frames, as its source. Additions or subtractions of these attempted speed boosts or retardations would simply alter the calculated answer of the properly unalterable (net) velocity of light. Light's speed was, as the 2nd postulate accepted, independent of the state of motion of its emmitting (or 'carrying') body (source). It wouldn't be boosted or reduced by any such increase or decrease in its reference body's speed and so applying the old transformations to its final speed would wrongly subtract an imagined amount of extra or lesser speed from that unchangeable 'total' - leaving as the net speed of the light concerned either a slower or faster, but invalid, value and thus prove inconsistent with a fully generalised principle of relativity which requires the law concerned to hold true.
178. A principle of relativity was thus required that would be verified by means of a new transformation equation which could 'handle' or incorporate both varying and non-varying velocities of bodies and their reference systems and remove all or much (ie a proportion) of the extra speed provided by the reference systems for variably-moving bodies but none of any such attempted increases (say) for such as light - since they were (it was) already at the upper limit of speed for any body. It would thus be the proportion of that upper speed limit (c) at which any body was moving that would determine how much of any attempted increase (say) could be manifested in the total speed measured (and later subtracted to leave that actual net speed proportionally restricted). Einstein was convinced that his new principle of relativity, incorporating the restrictions required by the law of light's fixed speed, should prove valid for all frames of reference whatever the differing speeds of the frame on which any body is placed into whatever degree of motion and that from which it may be measured (recalling that no motion can be meaningfully assessed without an agreed frame (or 'body') of reference). By holding fast the factor in the transformation equation which had to remain constant (ie light's speed), he was able to see how the other factors inherent in the total/combined velocity of the reference frames and bodies concerned (viz: time and space) would in theory have to be affected. And he saw that they would have to become variable, effectively according to the speed still available. They would have to adjust or become 'malleanble' according to the speeds concerned to provide an equation that could 'handle' all types of moving bodies; that is, those which can vary in their speeds (at least up to some limit) and those that can't (go either quicker or slower). By 'handle' is here meant could provide a transformation equation that when applied would confirm the requirements of a generalised principle of relativity for all laws of motion - for all bodies perceived from elswhere.
[Note: Had either or both mechanics and electrodynamics required a concept of absolute rest as the only ultimate reference frame, then his new principle of relativity would not be the case. Instead, assertions concerning their operation would be based on a principle and theory of 'absolutivity' where different laws would have to apply with respect to the motions and frames of reference of mechanics and electrodynamics. And the value of c would have a potential to be be infinitely fast or instantaneous.]
179. The central problem thus focused not only on the derivation of the correct equations (as by 'holding fast' to the restriction inherent in the role of light's speed) that would reflect the actual relationships that must somehow pertain to the motion of all bodies (including light) over time and space - whatever may be the reference bodies to which that motion is referred - but on a logical explanation of how and why the resulting alterations in the time and space elements of this new equation (which appeared to be the case) actually came about. What was the 'mechanism' by which the limitation of velocity (that of light) placed on the relationships that Einstein was seeking to clarify apparently came to 'cause' time and space to vary as so required (by that 'forced' equation/relationship) ? To determine the elements of this relationship and derive the associated equations that validly apply when differently moving frames of reference are so involved, Einstein appreciated that one had to gain a new perspective on just what was involved when perceiving and measuring any body's motion (and thus the speed/time and space implied) in relation to the relevant frames of reference - whatever the extent of variation possible for the body concerned - whether maximal or minimal (including, for light, nil) - depending on what proportion of the available speed possible (c) was being attempted. The greater that proportion, the less would be the amount of any extra speed sought that could be added (boosted). [The physical (?perceptual) basis ('mechanism') for this increasing restriction needs to be made obvious. I would assume that the equations derived by Einstein must incorporate and take account of the effect of such a mechanism on any delay, for example, in the receipt of the information (by any observer or recording system regarding the time and distance measures involved when a body's motion is measured at a distance (especially a varying, remote distance).]
180. To do this, he realised that one must first gain a better understanding of just what is meant by the time and the space (distance) over which any body so moves - firstly as assumed in classical mechanics and then as must be the case, as he eventually saw it, under a generalised principle of relativity. By this means, he seems to have believed that he would gain that understanding of the ultimate 'cause' of the resultant (and previously unsuspected) variations in time and space that appeard to be the case if the constant velocity of light must be 'held to'. [Thus, was the 'cause' simply perceptual; that is, did it depend only on the observer's (or the measuring apparatus's) 'viewpoint - vis a vis the location of the motion concerned ? And if it did, was there not some logical continuity between the two 'explanations' of how and why the relativity of time and space are the case ? Do they (ie as (1) follows from forcing the principle of relativity to accord with the constancy of the velocity of light and (2) as follows from an inevitable delay ('lag') in receipt of the relevant information by way of non-instantaneous light signals) not both imply the same one 'cause' ? If so, one would assume it had to do with the fact that light, as a reliable signal to convey, necessarily indirectly, information of both time and space regarding that motion was not only a constant in its speed but that constant speed was the maximum possible speed of anything, including (almost coincidentally) itself. There had to be some delay, therefore, in the receipt of that information so that there was no other means by which the reality 'out there' could ever be obtained in any other supposedly more accuate or valid form. That was the only available reality. I would assume that this would still be the case had there been no evolution of animal or plant life as well. Are observers or recording instruments crucial or not ? What about clocks and distance-measuring instruments ?]
181. He thus first examines Time in this regard in order that its nature (as measured by clocks) may prove consistent with his main postulate and so relate validly to the other elements involved - all moving bodies measured over space - in relation to a relevant reference system of coordinates. These relationships are represented by the constituents of the transformation equations which serve to confirm the new principle of relativity for the motion of all bodies - this now being recognized as the way nature works rather as it was previously assumed to work - when various anomalies and difficulties became apparent under the limitations of the former principle and its transfomations. When focusing on how the elements of motion (time and space) may have to adjust to allow his two light-relevant postulates to prove mutually compatible, Einstein had a sudden insight or breakthrough regarding a posible 'mechanism' by which this theoretically-concluded outcome might actually transpire - at least with regard to Time. This was, as alluded to, that information about it (ie from a clock) could only be conveyed by signals and that while light was the fastest and most reliable such signal, it did entail a time 'lag' between when any event concerning moving bodies actually occurred (as the clock reading there) and when the necessarily delayed information about that clock reading/time was actually perceived/recorded (elsewhere). That is, that time only exists in terms of our indirect perception of it via signals from clocks. There is no other 'real thing' out there called time that can be measured instantaneously (as might the perception of the heat or weight of a body pressing directly on one's hands, say). It only exists as an agreed position of clock hands relative to the clock's numerals (or a digital equivalent). In a 'frozen' universe, there is no time, whether absolute or relative, as there is no motion to measure in terms of time and space. [Is there any meaningfiul space though ?] He thus begins his analysis of the crucial relationships required for revealing the new transformations mentioned above (which will underlie the new principle of relativity needed to resolve the former 'difficulties') - by examining what we mean by (and thus what 'is') Time.
182. The time at which an event (as a moving body) occurs is typically determined by an observer (or recording instument) noting the position of the hands of a watch or clock located very near to where the event takes place - ie virtually simultaneously. The information concerning the occurrence of both is conveyed by reliable and fast light signals. To the local observer, these may be considered to be virtually simultaneous but the perception of same by a distant observer would not occur until after a time lag which depends on the distance involved (if known).
183. To analyse the crucial relationships, Einstein next considers the motion of a material point (in lieu of a small body, say) as the event of concern, typically over a greater distance than is the case with the motion of a clock hand per se. As mentioned, such motion can only be assessed in relation to an agreed body of reference which in this case may be viewed as a (relatively) 'stationary' system of coordinates in which the initial position of the point may be accurately described (as at od near the origin, say). Its subsequent motion (along the x axis) can then be described as a function of time as noted on a local clock (itself positioned near the origin and point) - providing that motion occurs only locally (over a short distance). It is thus the local event of concern. As such, we could record the time at the start and end of the motion by means of our one local clock and therby determine the velocity of that point's motion, over that local distance. If however, we wish to determine the motion of the point over some interval located much further away, it may be necessary to evaluate (judge) the times of the start and finish of the interval at locations that are remote from the position of our one local clock and observer near the origin of the coordinates. But the simultaneity as between the occurrence of the distant events and the local clock's hand positions would no longer be the case - within acceptable differences of the time lags entailed; the reception of the information from the distant events being much more delayed than that from the local clock. They would be much too out of 'synch', as it were. Different clocks would be needed to establish the times near to such distant events measured along the axis of the stationary frame of reference (no matter how long that may be). And, if occurring on a moving frame...??
184. Einstein suggest that one might judge the times at the distant events by coordinating the hand positions of the local clock (ie near the origin) with light signals sent back from the various distant events. But such a coordination has the disadvantage that "it is not independent of the standpoint of the local observer (ie it would be too dependent on that standpoint at the origin and as a consequence not accurate or valid.). I think this means that allowance is not made for the uncertain lags in time that receipt of such signals would entail. In any case, it may be worth recalling this phrase when we come later to conclude that time (and space) at least 'are dependent' - on something to be later described (ie are relative, or vary, accordingly). This disadvantage and inaccuracy can however be overcome, he notes, by arranging for each event to be timed by their own local clocks and to synchronise the times they display so that there would be only the one synchronous time along all of the common (still stationary) reference body. Such synchronization can be established by means of light signals such that the time for such signals to travel from one clock to another, showing a given time, is equal to that taken for the return journey showing that same time. This procedure assumes that the speed of light is a constant (c) which thus equals the distance the light travels from clock one to clock two and back again divided by the difference between the start and finish times.
185. By this means, one is confident that any event so measured anywhere in the stationary system will be simultaneous with the stationary synchronised clock in its vicinity and thus with any given specified stationary clock anywhere in that system, including the one at the origin with its observer). As such, the timing of all such events will fulfill this definition of time in the stationary system and thus be valid therein - time always entailing this fundamental and necessary element of simultaneity, by definition. [Note: as velocity equals distance per time, any calculation of velocity would require measures of both distance (or length) and time. If it is the case that a new conception of time is accepted as necessary, the proper evaluation of velocity would normally require a new conception of distance as well. But if one already had an acceptable value for this new conception of velocity, it would be possible to produce the new conception of distance (or at least its new value) in terms of the already shown new conception of time - as it would equal velocity divided by time. But do we have this new valid measure of velocity ? - so that we can derive the new value of distance only in terms of the new value of time (as justified previously) rather than having to justify a logical basis for a new conception of distance per se ?? Answer:............]
186. Einstein then seeks to show us that kinematics as normally applied in the past, without the influence of his two new (light-speed relevant) postulates being applied, would expect to find that the magnitudes of space (distance) and time as viewed in a stationary and a moving reference frame respectively - when viewed from a neutral position - would be no different but when analyzed instead by a more broadly-based kinematics (based now on his two 'determining' postulates) would be found to differ. That is, their perceived magnitudes would thus have to prove to be relative to (dependent on) the difference in the speeds of the relevant reference frames concerned - rather than, as tacitly assumed before, being unaffected by same (and thus absolute and unchanging in whatever circumstances). [Note: he has suddenly implied a now moving reference system, as well as the stationary one - without (yet) developing his discussion about synchronised clocks as may also have to apply to this moving system. But see now below at ***, possibly.] Thus, his analysis begins instead by re-stating his two principles or postulates in a more general but precise manner (as the basis for this 'improved' kinematics for analysing the motion of all bodies over time and space). He thus defines (1) his new principle of relativity, as follows:
187. One of these physical systems in nature is that of light which, when it undergoes change in its motion (ie which it does whenever it is emmitted), does so according to its law which states that its subsequent speed (in a vacuum) is always a constant. The new principle thus incorporates this fact already (as reference to differently moving systems would not in its case alone be expected to manifest any such change) but Einstein felt it needed added emphasis (as its very constancy might appear to be inconsistent with the changes of state of almost all other physical systems to which the principle applies as these all show variation rather than constancy in their subsequent speeds whenever so referred - their unchanged net values as promised by the principle only being manifested after the application of the approriate transformations). He therefore states the constancy law of light as a 2nd (sub-)postulate:
He adds that (paraphrased):
188. With this basis, his new kinematics proceeds by showing (within its terms) that the magnitude of space (length) occupied by a rigid rod and the magnitude of the passage of time shown by synchronized clocks, each in the stationary system, will each appear to differ depending on the motion or otherwise of the system on which they are measured - that is, as noted from the same neutral viewpoint. [Again, a now moving system is suddenly introduced.] These mathematical results (must) come about (in theory) when one holds fast to the mathematical implications/requirements of the two postulates - ie as a kind of theoretical prediction or demand. Any subsequent evidence that such results are actually so found in nature would thus strongly support that theoretical (postulated) basis of the associated theory. [And, these would be further supported if one had a reasonable 'mechanism' by which this otherwise only logically deduced outcome actually comes about.]
[***Is this next paragraph where this 'mechanism' (ie the inevitable 'lag' in perceiving information about time and space from a distant/moving system) is finally considered and described in a more explicit manner ? Or, if not made so explicit, is it not at least implied as operating by way of (within) the equations concerned such that the distance that the time signals must travel to convey the required information regarding the velocity (distance per time) of the body concerned is somehow fully represented therein?]:
189. To demonstrate (theoretically) that length and time are not 'seen' to be of unvarying magnitude from wherever viewed and measured (as formerly assumed), Einstein begins by considering a stationary rod of length l as measured by a stationary measuring rod or ruler. [Note: the previous sentence differs from one in which the words 'seen to be of' and 'viewed and' do not occur.] The rod is then caused to move along the x axis of the stationary system, parallel to it. (This stationary system could be (on) the Earth - as a long straight railway line, say, on which a train could, if required, move and so represent an associated 'moving system' thereon.) Indeed, for we then imagine the rod's length to be measured as it is moved on such a moving train by two methods: (a) An observer moves with the rod and the ruler (ie on the same moving train) and measures it directly just as he would if all three were stationary (on the railway tracks, the embankment or on the stationary train carriage), and (b) The observer determines two points along the x axis of the stationary system (eg the train tracks) where the two ends of the rod (immediately above) are located during its journey - at a definite time - as shown on synchronized stationary clocks (arranged as described above) and then (subsequently) measures the distance between these now stationary points with the stationary ruler. This would provide an indirect measure of the length of the (moving) rod - in the stationary system (ie by the use of time).
190. Now, according to the new principle of relativity, the length discovered by the direct method (a) - where all three elements concerned are moving together - that is, the length of the moving rod by the moving ruler and observer (a kind of 'sub-system' moving on the stationary system of coordinates, as the tracks) must equal the length l as determined initially by the stationary ruler measuring the stationary rod (also directly) on the stationary system (tracks). On the other hand, according to that same principle of relativity, combined with the law about light's constant speed (which in any case is implied in the 1st principle), the length as determined by indirect method (b) will not be equal to l - as determined initially on the stationary system. The former kinematics (based on the original principle of relativity and an assumption that light's speed was not necessarily constant), tacitly assumed that the lengths determine by the two methods (a) and (b) would be precisely equal (to each other and both to the length l - as determined initally by the totally stationary direct method). That is, that the moving body (measuring rod or ruler) at time t would be identical (with respect to its (?perceived) length) to that same body at rest in a definite stationary position. But by the new kinematics, the bases of that tacit assumption are questioned - as application of its new principles reveals that basis to be in error. At this point, the unexpected difference in the lengths concerned is simply a theoretical assertion, although it has been based on an analysis entailing the two principles which we are told would, as it does not incorporate the bases of that tacit assumption, produce this outcome. (Again, one's perception (or an instrument's record) of the differing lengths would seem to play a part in that difference.)
191. Having so described/analysed (if not in detail) how the magnitude of space (length) may not be seen as previously assumed when viewed on a moving frame, Einstein then describes the similar situation with respect to the passage of time - but with a little more detail of the kinematic analysis and justification of the conclusions. He begins by imagining that at the ends (A and B) of a rod in the stationary system, clocks are placed which have been synchronized with the clocks in that system; the times they show at any instant would thus correspond with the time shown by any and all other synchronised clocks in that stationary system, including these two. There is also to be a moving observer positioned at each of the rod's ends, with the clocks. (This seems to imply that the rod and its clocks will themselves also be moving (on the moving train 'over' the stationary tracks (system) even though he hasn't specified this for either the clocks or the rod, only for the observers.) A ray of light is then emitted from the A end of the (moving) rod at the time (in the stationary system) called tA (as observed seemingly on its attached clock at A) which is directed to the B end where it is reflected at time tB (as observed at that end on its associated clock) back to A, which it reaches at time t'A (again as observed on its own clock). (Note: he hasn't specified that this light journey between A and B and return occurs in conjunction with the moving rod or the stationary rod, or both. To maintain the analogy with the arrangements described for the assessment of length above, however, we may assume that it is the moving situation that is implied.) Accepting the principle (or law) of the constancy of the speed of light (c), one would, he states, then find that:
where rAB denotes the length of the (only now specified) moving rod - as measured in the stationary system. Any non-moving observers in the stationary system would, says Einstein, declare that the clocks at positions A and B of the (apparently ?non-moving) rod were synchronous while those moving observers would find that the moving clocks attached to the presumably moving rod would not be (viewed as) synchronous. (Do the two observers in each case each observe only their own or do they also observe the other's clock times ?) He then concludes that "we cannot (ie therefore) attach any absolute significance to the concept of simultaneity (since) two events which viewed from a (?stationary) system of coordinates are simultaneous (?synchronous) can no longer be looked upon as simultaneous (?synchronous) when envisioned from a (sub-)system which is in motion relative to that (stationary) system." (What exactly are the events of concern here?)
192. Presumably, this conclusion is consistent with the assumption that the rod observed by the stationary observers was then itself stationary. This would be consistent with Einstein's earlier description of the moving observers measuring the moving rod with their equally moving ruler - in the stationary system - where these three elements were described by him as the 'moving system' (and I called it a moving 'sub-system') - ie within or on the stationary system that he consistently and only refers to (as though he will later introduce a proper moving 'full-system'). In any case, we can assume that the conclusion regarding simultaneity can be generalized to equate to time per se (as the latter is defined as always entailing the simultaneity of the event concernd with the time shown by the position of the local and synchronised clock face hands). Generalising this further, one would assume that as with time, so the concept of length (space) too (as shown above) never represents an absolute magnitude; both concepts have only relative values - the magnitudes of which are as viewed on the moving system from the stationary system and thus entail some lag in receipt of the relevant information re time and space.
193. We might point out here that while the analysis of the relativity of time entails the use of the italicised terms 'appears', 'indications', 'find', 'declare', 'viewed from', 'be looked upon', 'envisioned', etc, they don't for some reason appear in the earlier case of the analysis of the relativity of length. Thus, in the case of time, the phrasing isn't of the unqualified form: '...the two clocks therefore were (or were not) synchronous...' or 'that the two events therefore are or are not simultaneous...' but, rather, is qualified as: 'obsevers would 'find' or 'declare' that...the clocks were or were not simultaneous' and that events which when 'viewed from' a (particular) system of coordinates...are simultaneous, but are no longer 'looked upon' as such (rather than simply 'being so') when 'envisaged' (viewed) from a system which is in motion relative to that (first) system.' This will be further considered below. [See also 'Some Thoughts...(on Seife's article).] It seems that these terms accompany the exposition regarding time more explicitly than in the case of space (distance; length) because his thesis is premised more in terms of time than space, with explanations about the latter subordinated to a kind of analogy with that for time and tending to a less specific or detailed analysis and thus with less qualification with respect to such as 'appears', 'is viewed as', etc which, in any case, tend gradually to be dropped by Einstein (and most of those later commenting on his theory) even when considering thr time aspects as though they can be conveniently forgotten even if sometimes implied. [Thus, see p 48 of Einstein's article where the terms 'appear(s)' and 'as viewed from', etc. are again utilised, but only initially, when describing that next section. As mentioned above, they may however become implied within the equations he derived to this end. This to be further confirmed one way or the other.]
194. But first, he continues by developing 'the theory of the transformation of spatial and temporal coordinates...' - from a stationary to a moving system - ie by deriving the relevant transformation equations which will confirm the validity of his two postulates - with both being mutually consistent and compatible when applying to the motion of all bodies in nature. [See also earlier account of this in Section 3.] The 1st postulate must apply consistently to the motion of all bodies, including light, within one all- encompassing category (as this and the assumption that time and space were always constant appears to be the source of the prior 'difficulties) and to confirm this, it is necessary to derive equations which include the unchanging value of the speed of light (c) which, in turn, 'forces' the values of time (t) and distance (as components of velocity (v) of the bodies concerned) to become malleable (changeable) or at least to make apparent the 'malleability' that had actually always characterized them, if unrealized. [Again, we must ask whether these equtions imply that the perceptions are thus actually affected by virtue of what is actually 'behind' them ?] Applying these new transformations based on the new means of determining the times and distances concerned thus shows that the motion of all bodies can indeed be so treated consistently within the one category (given acceptance of this apparently inevitable 'malleability' of time and space). Empirical evidence would apparently later show that such alterations in time and distance are indeed the case, providing the relevant measurements are made indirectly - that is, by way of light signals conveying the information about the motion concerned. (And, as it happens, such measurements can, it seems, in any case only be made by this method.) And that, therefore, some mechanism must apply that accounts for such effects - ie other than the conclusion arrived at only by way of the logical deduction which preceded this realization.
195. The values of these measurements (by observers or instruments) will vary according to the extent of the difference in the speeds of the frames of reference involved - the one being where the body concerned is put into motion and the other that from which that motion is observed and measured; the greater the difference in their respective speeds, the greater the effects on both time and space perceptions - from 0% difference when the there is no difference in their speeds, through an intermediate 50% difference, say, when the difference is very considerable (and thus space is contacted to half its otherwise expected value and time becomes slowed/dilated to a comparable extent) to a 100% difference - when that difference is the maximum theoretically possible (at the speed of light). Thus, the 'viewpoint' of the observer (or of the 'perceiving and measuring apparatus') is crucial and unavoidable as a factor in the overall logic and reality of the electrodynamics of all moving bodies. Time and length of any moving body (event) on one reference frame can only be measured from the viewpoint of another moving at a different speed, through the intervening (indirect) auspices of time-consuming light signals needed to convey the relevant information about these two components of a body's motion/velocity. It is this latter feature of motion in nature and our perception and measurement of it which connects the two postulates - namely, that the equations which confirm the first arise by holding fast to the inclusion of c (the second postulate) therein and the explanation of why or how the malleability of time and space, as so concluded, proves actually to be the case. The validity of this conclusion is at least suggested by the fact that the perception of these varied measurements applies equally and symmetrically to those observing same from the 'other' point of view; neither have precedence and thus both perceptual stances are equally valid. The communality, compatability and connection of the two postulates are all conveniently represented by the symbol (c) !
A Further Review and Analysis of the above matters.
196. Before 1900, the physics of moving bodies (mechanics) was, according to Einstein, encountering certain 'difficulties' (as we've noted), particularly in regard to electrodynamics - then still considered a part of mechanics. While considering such problems, he became aware of a physical principle pertinent to this general area - namely the 'principle of relativity' (as recently so named by Poincare, I believe, around 1902). The essence of this principle was in fact first described by Galileo in the early 1600s and further elaborated by Newton later that century. It pertained exactly to the matter of the motion of bodies but to a large extent had been so long accepted that it was essentially taken for granted and, until Poincare, little referred to by lorentz's or Einstein's time, I believe. In its traditional form, it stated simply that the laws of mechanics (and thus their measured outcomes) were not affected by any difference in the velocities of the body (or reference frame) in which such laws operated on bodies - from wherever their outcomes may be measured. This appeared to be the very area in which the difficulties had arisen. For certain outcomes did appear to be so affected, namely those concerned with one particular class of 'moving body' - that of the 'packets' (quanta) of electromagnetic waves (including light) - for which there was a fixed, unalterable velocity. Applying the old transformations to outcome involving the motion of light gave faulty (or at least confusing) results.
197. At about the same time (ca 1903), Einstein also became convinced that the motion of such waves differed significantly from that of all other bodies in that their velocity (according to Einstein's analysis of Maxwell's equations and laws of electrodynamics) was a universal constant. Applying the traditional principle of relativity to the more general class of moving bodies - whose motion was, according to their laws, always variable rather than constant - he could see that its central tenet did indeed appear to prove valid when its associated transformation equations were applied to the motion of same (which application and analysis serves to prove/confirm/reveal that apparent validity). In other words, when that portion of such bodies' total speed that was due to the speed of their reference frames (as measured from some neutral viewpoint) was fully accounted for (eg by being subtracted) by applying those equations, the net speed remaining was apparently in complete accord with the laws concerned - whatever may be the difference between the (uniform) speeds of such reference frames - relative to that neutral viewpoint. Such speeds made no apparent difference to the outcomes of the laws of motion concerned - just as expected by the principle of relativity.
198. Thus, in his book 'Relativity - The Special and General Theory' (1920) (to be analysed more thoroughly below), Einstein describes briefly in Section 6 an example of what he referred to as 'the addition of velocities' - of classical mechanics (introduced earlier in the section on the 'Composition of Velocities') . In this, he imagines a train (the Moving reference frame) travelling at a velocity v (say 100 mph) in which a man walks forward at a steady velocity w (say 10 mph) and asks 'at what TOTAL velocity (W) is the man moving relative to the embankment or station platform (the Stationary reference frame) ? While he indicates that the answer would appear to be equal to v + w = W (ie to W = 110 mph), he notes that this answer of classical mechanics will actually prove to be wrong (as later explained). In addition, he seems to feel that it may be useful (possibly to that later explanation) to give this early answer in terms of the components of velocity - distance per time - rather than in terms of the resultant velocity per se. Thus, he notes that if the man stands 'still' (ie relative to the interior of the train) on the moving train for a given time - eg one second, say, he would yet move forward, relative to the station or embankment, through a distance (in miles or feet) equal numerically to the velocity of the train (ie 100 mph/60x60). After walking at velocity w, he would traverse an additional distance w relative to both the train and to the embankment, numerically equal (at 10 mph/60x60) to that walking velocity w. He therefore covers the total (now distance) W which equals the sum of the component distances: v + w, relative to the embankment, in the one second of time being considered. And, as with the velocity equivalents of these classically derived distance and time components, Einstein will later show that they are necessarily also incorrect. These same conclusions would apply equally to the motion of any other tangible body (as a thrown ball of a fired bullet, say) within the moving train.
199. In all such cases, the requirements of the classical principle of relativity would appear to have been fully met in that the motion/speed of the train would have no apparent effect upon the net velocity (within the train carriage) of the motions of the bodies concerned (as their laws would expect) - after the velocity of the train was subtracted from the total velocities calculated - as per the Galilean transformation. This same conclusion would follow with respect to the values of the velocity components (times and distances) concerned. It is the application of such transformations that confirm the apparent validity of the (classical) principle of relativity (that the laws are not affected by the motion of the train, or whatever may be the moving reference frame. [See also here the paragraphs below where the addition of velocities is derived similarly from the transformations - both for the classical case and (as in next paragraph) the light/relativity case.]
200. However, when applied to the motion of light therein, these usual transformations (eg subtractions) would not yield the true and valid net result required by its rather unique law (of always manifesting an unchanged, constant value) - again as required by the principle of relativity. If it somehow had, it would of course have further verified that (traditional) principle of relativity, but this would not be the case, with those particular transformations (subtractions). Both Maxwell's laws and the findings of de Sitter are quoted by Einstein, as well as those by Lorentz, to indicate that the velocity of light is in fact not affected (eg boosted) by the velocity of the moving body emitting the light as measured from elsewhere - ie there isn't the usaul addition of velocities with it. Thus, the example given for the addition of velocities above (the man walking along the train carriage) would conflict with this conclusion about light since the (unchanged) velocity of light within the moving train carriage as measured from elsewhere would, after accounting for the velocity of the train by means of the usual transformation (subtraction), have to equal c - v, a value less than that at which the law and the principle of relativity states light's velocity must always be; that is, the same value whether measured in a stationary train or a moving one. The requirements of the principle of relativity as it stood would not be confirmed by applying the usual transformations. Either it (with its usual transformations) or the law about light was incorrect; they weren't compatible as they stood. Einstein would focus on the transformations and the implications they had concerning the components of the velocities involved in their application.
201. Reasoning that the principle of relativity in its essence was that general and fundamental a principle of nature that it should apply validly to (accord with) the motion of all bodies, including light (the law concerning the fixed velocity of which was also well founded), Einstein decided that he must examine more carefully all elements of motion which entered into the seeming incompatibility. He examined the components of velocity as they pertained to the traditional transformation equations and noted that these (time and space) had always been tacitly assumed to never vary whatever the circumstances of the motions of the bodies concerned. Was this assumption consistent with holding fast to the two 'laws' (of light and relativity) ? He effectively asked himself 'what, in theory, would have to be the case (the form of the equations entailing the required relationship) for this requirement to be the outcome?'. He thus analyzed the equations that would arise when the net value of the speed of light was 'held to' its one constant value (after any relevant subtractions) - despite having different velocities for the relevant reference frames. For that would be as the principle required - of allowing light's law, as all laws, to always hold true to itself - in its case, in relation to either the moving or the stationary reference frame.
202. He found that to provide this result, it was necessary to accept instead that the values for the components of the velocities of the moving bodies concerned and their reference frames - ie time and space - must themselves be the variable factors - so that the velocity of this particular body (light) could remain, as its law and the principle of relativity requires, unchanged. The values of (a body's) velocity components (as entered into the traditional transformations) had previously always been assumed to remain unchanged in all circumstances. And at the usual velocities concerned, this seems to have been approximately justified and correct. But, at much faster velocities, some doubt would arise regarding this long unquestioned assumption.
203. This incompatability can be shown (says Einstein) to be due to the tacit assumptions on which classical mechanics and the associated Galilean transformations had previously been based; namely, that the time and space values comprising the velocity of any moving body are not dependent on the motion of the relevant bodies of reference. They were assumed to always be the same regardless. He seems to have examined this basis of the traditional transformations, and come to this conclusion, after first considering what would have to be the condition of the time and space variables in the transformations if one 'held fast' to the requirements of both the principle of relativity and the law of light's speed. That is, when the amount of the 'extra' velocity of the moving reference frame that was subtracted from the velocity of light measured in that frame (still c) necessarily left a value for light's speed (ie was'held to') that indeed remained as c. To effect this, the time and space variables in such adjusted transformations would have to prove variable (not constant and never varying as thought before) according to what proportion the velocity of the moving frame (which effectively 'sought' to provide a boost to the total velocity of the body moving within that frame) was of the maximum possible velocity of any body (including light). This meant that the boost given to any tangible body (ie other than light) would necessarily have to be some proportion less than the extra speed attemped by that moving frame. For light itself (or anything that might travel at of near that maximum possible speed), this proportion would have to be total (ie 100%) or virtually so.
204. The conclusion regarding the addition of velocities concerning the man walking on the moving train was based on the times and distances underlying the velocities involved relative to both the train and the station. The net velocity of the man relative to the station due to the walking itself (as expected by the classical principle of relativity) was obtained by removing that part of his total velocity that was due to the velocity of the train - by applying the Galilean transformation. This net value appeared to confirm that principle's requirement - that the velocity of the train did not affect that net value; the distance he covered in a given time (ie his velocity) within the moving train (for his particular effort or applied force) was no different than what it would have been had the train not been moving. That is, the law of motion applying for the application of that force was, as required by the principle of relativity, not affected by the motion/velocity of the train when the man's net velocity (distance per time) was measured from the stationary platform - ie after subtracting the full velocity of the train by means of the traditional Galilean transformations .
205. [As alluded to above]...Einstein points out that the conclusions concerning this addition of velocities (ie for classical mechanics) can also be deduced from the Galilean transformation itself. For this, he replaces the walking man with a point moving along a moving reference system (K') where the distance (x') covered = w.t' (that is, equals the velocity (w) times the time (t')taken on that system. He asks what values x' and t' have when expressed in terms of x and t (on the stationary system (K). This is provided by the Galilean transformation equations 1 and 4 from which 'it follows' that x = (v+w)t, where x is the equivalent of the distance walked by the man relative to the embankment or station. His motion or velocity (W) then = v+w. [Note: the phrase 'it follows' actually masks other steps in the logic of this important derivation or proof which we should later clarify.]
206. But when the law concerned was that pertaining to the motion of a light particle measured on such a moving train, subtraction of the train's full speed would not in that case leave that law concerned unaffected - as the principle of relativity required - but would leave a value that was less than should be the case. The law and the principle were thus not compatible when the velocity values concerned were subjected to the usual Galilean transformations. In other words, the principle of relativity in its classical form was not confirmed thereby with respect to the law of light's constant speed, as it apparently was in regard to the mechanical law regarding the moving man (or any other tangible body). In this regard, Einstein asks an apparently related question: 'How might we find the place and time (?velocity) of an(y) event (the movement of any body, including light) in relation to the train when we know only these values with respect to the station such that the principle of relativity is not contradicted but indeed confirmed by the appropriate transformations - that is, where the laws of the motions of all bodies meet its requirements ?
207. Such transformations would have to incorporate a relationship between the components of velocity (distance per time) of any and all moving bodies, including light, relative to both moving and stationary reference frames which would thereby confirm the validity of the more generaqlised principle of relativity with respect to all such moving bodies. Can we, asks Einstein, conceive of such a relationship ? That is, how to determine the unknown values of distance per time (for the velocities concerned in the moving frame) as functions of the values in the stationary frame that we do have ? What is their relationship (ie the function that one set of values are of the other). For they are some variation of those known values according to the factors which cause or require them to vary. By knowing this relationship, we would be able to determine the velocity of any body with respect to the moving frame (eg the train) if knowing only that with respect to the stationary frame (the station or embankment) that proved consistent with the generalised principle of relativity; eg that would find that the velocity of light is concluded to remain at the (unboostable) value c whether in relation to the train or the station when the 'proportional' transformations are applied (and that the velocity of all other bodies are concluded to vary only accordingly, depending on what proportion their velocities are of that maximum possible velocity of c.
208. For an easier comparison with these present developments, we may also show the equation W = v+w (as above) as W = v+w/1 (which still = v+w). For when we consider this velocity W instead from the point of view of the generalised principle of relativity of the theory of relativity (rather than that of classical mechanics with Galilean equations), we find that expressing the equation x' = wt' of the moving frame in terms of x and t of the stationary in terms of the generalised transformations (rather than the Galilean ones), the above equation for the velocity W 'then becomes' = v+w/1+(v.w/c2). [Note: we should seek to clarify just how this equation 'then becomes' the form thus derived.] The velocity W thus becomes the product of v.w divided not by 1 (to give v.w itself) but by a value larger than 1 (and does so to the extent which the product of the two components of the total velocity of the body on the moving platform (v.w) is a proportion of the product c.c (ie c squared). [Expand on the rationale for the necessity of determining what proportion the v x w product is of this latter product in particular.] The greater is v.w, the smaller is the velocity W (with consequent variations in the values of its constituent components of distance (shorter) and time (longer)). [Or is the velocity consequent upon these constituent values' prior determination ?] So, asks Einstein, which example of the value of W better accords with our actual experience of nature ? He suggests that the experiments of Fizeau supports the latter version. (see p 39-41.)
209. By means of these new transformations, the motion of all bodies could be dealt with thereby within 'the same one conception of the principle of relativity' - with light and its maximum and unchangeable velocity simply falling at the far extreme of the motion of all (slower) moving bodies - which would continue to display some variability (changeability) of their speeds (as their laws allow) whenever give a boost (say) by a faster moving reference frame (as the train). Subtraction of the now correct proportion of that attempted extra speed (depending on that v.w to c.c ratio) would then leave a net value that was consistent with the laws of motion of the bodies concerned, be they tangible or not (ie light and other electromagnetic wave bodies/packets).
[Note: It is at about this point that I should enquire whether the effect on the variable values of time and space that are provided in the new transformations (as based on holding to the value of c in the equations and the consequent algebraic manipulations), are equally the result of the lag in the information according to the extent of the difference in the velocity of the different reference systems concerned. Is this effect 'built into' or implied within the equations ?? And thereby account for the variations arising ? But the apparent variations in time and space that are concluded on the basis of holding fast to the value of c aren't at all explained by such a conclusion; they are on that basis simply acts of faith. How do we actually account for them 'at the coal face' of empiricism ? What else, if anything, is actually going on there ?]
210. He had thus concluded that the traditional principle of relativity and its associated transformations must in the past have tacitly assumed (for centuries) that the values of time and space in velocity measures for all normal bodies in motion and their reference frames (ie the values so subtracted) were always the same (unvarying). This was helped by virtue of the fact that any such variations at most normal speeds (if they were indeed the case) would be so minute that they would be both unmeasurable (then) and also of little or no practical relevance in any case. If such variations were there, they would no doubt have been completely overlooked. It would take the anomalies arising when the velocities concerned were at or near that of light to sufficiently maximise and make apparent these theorized unsuspected effects on the magnitudes of time and space. But how could Einstein account for such theoretical effects actually arising in nature itself ? He focused his mind on how the values of time and space - as basic elements of the motion (velocity) of all bodies - were determined.
211. He eventually concluded (around 1904/05) that they can only take on their possible values by indirect means of measurement - at least when the motion of the bodies concerned are being observed and measured from a viewpoint moving at a different speed to that on which they are forced into motion. That is, the information about such motion - ie as and when it actually happens - can not be instantaneously conveyed to and received by any observer/measurer at some other, often differently-moving, viewpoint. Rather, it would be received there and appreciated only after some time lag - time for the signal conveying that information to travel the distance involved - a distance which may not always be known. And if it is unknown, then the actual time at which any event may have occurred can not itself be known - only the information received about it could be treated as validly representing the event itself. [relevant about here also is the fact that all motion is relative such that the perceptions of motion on one reference frame (B) from another moving differently (A) would be exactly comparable with and equally valid as those of (A) from (B).]
212. Einstein's 'eureka' moment (when he could finally see a way by which his foregoing conclusions could, if still in theory, actually come about) came when, in about May 1905, he imagined zooming away at the speed of light on a tram from the main square in Berne when the clock there showed 11 pm (say). At that speed, the light reflected off the clock face would indicate the position of the clock's hands as remaining at 11 o'clock. For, as the hands actually moved towards 11.01, say, the light information conveying that (and later positions) would never reach him in his speeding tram; it would continue to show him it was still 11 pm, while his own watch, travelling with him, would show it continuing to approach 11.01, 11.02, etc. In this thought experiment, Einstein would know that the clock in Berne would have continued to tell the correct time there, but this would not be the case for an observer on the tram who had not been aware of its origin, time of departure or immense speed (nor would any observer back at Berne be able to conclude anything about the time he might see on Einstein's watch other than that it continued to show 11 o'clock there in the tram). If the tram's speed was only half or less than that of light, then the slowing (actually 'stoppage') of the apparent time back at Berne wouldn't be so complete (as in the first example) but would still be slowed by some proportional extent. There would be an unavoidable time lag in receiving the information as to the time concerned. [Interestingly, this lag becomes effectively infinite when the difference in speeds between the two reference frames equals that of light.]
213. The idea that the existence of an upper limit on the velocity of anything (ie at c) somehow provides 'restriction' or 'pressure' on the full increase in velocity attempted for slower-moving bodies when moved by their own frames of reference (thereby so limiting those increases to some proportion of that possible upper limit) must be another way of saying that by holding fast the extent of the velocity of light to its own limit (at c) when determining/analysing possible new transformation equations - where its reference frame similarly seeks to boost it - 'pressurises' or limits the values of time and space to some proportion of their usual values ? If it is, we still require the 'mechanism' by which such 'upper limit pressures' effectively 'back-up' to so effect this action. Einstein said it came about due to the above-mentioned unavoidable time lag in perceiving the information concerned - which was symmetric in regard to which reference frame was observer and which the observed. Does this really equate to 'pressure'? No! The concept of 'pressuring' used here and elsewhere is not really the case; the action is more akin to being the result of what's left or available - from a decreasing fund. At the speed of light, there is nothing left in the fund for a faster moving frame to contribute; no further 'scope' for contributing additional velocity.
214. Thus, the true net speed of a bullet fired at 200 mph (rather than a pulse of light at c) on a train travelling at say 100 mph relative to a viewing platform wouldn't be 200 + 100 = 300 mph from which the full 100 mph is subtracted to leave the bullet's unaffected speed at 200 mph (as the traditional principle of relativity might predict), but rather only 99 mph (say) would be subtracted from a total of only 299 mph, thereby leaving the correct value of 200 mph for the bullet. That is, the lesser value of W = v+w/1 +(v.w/c.c) is applied (rather than the full value of v+w/1 .] The reduced 'scope' for the moving frame to add further velocity isn't however directly due to that ultimate 'barrier' of an upper speed limit but to the increasing lag in receiving the information regarding the total velocity of the body concerned. No doubt there is some mathematical connection between these two correlated 'explanations'.
215. If the measures of space (length or distance) were somehow affected comparably to measures of time, then Einstein would have found the complete 'mechanism' (albeit still theoretical) to account for his conclusions about the proportional variations in these components of velocity that must occur if the speed of light always remains at its one constant value (as inevitably perceived!) despite any attempted 'boosts' in its speed by means of being released on what is seen as a faster frame of reference (when measured from a slower moving one) and this increase in velocity then wrongly subtracted to confirm the unchanged value of light (at c). [And other slower moving bodies wouldn't gain as much of a boost thereby either, as formerly assumed, but a proportional one only - the proportion becoming less the greater the boost attempted - due again to the greater time lags involved in perceiving the relevant information. He would later (after publishing his theory) exemplify the time aspect of this 'mechanism' in more graphic ways - as with his analysis of simultaneity using lightening strikes, such simultaneity proving an inevitable/fundamental element in time preception.] All he needed now was to find some empirical evidence that this was indeed how nature (vs theory) actually worked.
Appearance and Reality
216. Sometime after writing most of the foregoing, I bought and read a small book entitled 'My Einstein'. It was comprised of a number of articles by different authors which provide interesting material on their views, opinions and appreciation of Einstein based on either personal contact or, more often, through their familiarity with his work and/or with those who knew him more directly. In most cases, these ideas are associated with various different aspects of his theories (of particular interest to the respective authors) which they usefully elaborate without too much technical detail but which may provide the layman (as myself) with some new slant on various fundamental topics in the physics of motion and relativity. I hope eventually to cover (analyse) each paper in turn but in this first instance, I want to address my thoughts to certain ideas expressed in the paper by Charles Seife, a writer for Science magazine, entitled 'The True and the Absurd' as it provides a convenient structure to help me better formulate certain difficulties I'm having in understanding aspects of the theory (as touched on above regarding 'appears', 'is viewed as', 'apparently', etc). It would seem to fit appropriately about here - near the end of the present sections analysing and reviewing the Kinematics of Einstein's 1905 paper.
217. In this article, Seife first addresses the background to Einstein's conclusions about the relativity of time and space. He begins by describing his early thought experiment (when just 16) of travelling with a beam of light at its speed, but doesn't elaborate upon this other than to imply that this would one day provide Einstein with part of the basis of his exposure of a flaw in Newton's laws of physics. He then describes how the Danish astronomer Romer had explained a discrepancy in the apparent position of one of the moons of Jupiter in terms of the varying distance that Jupiter and its moons are from the Earth. Knowing these varying distances allowed him to calculate the speed of light - to be about 300,000 kilometres per second. The time the light takes to reach the Earth from Jupiter or from its moons thus varies slightly according to those varying distances (at least this would be the case if that finite speed was also constant, something that was established but only much later). Thus, says Seife, with those varying distances and times, "...it's as if someone were playing with your clock, making it run faster and slower...". A key phrase here would seem to be 'as if'; one's local clock doesn't actually vary in the structure or function of its running speed (I believe) nor, therefore, would the time that it actually measures vary - at least on that basis.
218. In his youthful thought experiment, says Seife, "Einstein dreamed he was zooming away from Earth at near-light speed, leaving he native Germany far behind. He imagined looking back at the receding planet and spotting a clock ticking away." Depending on his increasing speed, the light from the clock (and the time it apparently thus displayed) would take more and more time to reach him. It would again, says Seife, "...be as if the clock were ticking more slowly. And if he actually reached the speed of light, it would seem as if the clock's hands (and thus the time they represented) had stopped moving entirely". (We may reasonably suggest that, as above, they hadn't actually stopped but, again, only appeared to do so to such a rapid traveller.) It was, said Seife, "...a consequence of Newtonian laws of motion and the finite speed of light". That is, it (the light) and the information it transmitted wasn't infinitely fast or instantaneous. We may thus assume that the consequence was the apparent slowing or stopping of the clock and of the time it measures, not any actual slowing or alteration in the passage of time at the clock-face. So, by accepting those two 'laws', "...you would, said Seife, have to concede that a clock would seem to freeze - if you moved away from it at the speed of light" and at least seem to slow considerably if one's speed was somewhat less than that.
219. [Note that this admission of such alterations being only apparent or 'as perceived' (vs actually being different) would nevertheless, I believe, be interpreted eventually as being 'effectively actual' since there was no other way for the information about it being transmitted any faster (ie instantaneously) than by light but that necessarily taking this 'lag' into account somehow still doesn't negate the apparent reality of this altered perception as being the (necessarily; inevitably) accepted and only case. The rest of this account will focus on how to explain or accept this apparent fact. It seems that the universal symmetry as between any two reference systems involved in the occurrence and the measurement of moving bodies means that any appearance of altered magnitudes of time or length (and thus velocity) so arising must be equally valid from either (symmetric) view point. By some logic, this undeniable symmetry somehow translates into such appearances equating to (?effective) 'reality' - as prove by subsequent predictions in this regard and as reflected in the general absence in explanations of relativity of the terms 'appears' or 'as perceived', etc.
220. Thus, Einstein would eventually show that by 'holding fast' to the implications of the law of the constancy of the speed of light, it must follow that time and space are actually variable - depending on the difference in the velocities of the coordinate systems involved in the measurement of any moving body. This difference is the same thing that accounts for the 'apparent' slowing of the passage of time and the 'apparent' contraction of the length of any moving body (as describe for time at least by Einsten originally in terms of his flying away from the Berne clock). And the necessary invariablilty of ('holding fast to') the speed of light (plus the symmetry referred to above) would seem to be the ultimate reason(s) why such 'only apparent' effects must, nevertheless, be accepted as being (?in effect) not just 'apparent' but (albeit gradually and mysteriously during such explanations) somehow 'actual' !?! Also, the addition of velocities must (proportionally) 'fit into' the finite limitation of the possible total velocity possible (ie c). Time and space thus adjust in accord with that necessary and particular proportionality and the lag in information's transmission (and that symmetry factor!) somehow provides a (?convenient) 'explanation' of this. Note: I now accept that the physical explanation of this isn't the limitations of that upper limit of velocity per seibut that this is but a correlate of the actual physical 'mechanism' concerned, namely the 'lag' in the light signal information regarding the time and distance comprising the velocity of the body being thus observed/measured from 'afar'.]
221. It was in any case my understanding that it wasn't the 'youthful' (ie 16 year old) Einstein who imagined looking back at a clock in his native Germany, when visualizing his trip beside a light beam but, rather, when he was about 25 and, in fact, in Switzerland - on his way home from discussing related matters with his colleague Michel Besso when (as I mention in the preceding paragraph) he noticed the large clock in Berne town square. But otherwise the logic remains the same: the clock's time would appear to him to be frozen at the time it showed when his imagined light speed journey began. A stationery observer back in the town square would similarly see the time (if he could) on Einstein's watch frozen at that same time, while both would see their own local clocks or watches show that the time had continued to progress at its usual rate, which of course it had. Had the information about the times at the distant venues (as opposed to the actual times there) been conveyable instantaneously rather than 'only' at the speed of light, it would of course have agreed with the actual times. But the fastest possible signal to convey such information - about anything - is light - which while extremely fast is not of course instantaneous. It takes time. So, there is a 'lag' to consider when analysing such a thought experiment - a lag between appearance and reality. Apparently, Newton hadn't taken that into consideration, believing as he did in instantaneous light and gravity (or at least in absolute time and space, in any case).
222. Seife then points out that "a few years later" (ie a few years after his first thought experiment at aged 16 seemingly), Einstein would refine such experiments and show thereby that a classical principle of Newton would have to be discarded - (seemingly in favour of that regarding the constancy of the velocity of light that was implied within the electromagnetic equations of Clerk Maxwell). Hidden within the implications of those terms 'refine' and 'show' was nothing less than the special theory of relativity itself - in which time and space would be so 'shown' (possibly very technically) to be variable or relative and not constant or absolute, as indicated within Newton's laws. But such a leap in the present logic doesn't explain in any degree whatsoever how we get from time appearing to be slowed (say), by virtue of the time lag in conveying information about actual clock time, to being slowed in reality (not just in perception) - as is apparently explained or implied within the special theory of relativity - possibly by means of the factors mentioned earlier. Also, we don't know to when 'a few years later' refers - for example in relation to the thought experiment involving the Berne town clock - in early 1905, say. However, to further exemplify the basis of the conclusion that time is truly slowed or even stopped, Seife does quote Einstein's later (post-1905) thought experiment concerning the perception of the simultaneity or otherwise of two lightening strikes when one is situated exactly mid-way between them - either in a stationery position or one moving (as on a train) towards one or other of the two locations where the lightening strikes. I don't believe this example was one that Einstein quoted as being a factor in arriving at his theory (during 1904-05, say) but rather was one he used some time after its publication - when later seeking to better explain or at least give the flavour of its basis or logic to the layman [Yes: see now below.]
223. [We may note here that a reason why Einstein was seeking to establish that time and space were in fact variable, that is, relative, rather than as Newton presumes, absolute, has not yet been indicated in this overview. As will be elaborated below, it was as I understand essentially to provide a means (the only means) whereby the principle of relativity (once generalised) could be shown to be (as indeed it was assumed by Einstein always to have been) compatible with the constancy of the velocity of light. Their apparent incompatibility - which Einstein felt shouldn't be the case as they were in his view both firm and valid principles of physics - appeared somehow to relate to a rigidity of the usual measures of time and space - as being 'absolute' and unvarying. [We should provide the details regarding how he came to recognise this probable relationship; it was quite possibly because there were no other 'variables' available that could be relevant or responsible. But some idea about how this came to him would, as always, seem important and useful here. Did it not follow from the consequences of 'holding fast' to the constancy of light's velocity when deriving more fitting transformation equations?] Any suggestion that either variable might in fact not be absolute after all, was thus focused on by Einstein for this specific purpose - that is, to 'find a way' (even if it (time always requiring light signals?) only suggested the seeming 'mechanism') that would allow both these basic principles to continue to apply without being inconsistent with one another. Eventually, he did note just such a suggestion (via lightening strike example..or...?the Berne clock?) and when he followed this up (as he explained in his book on Relativity - a critical resume of which is provided below), it seems to have allowed us to better appreciate the actual (but thus obscured) compatibility which he felt must have been the case all along. The principle of relativity could be satisfied after all in that the law of the constancy of light's velocity would prove to apply whatever the difference may be of the velocities of the reference system it may be measured/observed from and that of its source.]
224. But first we may continue describing part of this same explanation (but as provided by Seife) regarding the basis of the conclusion that time (in particular) was really relative (ie depended upon...something), not absolute. Thus, says Seife, Einstein points out that while a stationary mid-point observer would report that the two lightening flashes occurred at the same instant, the observer travelling past that same point when the two bolts struck the two distant points simultaneously, would report that the flash arising at the location towards which he was moving would be reported by him as occurring before the one arising (at that same instant) at the location from which he was moving away. That is, between the instant that the lightening struck in the two locations simultaneously (as we have been informed in this thought experiment) and his perception of either flash, only moments later (and now 'out of synch'), he would have travelled towards the approaching flash and away from the other. The distances that the two flashes (being the light information about the simultaneous striking bolts), starting out at the same time (as defined in the experiment), had to travel to reach the moving observer were thus different over those two 'legs' of their journeys, whereas the distance of these two legs were exactly the same for the stationary observer.
225. Seife notes that the stationery observer would thus report that the two bolts had indeed struck their tracks at precisely the same instant and were thus simultaneous (as would be the equally slightly later information flashes of same reaching his retina on which he based that conclusion). But for the moving observer, he points out that the time taken for the two information flashes about such strikes to reach his retina will be different depending on whether he is approaching or moving away from the relevant bolt when they actually struck simultaneously (as the distances that light information must travel would be different in the two cases - as light always travels at the same speed. He may thus report, says Seife, that "the two strikes were not simultaneous" - as based on the non-simultaneity of the two information flashes. I would suggest that for consistency and accuracy, he should really have reported that the two strikes would appear not to have been simultaneous (when in fact they were - as we have been told); the unqualified conclusion that they were in fact not simultaneous was thus wrong. One or the other bolt would thusappear to have struck either before or after the other one. They may well have been simultaneous - especially as per this particular thought experiment.
226. [We may have to consider whether there is any way he could report (or calculate) the actuality of the situation or is he 'doomed' forever to be constrained to report (as his seeming reality) only that illusory (and momentarily delayed) perception ? But, if Einstein's theory is verified consistently, it would seem to imply that such a constraint is somehow always expressed as 'the actuality' nevertheless !!? If so, then I would like to have that 'somehow' broken down into its logical steps and be so accounted for! It certainly hasn't been thus far. What is the 'mechanism' at work ? We might also recall that any effects on time and length as proposed (theorized) may well not be universal but are in fact restricted to measurements (judgements) made of activities on one moving frame of reference from one moving at a different velocity. But are most of our judgements not made within the confines of our usual one local frame only ?? There seems to be a general acceptance (after 1905) that time and space are not absolute or constant but 'are relative' - period ('full stop'). Is that the case really; should it not be qualified as always depending on whether the situation entails such differently-moving reference frames, but not otherwise? But of course even our own seemingly 'still' frame, may be seen as the one that is moving by those on (what appears to us to be) a differently (ie 'actually') moving frame. ]
227. Seife summarizes the imagined results of Einstein's thought experiments, which entail such configurations of moving objects as above, as "very odd" in that they (the foregoing set of differing perceptions) wouldn't be predicted by the existing laws of classical physics (when, seemingly, they should have been). Had they been differing realities, I could see how such laws might well require some variation in Newton's laws to account for such, but simply differing perceptions, at least as described, seem to me to have been quite adequately accounted for by those existing laws - as we have differentiated the realities from the perceptions. Nevertheless, says Seife, "Einstein realized that it (ie such differing perceptions) meant that 'the concept of simultaneity' had broken down." [Note: not 'the concept of apparent simultaneity' or...'appeared to have broken down'...but somehow real simultaneity had actually done so - ie had broken down - and hence (it seems) a new concept of (?real or ?apparent) time was needed.] From this, says Seife, it apparently follows that one's perception of time becomes altered - when moving relative to one who is not so moving (or is moving differently). This at least does seem to support the idea that it was indeed simply one's perception of simultaneity that had broken down, not that it had actually done so (whatever that may mean), and thus that it was this same conclusion that must apply to time itself - that is, to its perception. And yet, on this basis, says Seife, Einstein nevertheless concluded that Newton's conception (not perception) of time as being absolute, with each actual tick of the clock being the same no matter how an observer of same was moving (and so receiving such perceptual information out of 'synch', as it were), must be wrong. [Note again: not 'must appear to be wrong' or 'perceived' tick but 'actual', seemingly. But just how does one get from the former to the latter??]
228. And with similar logic, says Seife, Einstein concluded that Space (length) too was not absolute as Newton believed but was also (not just appeared to be...nor that the perception of length was...) dependent on one's motion and thereby really varied, as did time. Both did not just appear to be dependent or relative therefore but really were - ie 'relative' - to one's motion. Clocks did tick mechanically slower and rigid measuring rods were materially shorter....apparently.
229. I don't quite 'see' it (yet). There seems to be a magical leap from the 'apparent' logic (to coin a pertinent phrase) associated with perceptions of altered simultaneity, time and length (which, granted, may serve as a kind of model of a more subtle conception of reality) to that of some almost unknowable 'real' logic, possibly buried somewhere within Einstein's more detailed theory of special relativity, by which the actual effects on these factors somehow arise and may presumably be shown, but to which most of us, still held in a world of 'appears','as seen from', etc, seem not to be privy. Maybe Newton meant that time and space 'appear to be absolute' and Einstein came along and said 'no, actually, they 'appear to be relative' (and we haven't access to whatever they may 'really' be) !? Maybe they're only 'effectively actual' !? [Note: However, I'm beginning to accept that the transformation equations do incorporate the quantitative outcome of the explanation Einstein has, I believe, given (somewhere) regarding the inevitable lag in receipt of the necessary information about time and distance (in observing a distant body's motion - including clock hands) and that these same apparent values arise as the equally inevitable result of holding to his two postulates; helpfully, these should be (ideally) two ways of saying the same thing! But that the possible effects of velocity of mass isn't yet another equivalent explanation.] [NB See now last paragraph in resume of Section 16 below as well.]
230. It may be pointed out that just as the term 'show' as discussed earlier, so too the term 'realized' in the preceding paragraph seems conveniently to mask that large leap over the intervening steps in the logical exposition. While the perceptions of altered time and length so described may well have (as a kind of heuristic 'model') stimulated in Einstein's mind a line of reasoning whereby he at least could 'see' how the apparent incompatability of the principle of relativity as it then stood and that concerning the constancy of the speed of light could be resolved - such that time and length (not just our perceptions of same) would indeed have to become really variable (or 'malleable' as he once said) in the same way that Newton assumed that invariable time and space were real, that line of reasoning (that 'realization') is not at all clarified (in my mind) by means of the thought experiments and the perceptual anomalies just described. [Note: the incompatibility thus referred to is a rather general form of expression which might be helpfully expressed more in terms of the specifics concerned - namely, in terms of the alteration in the values of time and space needed to allow light not to be increased in speed by a faster reference system, etc. (see later)].
231. The time that any signal, including light (which, admittedly, is (a) extremely fast and (b) constant and thus reliable) takes to convey information (eg regarding the actual time or length at a different place) must, one assumes, be always taken into consideration. Quite possibly, we never have access to anything but such perceptions ?? Moreover, one must always keep in mind that the perceptions so based are equally valid from the 'other point of view'; that is, in terms of both mathematics and physics, there is no difference in the precedence or validity of perceptions made between any two coordinate/reference systems moving relative to one another. Therefore, no conclusions can be accepted that prejudices either perceptual stance. Thus, with respect to Einstein's tram journey, his perception of the time back in Berne (remaining at 11 o'clock, say) while locally his own watch showed him it was now 5 past 11, can not take precedence over the man in the Berne town square who notes that in Berne it is 5 past 11 but on the tram, it appears to him to still be 11 o'clock. Who is right ? The answer is...BOTH. At least this must be the answer if neither knows all the details of this 'thought experiment' (as the distances or the times so travelled).
232. Similarly, in the case of the lightening strikes, the magnitude of the equal distances from the mid-point between the two simultaneous strikes (at A and B) is irrelevant with respect to the validity of the report that they must have occurred at the same moment some very short time before they were perceived to be simultaneous. Without knowing those equal distances, however, one couldn't say exactly when they had actually struck the tracks but one could at least say that they did so simultaneously - at some uncertain but equal distance and time from the observer. But if they were reported by a moving observer to be perceived at slightly different times, one couldn't say at what distance they strucknor whether they may actually have been simultaneous - since there is no way for the information about them to be conveyed instantaneously to that observer that could answer that question with certainty (I believe). [Do we assume that in the thought experiment both observers know the locations of A and B but that in 'real life' this is what isn't known and thus why we can (?typically) have only perceptions as the 'experiential reality' that Einstein so stresses ? If so, why isn't this very relevant feature emphasised (or even mentioned) ? But, to assert that two events are not simultaneous because our only possible perception of same indicates that, even though they might in fact actually have been simultaneous, would seem too insecure a basis on which to build a reliable physics of nature. However, one might again cite the problem that arises when there are two possible perceptual stances and neither is warranted to take precedence! That is, due to the symmetry factor mentioned above.]
233. We may also point out here that, more to the point, the problem of the appearance vs the reality of the alterations in time and space (and thus in velocity) as described in Seife's paper (and which we are seeking to resolve) also manifested itself in Einstein's own paper of 1905. For we find (just as with Seife's exposition) that in its Section 2 of the Kinematic Part above - 'On the Relativity of Lengths and Times', we may recall that he expressed the following conclusion about Simultaneity (and thus Time) near the end of that section: "So we see that we cannot attach any absolute significance to the concept of simultaneity, but that two events which when viewed from a system of coordinates are simultaneous, can no longer be looked upon as simultaneous events when envisaged from a system which is in motion relative to that system." Thus, he doesn't say, simply, that such events when so viewed 'are no longer simultaneous' in any absolute sense but, rather (in effect), 'no longer appear to be so' - ie as now 'viewed', 'looked upon', envisioned', etc. Rather, they are actually relative. What are the implications of this selection of terms (vs the use of various forms of the verb 'to be' (rather than 'to appear') for his actual theory ?
234. This interpretation was further supported when we looked earlier (above) at Section 4 - 'Physical Meaning of the Equations Obtained with Respect to Moving Bodies and Clocks' where in another conclusion (after careful reasoning), he noted that "..whereas the Y and Z dimensions ..of any rigid body do not appear modified by motion (in the X direction), the X dimension (of length) does appear shortened - in the ratio of 1 to sq rt (1 - v^{2}/c^{2}). However, within this very same sentence (after a semi-colon), he then adds the elaboration - "ie, the greater the value of v, the greater the shortening". That is, he doesn't now say - "ie, the greater the value of v, the greater the apparent or viewed shortening". Why not ? For consistency, objectivity and clarity, he should have done so, I believe (unless there was some confusion regarding the term 'appear' in the translation?). Equally, he continues by noting that when "...v = c (and thus v/c = 1), all moving objects - as viewed from the 'stationary' system - shrivel up into plain figures.." (that is, 'lose all of their length dimension' presumably). Again, for consistency, should the phrase not have been 'appear to shrivel up' ? But, while it wasn't thus made explicit, it was presumably still implied(!) (unless he somehow wants it 'both ways' - ie first one way - as 'appears' - and later another way - as 'is'). For one understands from other accounts of this phenomenon that the bodies concerned don't actually alter in their 'rigid' physical structure. Finally, in this section on length, he notes that the same results would apply with respect to bodies at rest in the 'stationary system' - when viewed from the system in uniform motion. That is, the 'apparent' effects too are symmetric and of equal validity and relevance from either viewpoint. This seems to be rather crucial and significant in the ultimate validity of the theory. Such symmetric perceptions of equal precedence would seem to be the inevitable reality (despite all the gradually understated 'appears') on which all the implications of the theory - as confirmed by later observation - are validly based.
235. The brevity of the only once admitted concept 'appears' within the discussion on Length (ie Space) is further exemplified in the next paragraph when he addresses the matter of the other component of velocity - ie Time. His conclusion in this case is "...whence it follows that the Time marked by the clock (viewed in the stationary system) is slow by... 1/2 v^{2}/c^{2}". That is, it isn't, to be consistent with the discussion on Length, described as 'appears slow...' by that amount - but is so stated that one would now reasonably assume it to be slow - ie as an actual slowing of this amount. Again, we may assume that such a result is symmetric as described for Length. But do the clocks concerned actually tick more slowly - in some physical sense - (and do the lengths concerned contract physically as well) or is it 'merely' a perceptual anomaly (even if a mutually symmetric 'perceptual reality') ? While the clock hands may slow, does the passage of time that it purports to measure also do so ? Yes - as with the Berne clock's time; it stopped ticking (moving) forward ! Does time exist - beyond its man-made measurement by the successive ticking of man-made clock-machines - or have we simply invented it as a means of differentiating different velocities of a moving body on different occasions - eg at the very same location ?? If the universe was created (or created itself) in little more than an instant, say, time may well have existed for that instant but if the universe then 'froze' and entailed no further change whatsoever, there would presumably be no time nor need for any. [Note that in the transformations first advanced by Voight, there was apparently some implication regarding time dilation, as there would be in Lorentz's equations. The magnitude of the latter were concluded by Ives & Stillwell (1938) to have been more precise than Voight's (and possibly nearer those calculated by Einstein despite the initial concern that Lorentz's was simply made to fit the results (ie exactly).]
236. Thus, observers travelling in opposite directions at a velocity difference of near the speed of light (if that were possible) may find that if their own watches showed them that it was 12 noon when their journey began, after 5 minutes they would both note that it was then 12.05 on their own watches but still about 12 noon on the others' watch (if they could see that far). What is the correct time then ? Neither has precedence; they are equal and symmetric. Neither is more 'truly moving' in relation to the other even if one or the other might appear to be; the mathemetics and the validity of the logic is identical. Time is thus relative to the perceptual viewpoint adopted. The same argument may apparently be made with respect to space (length or distance). Now in the example just given, it may be the case that each observer already knows that his counterpart's watch shows the same time as his own just before the journey begins. They may also know what the difference in their velocity may be and at what velocity (c) the delayed time information (via light) from the other's watch will come to them. They could thus calculate what delay may be expected in the perception of each others' times. But, where these factors are not known, both could only trust what information they do have. This seems to relate to what Einstein describes as the reality (and validity) of 'our experience'. And where the velocity difference was (more realistically) at any other (lesser) magnitude, the extent of the difference in the perception of each other's times would be correspondingly less - according to some function of the ratio of v/c. But why is there no mention of the apparent importance of this crucal effect of this ignorance of the associated knowledge ?
237. So, we see that for both Length and Time, the interpretation that the alteration or modifications of these constituent factors of velocity in terms of perceptions or appearances which seems initially accepted by Einstein and those interpreting his theory, is generally replaced by the idea that such modifications tend to be described 'as if' actual or, if not actual, are gradually portrayed as 'effectively' or 'virtually' so - with such qualifications also quierly and conveniently dropped. While the subsequent examples cited by Einstein and others that indicate that one or other of the latter forms - of actuality is the case, appear to be valid (and thus the underlying hypotheses regarding Space and Time are consistently supported by later examination), it seems to me that there is nevertheless a very strong need to more fully explain how the conception of 'appears', so often introduced at the start of such discussions, turns so magically into 'is' in this or many other expositions of his theory as they are further elaborated. Surely it's rather crucial and fundamental. We may note that there appears to be no symbol in all the mathematical reasoning and equations underlying the theory for the verbal concept of 'appears' vs 'actually is'. Rather, there seems to be an assumption that any such reasoning, with its various equations and formulae, always implies that latter reality (unless specifically excluded?).
On Einstein's Insight about Time - (from books by Michio Kaku and Albrecht Folsing).
238. These two recommended authors describe how Einstein finally found the source of the time (and later space) adjustments he felt were needed to fulfill the demands of the principle of relativity (as he believed to be the case) - ie to allow the law of the constancy of the speed of light to hold, devoid of an ether, and, as a law of nature, to accord with that principle in all possible circumstances. This was apparently 'the problem or difficulty' he had been addressing latterly - rather than trying to answer Michelson directly - as Lorentz had been - regarding a somewhat different problem, with different premises (involving a still ether and the traditional mechanical model to explain light's behaviour). It should also allow him to 'see' why he couldn't after all race with (or match the speed of) his sunbeam and see it as a 'frozen' wave - as Newton's mechanics appeared to have allowed - which was his original concern and problem. As touched on above, if it turned out to be the case that Einstein failed in finding this solution, science would have had to continue to 'live with' some unacceptable 'anomaly' that was either apparent because clear empirical evidence pointed to it or was generally accepted to be the case as established by convincing theoretical considerations. Just what was this 'anomaly'. Did Lorentz, Poincare and Einstein all agree on what it was? Was it that obvious and incontrovertible, or was it still unrecognised or not fully accepted ? How was it described by Einstein in his 1905 paper (if it was)? But first we consider the factor by means of which this elusive problem/anomaly was in part resolved - Time. It may help answer the foregoing questions.
On Time - 1. Based on Michio Kaku's book.
239. [Note: In this account, I have integrated certain ideas I've concluded myself - some from other readings - so it is no longer just Kaku's framework (as excellent and useful as that is). This focus on Time by Einstein would appear to have been an important step when he sought to resolve the seeming incompatibilty problem. While eventually appreciating that the latter situation could define 'the problem' more succinctly than had been the case initially, it apparently proved difficult to resolve until he had this 'insight' regarding Time (as well as his conviction about the lack of any absolute rest or motion in the universe, or need for such an ether medium, still or otherwise). One should be able to integrate both elements into Einstein's later (1920) account at an appropriate point (see below). Ditto for any of Folsing's remarks on these same topics which follows below. Both authors provide some of the pertinent background material on Einstein's early career as already touched on above.]:
240. As a teenager, Einstein apparently read a popular book about science in which the author asked the reader to imagine travelling within a copper wire along with its flowing electricity. This was no doubt an important influence on his own imaginary trip, conceived about a year later (ca 1895), of travelling along with a beam of light waves at their enormous speed. Would they, he apparently asked himself at one point, appear as still, frozen (unmoving) waves beside him? At first, he apparently thought they would, at least according to the then prevailing mechanics of Newton, since the apparent relative speed of anything, including light, should in that model vary according to the speed of its source or of the one measuring it. To a fellow traveller therefore the forward speed of the light waves originating from a relatively stationary source should appear as virtually still. If released from a very fast moving source they should on the other hand flash by (or appear to flash by) a relatively stationary observer even faster than light's usual speed. However, Einstein would later change his mind about these early conclusions, as we shall see.
241. For shortly after this, he began his course in physics at the Zurich Polytechnic (1895) and, within a year or so, became aware of Clerk-Maxwell's famous equations regarding electricity and magnetism and their relationship with light. Being an excellent mathematician, he analysed these in depth and, recalling his earlier thought experiment about how light waves would appear when one travelled with them at their speed, eventually realised that they probably couldn't, in Maxwell conception, appear as 'frozen'. For his analysis indicated that not only did such electro-magnetic waves travel at the same speed as light and that, as Maxwell had concluded, light was itself comprised of such waves, but that a ratio of the electric and magnetic components of all these waves, which was a constant, eventually indicated to Einstein after some vascillation that their speed, and thus that of light, was itself a universal constant - later symbolized as c. In this interpretation, light could not vary its speed to reflect a faster speed of its source or allow an observer accompanying it to perceive it at anything but its usual immense speed relative to such an moving observer (amazingly). [The principle of relativity may have predicted this had the 'law' of light (c) been first established and accepted, for it would require it to hold fast to its nature and so not be influenced by the speed of its source nor that of any fast-moving observer - as all other moving bodies were (to a large extent) - as their 'natures' would allow that.] Whether Maxwell and others a little later (as Hertz, Lorentz and Poincare), had reached the same conclusion about light's constancy seems uncertain. If they had, it appears that no one else appreciated any crucial implications of this. [Lorentz includes the idea of c in his later equations but must have still assumed (inconsistently?) that this would be somehow reasonably 'over-ridden' in such as Michelson's experiments and yet 'masked' by influences his theory would introduce.]
242. This constant speed, of 186,000 mps in a vacuum (as space), was thus, especially in Einstein's analysis of Maxwell's theory (almost by definition), unaffected by the speed of its source, target or of those observers measuring it; it was an inherent feature or 'law' of nature. This was a most unique conclusion. [Indeed; some physicists to this day still have their doubts apparently.] For no other moving phenomena (as governed by the laws of mechanics) appeared to possess this strange property. It meant that if a pulse of light was released at point x from a relatively stationary source at the same instant as a similar pulse was released from a source travelling past the former one at some immense speed, they would still both travel at their one and only speed and so reach the same target at the same instant. The 'boost' in speed given the second pulse when released from its fast moving source would provide absolutely no advantage to it; it was as though it had one upper (and lower) limit and that speed was a constant - being both its maximum and minimum possible speed. And if an observer was travelling at say 3/4 the speed of light [relative to some agreed reference point] when he released that second pulse, he would nevertheless see it speed away from him not at just a 1/4 the usual speed of light (in relation to himself at his own very fast speed) but still at its usual full 100% speed of 186,000 mps from him - as though he were virtually stationary. [And while to other seemingly stationary observers he appears to be moving away that fast, to that traveller, such distant observers may (or could) appear to be moving away from him at 3/4 the speed of light and to them, from that point of view, he would indeed appear as stationary (which, in one sense, he is!] At least, this would seem to follow if anyone had at this stage yet pursued these ideas to this extent - at least in theory. Possibly Einstein had but no others it seems.
243. These conclusions by Einstein of Maxwell's theory (when eventually fully arrived at about 1903/04) would be in contradiction to the original mechanics of Newton (and to the electromagnetic mechanics of Lorentz?*) which would predict that the resultant speed of any phenomena, including light, must reflect its own innate speed plus that of anything conveying, boosting or hindering it. And that, similarly, if one could travel near the speed of light, its speed would to that extent appear much slower than otherwise - even 'frozen', if the speeding observer was able to go at light's own speed. This paradox and of the different perceptions that theoretical observers of same would report in terms of these two theories, depending on their own speeds, troubled Einstein for many years. Both couldn't be right. But before 1905, he would not appreciate why Maxwell's ideas, at least when shorn of the ether, would eventually prove correct - with one important revision. {* I'm not certain when Lorentz accepted that c was indeed a constant.]
244. After graduating (1900), Einstein was unable to obtain a position in a university, as he had hoped, due less to his grades than to his reputation as a rather difficult personality who didn't take kindly to authority. Through a friend, he eventually obtained a post as a Patents Examiner in Berne, Switzerland in 1902. To keep himself abreast of developments in physics, he and some friends formed an informal scientific discussion group. They discussed amongst other things Ernst Mach's book on 'The Science of Mechanics' and certain writings by the French mathematician Henri Poincare, in both of which, amongst other things, Newton's ideas on the absoluteness of time and space were discussed, as was the role, if any, of 'the ether' in light's propagation. The aforementioned paradox about light was also a frequent topic, it seems. For a time (ca 1902/03), he had apparently considered it could be the case that light may in fact not be a true constant and also that there may actually be an ether - in an attempt to resolve these persistent difficulties; but he later abandoned both these ideas more completely. By about 1903/04, he had apparently become aware of Lorentz's transformation equations and of Poincare's thoughts on both Time and on the principle of relativity and these likely informed his thinking increasingly over the next few months. Before about 1904, Einstein hadn't, it seems, yet (re-)discovered this particular 'general principle' (of relativity), although he had recently decided that he must find some such general principle to resolve 'the difficulties'. For his other more usual attempts to do so (trial and error experiments in which the various elements concerned were mentally manipulated - in the abstract) hadn't worked and he saw that by arguing from some more general overview or principle was more likely to prove successful - that is, to find some inconsistency or anomaly; some general principle with whose over-riding logic the more specific factors involved would have to come into line.
245. Thus when, after noting Poincare's reference to it, he gave deeper thought to the probable role of a principle of relativity (with its particular demands), he seems to have concluded that this was indeed the principle for which he was searching. By analysing how it should and could apply with respect to all laws of nature (including that of light) he eventually saw that some kind of 'adjustments' (specifically of time and space) were clearly needed but ones that could be better justified than those advanced by Lorentz of which, we must assume, he had recently (? ca 1903/4) become aware. But they were, he felt, inadequately explained and came about more as necessary if passive adjustments to the arithmetic of his newly elaborated transformation equations (with the ether as a causative factor) designed to explain rather too exactly a different (if related) outcome than that for which Einstein was seeking to account. But, while Lorentz in effect did so also, Einstein needed to discover a more realistic, more explicable basis for such apparent variations.
246. And then, one evening in May 1905, he had been visiting his friend Michel Besso - going through this particular matter yet again and, as before, had to admit defeat; there seemed no way to resolve the difficulties and decide just how the differently moving observers would report their perceptions about light's speed. Was it truly constant whatever one's own apparent speed or the speed of light's source, or would it actually vary as most others still assumed? [As mentioned earlier, this problem was at this point still a theoretical one; the empirical evidence concerning a phenomenon whose velocity was 186,000 mps was not yet accessible for clear empirical confirmation.] On his way home, he apparently passed the famous clock tower in the Berne town square and looked up at the time; for some reason, he wondered what it would be like if the tram and he suddenly sped away at the speed of light ? If it was then 11.00 pm, say, then after one, two or five minutes, what would the town clock show him as the time (assuming he could still see that far behind)? He quickly realised that it would continue to show him that it was still 11.00 o'clock whereas his own watch travelling with him would show it was respectively 11.01, 11.02 and 11.05. That is, that the light reflected off the Berne clock's hands showing 11.00 pm would travel at the same speed as he was travelling - continue to convey that same information - but that the light leaving the hands just after that, as they continued to move gradually towards 11.05 back in Berne, would never reach him as he theoretically moved away at the speed of light. It would continue to show him it was still 11.00 o'clock there.
247. This realisation about the apparent slowing or even stopping of time, coupled with various conclusions he had been slowly gathering through his reading of current literature in this general area of electrodynamics (as by Poincare), plus his crucial talks with Besso - eg on the asymmetry problem in the related sphere of electro-magnetics (and ditto in mechanics) which implied there was only relative motion - as the principle of relativity had always implied, somehow led him to the resolution of his long standing paradox over the next few hours. That is, to allow the constancy (c) of light's speed (as just another general law of nature) to meet the demands of the principle of relativity (with which he was now convinced all such laws had to accord) - by means of the application of the equivalent of Lorentz's 'adjustments' (transformations) - but now with a more logical basis/origin for the necessary variability of time (and space) that Einstein recognised was inherent in those transformations (and for which he apparently now had the true source). The solution thus seemed to lay in his sudden insight that it was the concept of TIME in particular that had been misconstrued for so long. He later described this as his final 'step' in his long struggle to understand the full implications of Maxwell theory of electrodynamics - of all motion in nature (which was always relative). Everything in motion must, in a sense, become a proportion of the speed of light - that constant and upper limit of the speed of any and all moving bodies.
248. While Maxwell had established with mathematical precision the necessary equations by which the constancy of the speed of light appeared valid, Newton had not provided an equivalent analysis regarding his assertion that time (and space) were absolute (rather than possibly varying according to certain conditions). Their accepted invariable constancy or absoluteness were, rather, simply unquestioned assumptions or assertions. As long as Einstein accepted both his own interpretation of Maxwell's ideas on light's one constant velocity (distance per time) and Newton's unchallenged one on absolute distance and time values underpinning differing velocities of all (other?) bodies, the paradox would remain. And while he appeared to favour Maxwell and was probably also persuaded to a lesser extent - by both Mach and Poincare - that Newton's universal time and distance measures were, like the ether, unsupported concepts, he needed some clue and guidance that might resolve this latter problem more definitely. He wanted a logical, non-ether means by which to better justify Lorentz's transformations within which such variations in time and space were now integral. The imagined tram journey led to a realization of the mutual reciprocity and equivalence (re their validity) of perceived times and distance in respect of the motion of all bodies by those moving at different relative speeds, with neither point of view having precedence. Once he did realize this - that time and space were the relative ones for all motion (except for light which was constant) - his problem was to make that latter unique constancy compatible with the principle of relativity - as he now felt (more securely) should apply to all laws of nature). As touched on earlier, this reasoning required a prior justification also of the primacy of this principle - ie in terms of the arguments referred to above. Otherwise, there would of course be no need to seek a way to allow the compatibility being sought! His new kinematics, seemingly evolved later that night, then show how this compatibility was the case and could indeed be realized.
249. Prior to this, he (like anyone) had great difficulty accepting that if one could (in one's own fast-moving frame of reference) travel at or near the speed of light, that nevertheless any light 'switched on' therein would still speed away from one at its usual speed in all directions, including 'forward' - in the same direction one was travelling so fast. [Again, this earlier confusion likely reflected the fact that he hadn't yet considered the relevant implications of the principle of relativity (until ca 1904) - especially as illuminated by the true implications of the idea of the actual symmetry inherent in not only current induction but in all motion. That is, that there is no fixed resting point in the universe and therefore only relative motion was possible - except for light - as implied by the principle of relativity.] There should, somehow, be no 'frozen waves' to be observed. All this would soon be conceptualised within a more coherent framework - as a matter of resolving an apparent incompatibility between the constancy of the speed of light and the dictates of the principle of relativity as it had previously applied to mechanics.
250. If the principle of relativity was to hold with respect not only to the laws of mechanics (all of which entailed potentially varying velocities of the bodies 'moved' thereby - ie due to the motion of their associated reference systems)- but also to this awkward and unique constancy law about light (with its never varying velocity no matter what may be the velocities of its associated reference systems) - at least, once the relevance and seeming universality of this principle was finally if belatedly considered, this would be quite as expected by this principle: every such frame of reference should (apparently) allow all activities, both mechanical and electrodynamic, to proceed as nature intended - as the relative speeds of such frames should, according to that fundamental principle, accord as required to all laws of nature. [This was thought to be the case due either to the effects of inertia - as applies to the motion of all mechanical bodies - or, more generally, to the dictates of Einstein's and Bondi's even more fundamental principle of the elegance and unity of nature which ideally should have no exceptions; see also Hawking's and Einstein's own justification for the necessity and rationale for the principle of relativity above]. When one assumed that the light racer was travelling so fast, it was so only in relation to some other frame and from the (apparently) faster moving observer in his frame, the other frame could in fact appear to be the one moving (away) just as fast and thus the acceptance by those in the latter frame that light should move from them at its usual very fast speed proves to have no more justification that that of the other observer in his frame! [See also Einstein's discussion on the asymmetry problem.]
251. Thus it was the frozen time of the tower clock, in contrast to the normally proceeding time of his own 'local' watch - whilst on his 'light-speed' tram journey - that provided the clue that such doubts about absolute time and space were probably fully justified. This clue pertained to the fact that in certain circumstances (ie where those concerned don't have any prior information regarding each other's time - as Einstein atypically did have in his thought experiment), it is not possible to determine with confidence at what time an event occurs or how long was its duration. It would depend on the speed of one's frame of reference relative to that on which the event occurred and their consequent distance apart. Equally, it may not be possible to determine whether two events at different locations occur at an identical time - ie simultaneously. He would later say that "...by a revision of the concept of simultaneity, (in which local time had to be defined and times on fast-moving frames of reference only established (via Lorentz transformations) from that defined time) into a more malleable form, I arrived at the theory of relativity". By 'malleable' (?variable) here, he seems to have meant that the answer to any such question had two or more different but equally valid answers, none being more valid than the others. It would depend on or be relative to whatever speed one was travelling in relation to some other comparison frame, and vice versa. Our laws of nature had to be formulated to account fully for the different but equally valid experiences of us all - including the different perceptions we may have of the local time compared to others. For life isn't a series of thought experiments where we make up and know the prior conditions; much more typically, we don't know them.
252. Although he no doubt appreciated that while the clock back on the square only appeared to show that time had slowed - due to the fact that the light conveying its true on-going, changing status couldn't reach its target (ie his eyes) - and that the time shown on one's own nearby watch indicated that the town clock's apparent time was probably not the actual local time back there, his new awareness of the actual equality of these two realities and its consequences for the concept of 'simultaneity', somehow led him to a solution of his paradox. For, as mentioned, it would normally not be possible to 'know' which time was more 'right' than the other as neither was in any absolute universal sense; each was 'right' for their own locality and the other's could only be calculated by means of knowing the distances and speed differences involved over which the other's information could be conveyed by reliable signals. To be confident that the event of the hands on his watch (in the travelling tram) and that of those back in Berne had really reached identical positions simultaneously it was necessary to know what their positions had been when the two clocks were in the same locality. This is another way of saying when the common time for them had been defined and accepted and either clock and its observer then moved away at some speed, one could calculate what the time would be at their common origin by considering the distance and speed of their separation...but only if known!
253. The idea that time was indeed not absolute, but variable - being dependent upon what was the frame of reference of the observer - (and ultimately upon a reliable signal (particularly light) to convey information about the measure of time, whether locally or from a (?moving) distance, was the crucial breakthrough. It meant a new physics was required. The faster one moved away from a given location, the slower the time in that receding location would appear and, by reasoning to be elaborated later, the shorter would the distance between moving positions appear also. Thus, the speed (distance per time) of a moving object there would also appear slower. And these same measures would be affected in exactly the same way when activities on the 'moving' frame were assessed from the 'stationary' one - or vice versa. Because of this equivalence, neither can take precedence and thus neither is more valid than the other and can't be accounted for by somehow 'knowing' that one or the other is incorrect due to a reasoned time lag in receiving the relevant information. Both are equally affected and thus correct (or incorrect?). There is no other 'more real' time (or length). (The term 'appear' above may thus not be accurate.) A physics was needed that could accommodate both realities (and not 'appearances'). At least the constancy of the speed of light allows us to perceive two events as simultaneous by means of defining a synchronised time for those in separated frames of reference and/or take account of the effect of any lag in information (conveyed by light signals of known, constant speed) regarding real time perceptions - to define an acceptable common time for a defined locality or region. But, both time and space remain relative (and thus variable) where such prior arrangements for a local situation can not be made and differently moving frames of reference are involved (I believe). [This may need clearer analysis.]
254. [Note: This next paragraph comes from the account on the asymmetry problem most of which has now been placed prior to the present discussion on the insight into Time. This additional portion is inserted here but will need an introduction that proves compatible with the present remarks]: ...'So Einstein concluded that his imaginary traveller on the tram and the observer back at the Berne square were completely equal in terms of how each saw their own and the other's local times. Simultaneity of an event at either location with the other (speeding away from each other?) could not be established (be it position of clock hands or any other event) and this somehow suggested to Einstein that Time needed to be re-defined such that its actual relativity (vs absoluteness) became acknowledged (as it would also for space). Besso would later claim that it was he who brought the matter of the aymmetry problem into their conversations and (also in retrospect) saw it as a practical anticipation of the relativity concept - ie as though this aspect (regarding time and space or just the principle?) of the overall problem hadn't yet shown itself by that late point!. But, we recall that at about the same time, this concept had entered into Poincare's thinking along with his ideas about time and simultaneity. Einstein likely put two and two together shortly thereafter (and so applied that concept to both the question of the relativity of all motion (and the associated equivalence of the two coordinate systems) and the matter of the variability (ie 'relativity') of time and space - possibly even as late as that evening and night!' [But see below when Einstein 'returned' to the Lorentzian view about the constancy of light's speed (apparently inherent in his theory even though that meant it contradicted the relativity principle and thus why there was the dilemma - which Lorentz sought to overcome in his various ways (with Poincare). But this would imply that this principle was already determining his analysis; was it or wasn't it? And was he (Lorentz) really yet convinced about light's constant speed and/or upper limit ??]
255. [Note: This next paragraph may address and answer the question I've placed at various points - namely - in which calculations was it that Einstein realized he could/should now replace the absolute values that he (and everyone!) had or would use (pertaining to 'his problem') with these now variable, relative ones - in order finally to resolve the problem as he saw it ?]: Thus, as I wrote herein earlier: "This new conception would provide the means by which to reconcile the idea that anyone travelling very fast beside a light beam would still find that light would move away at its usual speed. Regardless of his speed, whether fast or slow, relative to the light's source, he could be quite unaware of this speed (assuming it was smooth and uniform) - just as we experience while moving so fast with our Earth - and so, in accord with a new principle of relativity, perform all normal activities in accord with the laws of nature (including now the law of the constancy of the speed of light). As such, it somehow must move away from him at its usual speed. A ground-based observer would see both the distant light beam and the reflected light from the nearby traveller slowed down such that the same relationship between them would still pertain. This was the reconciling factor. The traveller's speed would not be appreciated by the traveller himself if he had no other clues and in fact the observer on the Earth would appear to the 'traveller' as the one moving away at speed. The crucial point is that neither is any more the actual traveller than the other and that, therefore, the laws of nature (including that of the speed of light) should prove equally valid for both situations - ie be to that extent 'malleable' or 'flexible' - to maintain the universality of the laws of nature for all observers in the separated, differently moving environments that are their respective realities and experience. [Ditto seemingly re my example of racing with a light beam from the Sun to the Earth in 8 minutes.]
256. We can assess the behaviour of moving objects within our local frame - using 'local time' and 'local distance' measures but when assessing the same situations on or in relation to a fast-moving frame, we would have to make allowance for the speed that frame was moving in relation to the local one, and the distance and speed the light signals had to travel to convey its information from there (and vice versa). These allowances became known as the 'Lorentz transformations' as they were first calculated by Lorentz before 1900 although, as menioned above, they were not derived in terms of the same physical principles that Einstein's equivalent equations were when he deduced them independently in 1905. Moreover, this new interpretation should result in it not being possible by utilizing mechanical or electrodynamic measures to determine which of two moving frames of reference was moving relatively the faster or slower - as was seemingly the case before Einstein resolved the problem. This was apparently not the case when Lorentz's interpretation was applied. Light had required a special (unique) dispensation under Lorentz's model - which was a source of some dissatisfaction to many physicists at the turn of the century.
257. If it was the case that one would never see 'frozen' waves beside oneself no matter how fast you appeared to travel relative to some reference point or system, as light would always race ahead of you at its usual speed, it should nevertheless also be the case that those on that reference frame could equally be seen to be the ones moving at such a speed (relative to the first frame - which would now appear as the stationary frame). Those in either frame could see either themselves or the others as the ones moving or stationary! From themselves, light would always move away at its usual speed. [What would each see in regard to speed of light in relation to those on the other frame?? Recall that each would see the other's time as slower than their own and distances shorter - their own remaining 'normal'.] There was obviously a need for some kind of major adjustment in the analysis of the 'geometric' (kinematic) variables involved. Something had to 'give' that allowed this otherwise inexplicable perception - of (seemingly) going at such a speed oneself and yet the light still racing away from one at its usual speed, not some greatly decreased or increased speed, as seen from the stationary platform - and vice versa.
258. The basis for this adjustment was what Einstein suddenly 'saw' when he realised that TIME from certain points of view depended upon one's speed in relation to the clocks used to measure it. [It seems to me that the most fundamental principle in this analysis is that one's speed, even near that of light if this were possible, has no absolute significance. All of our movements are at some speed, whether 'slow' or 'fast' - but only in relation to something else - but have no bearing on how we arrive at our model of reality. We are, for example, mostly unaware that we are travelling around the Sun at a great speed (30 kps). Most of our perceptions of things moving near us entail an assumption that we are effectively 'stationary' or, if we are not, we can easily 'make allowances' for this vis a vis some other 'stationary' reference of interest. And even if we suddenly move at 3/4 the speed of light (in relation to some environment), our laws remain exactly the same. For that seeming speed is 'merely' in relation to something not going that fast. If everything around us was also going that fast (as it may well be), we'd be much less aware of it - especially if everything 'worked' just the same as otherwise - which it does - including the 'law of light' [providing we recognise that time and space vary when ws adjust for light's constancy when we view others' apparently moving platforms (or is it ours that is moving?). Otherwise, the principle of relativity doesn't hold good and we could definitely tell which frame was moving or not.]
259. The speed of light - relative to ourselves - reflecting another law of nature, would therefore always be 186,000 mps - ie 'from us'. [This is what accepting Maxwell's equations demands. It does seem to place the observer as a rather overly relevanr and/or important element. Surely light moves from the Sun at its one speed whether we're here or not?]. Our own relative background motion, whether slight or massive, relative to any other frame of reference, doesn't change anything. Once moving, we are virtually unaware of it and that 'smooth' motion, due to the law of inertia, can be effectively ignored. There's something very fundamental and crucial in all this. In Galileo's day, this was essentially the 'principle of relativity' as it applied to all mechanical motion and the laws governing same. They would apply identically whatever might be our own uniform motion relative to anything else. Einstein eventually realised that the 'law of light's speed', as a general law of nature, should (?must) accord with this same essential and most fundamental principle (for reasons elaborated previously; but, in a sense, this is still a hypothesis of the theory being presented). To the apparent (but not really) fast traveller, those left back on Earth become the ones (apparently) moving away, and at this same great speed, and so might be assumed (again wrongly) to be approaching the speed of light and so they could be equally the ones to see it's waves as 'frozen' - if, that is, either observers were really moving at near to the speed of light - in some absolute sense - but of course, they're not.
260. As the reasoning behind these two conclusions are equally valid, however, neither would take precedence and the laws of nature must somehow apply equally validly to both points of view. But, what Einstein first realised was that both observers would see the other's time and distance equally different from their own. The implications of that appears to be what was only THEN analysed in terms of the principle of relativity, I believe. Thus, if the principle of relativity should hold for both [as concluded on the basis of....(what?)] - from whichever frame any moving object is assessed - then Einstein's recognition of the different times perceived by differently moving observers would seem to provide him, finally, with the means by which this could indeed be the case. Or did he arrive at the latter recognition only after setting a principle of relativity as the more basic determinant?] In any case, it did so by indicating that the valid measures of time and space therein are not those absolute, unchanging ones used in the original principle of relativity but the now corrected, malleable, relative ones that depend upon the speeds of the associated frames of reference (ie that of the observer relative to that where the event occurs) and the inevitable lag in time for signals to inform each other of the other's times and distances. There is no other absolute time or space, only relative ones. If the observer is in the same frame as the event, then both time and length are as expected.
261. As Einstein later reported it, "...a storm broke loose in my mind....the solution came to me suddenly with the thought that 'our concepts and laws of space and time can only claim validity insofar as they stand in a clear relation to our experiences'. Einstein realised that our experience of time was always predicated ultimately on the conveyance of information about it by means of signals of one kind or another. The fastest signal was light (although such information could be conveyed by various other means as well seemingly - but not necessarily with such reliable constancy of speed which was apparently crucial in establishing a common definition of time that remained reliable). There was no other way 'to know' the time other than by means of signals and it was thereby (?always) dependent on (relative to) any lag in the receipt of that information according to the speed of its source and the distance involved. As such, it would vary accordingly and so not be absolute, real and constant. The same should apply to our measures of space. As such, it was possible for two or more observers to report that a given event (such as a certain position of a clock's hands!) could be seen as occurring at different times, not simultaneously, depending on their respective speeds and positions in relation to any clock (and thus that of the different distances signals may convey that event or time) away from or towards their respective clocks or observers. With greater speed away from the clock, time would appear to slow (be dilated) relative to more local time (and distances shorter). [Consider the example here of the moving 'light clock' (with the distance travelled by the light signals and hence the time between 'ticks' depending on the relative motion of the observer) and how the time indicated by any clock is somehow similarly affected and depends on whether one is travelling with the clocks or observing them as moving away from one.]
262. In Einstein's fast-travelling tram example, he 'knew' his local time was what it really was back in Berne (despite what the clock there showed him) but in normal situations, which one of such different times, even if known, is the more valid, if either, is unknown.] But this applied to the other observer equally; there was no way to establish that either view point could take precedence. Our laws have to be consistent with this mutually valid, reciprocal reality ('experience') and incorporate/accommodate all such equally valid situations. Neither was the more 'real' time; all 'local' times are valid to those in its vicinity, or to those further afield but not moving away - if a common time throughout an area has been reliably established. [Is this our 'simple/direct' measures of speed?] To each observer in their own environment, everything would appear as normal but to the distant or differently-moving observer [our 'indirect/complex' case], things would appear distorted and be accounted for by inevitable adjustments in the perceived length of bodies and the passage of time - by and of those on both (or all) differently-moving frames of reference. These differing perceptions worked both ways and hence neither was more valid than the other and neither was a subjective assessment of some more objective, local reality; both perceptions were equally 'objective' and real to those inhabiting them. [This may account for the ideas about ageing differently if moving very fast relative to another. and of slower times on clocks moving at different speeds. And, it works thus for both, indeed for 'all', differently-moving, frames.
263. Later, Einstein would appreciate that the same thing applied to SPACE. The faster one moves away from a meter ruler, the shorter it would appear - it (ie space) would apparently contract. [Consider a giant ground clock with a 10 mile circumference in which the minute hand makes a full circle of 10 miles in one hour (10 mph) as seen by an observer standing nearby on a hill above. But a fast traveller moving away at 90% of the speed of light on a 'magical' rocket ship would see the tip of that hand move just one mile (say) in that time (one hour) as noted on his own watch. That is, it says it is only 5 minutes past the hour it started at. Thus both time has slowed and distance (space) has contracted from his point of view (of the ground situation). According to one's speed, therefore, something may appear to travel a given short distance over an apparent long time in relation to one's own local timing and measuring instruments (which however, we may recall would not be local to any observer in a distant frame!). But to himself, he appears to be moving at his normal speed, and vice versa. These extreme 'thought experiments' were useful to clarify the principles involved but their implications were much more general and significant than applied in such odd cases. The 'geometric' variables that had to adjust and were thus identified during Einstein's imagined tram journey were of course those of time and (later) space.
264. This variability of time and space (when assessed from 'afar') had of course) been the case all along but was so extremely slight at our normal speeds of everyday life that they were completely obscured. No wonder neither Galileo nor Newton suspected it. But at the extremes of speed, as with or near that of light, this variability would (in theory) become manifest and its reality more obvious. It was at such high speeds that the strange effects allowed a distant observer to see space and time proportionately shortened and slowed to maintain the expected relations as required by the peinciple of relativity. To the faster moving observer, everything would appear as normal locally. The 'adjustments' necessary were those as theoretically perceived by any observers not travelling at or near those speeds. For all laws of nature to accord with a principle of relativity, including now that of light's speed, it was necessary to accept that time and space were variable as perceived by either party of the other - and dependent on the difference in the velocities of (and distances between?) the frames concerned.
265. Presumably, when Einstein applied his new ideas to his old problem of racing with a light beam, he was now able to appreciate that just as light moved away from him at its one constant speed when he was travelling on the Earth at some very fast if generally unappreciated speed (relative to the Sun, say) even though it may feel as though he was virtually stationary, so it would continue to do so sinilarly no matter at what speed he theoretically moved (whether aware of same or not) even towards that of light. For such speeds were only relative and had no absolute meaning or value; only light's speed moved at one absolute speed! Thus, while his previously imagined speed near to that of light, if really so, would mean the principle of relativity would have been (offended), he was now able to see that it would not have been (offended) by virtue of the variations that actually apply in the associated measures of time and space (when viewed from elsewhere). When bodies move at any relative speed (as perceived by those not so moving), these adjustments still apply (even if slight at slow relative speeds) while those bodies moving along with one locally are not so adjusted, nor is time. This allows all laws of nature, including that of light's speed, to be compattible with the proper principle of relativity and to meet Galileo's original criterion of being unable to differentiate which frame is moving by the measures of any law - whether of mechanics or optics (later coming under the one umberella of electrodynamics).
266. [I still have a bit of a problem reconciling a theoretical situation in which an observer comes 'flying' past the sun at near 186,000 mps just as a giant filter lets a pulse of light through from the Sun. The observer and the pulse both race away towards the Earth at almost the same speed, one would assume, and each takes about 8 minutes to arrive there. Which gets there first ? And how does an Earth bound observer perceive this and how the fast traveller - on (a) their own local watches and (b) the others' distant watches - with the known distances concerned reducing continuously? I'm sure the answer is revealed somewhere in the foregoing!]
267. In Newton's conception, the resultant speed of any object is equal to its initial speed (ie distance travelled per unit time) from an agreed starting point, plus any additional speed given it - as, say, to a bullet fired at 500 mph from a gun out of the window of a train moving at 100 mph. The final speed of the bullet in relation to the embankment would thus be 600 mph. But because the ultimate speed of any object is that of the speed of light, this simple additive calculation has an in-built constraint. Its as thought there must be a proportional ''backup' of this constraint - even if very slight where such speeds are a very small proportion of that of light, and conversely the case at very high speeds. If the train was going at the speed of light, the bullet fired from same would not go 500 mph faster but, rather, would remain at that same speed. At slower train speeds, the bullet would manage to move at its usual speed plus a little below the speed of the train - at least as viewed from outside the train. The amount less would be a proportion of the speed of light which depends on the train's speed. That is, would depend on the ratio of v to c. Thus, even at the normal train speed of 100mph, the final speed of the bullet fired from same (as seen from eleswhere) would be its 500 mph plus some speed a little less than the boost provided by the train's speed (as, say, 99.99997382 mph) giving the final speed of 599.999 etc mph. The same constraint applies to the addition of velocities for all moving objects, whether at slow or very fast speeds. If the bullet was fired from a platform moving at just below the speed of light, its total final speed would be that latter speed plus only a minsicule amount more.
268. The adjustments which account for these effects are in the variation in the time and distance elements of all motion and speed. Thus space contracts and time slows as a proportional adjustment or accommodation to the ultimate constraint of motion - that of the speed of light. It's like an invisible ceiling below which everything must make its proportional adjustment, that distant ceiling being less constraining at the slowest speeds but increasingly so as that speed approaches that of light - the ultimate constraint - with its now much nearer ceiling - which will never adjust (upwards). If this effect acted on the motion of objects moving at any and all speed simply by virtue of some such physical downward 'pressure' (due to that rigid ceiling) and being of increasingly less effect as those speeds became less, this explanation would be quite straight forward. But this proportionality is effected by more complex influences. [This factor (the maximum possible speed) as a constraining factor appears to be introduced into the explanations of relativity as a kind of parallel if important adjunct and not as an element that seems to evolve naturally within the argument. It must relate to the symmetry between the perception of speeds of relatively moving frames of reference, neither actually being truly stationary and both of equal 'merit'; there is no absolute, only relative, motion.]
On Time - 2: Based on Albrecht Folsing's Book.
269. Like Kaku, Folsing mentions that it was probably on some uncertain date in May 1905 that Einstein visited his good friend Michel Besso to again discuss his perennial problem with him. They apparently discussed every facet of it but could still not resolve it. But unlike Kaku, Folsing doesn't then mention that on the way home, Einstein had his imaginary journey on the tram during which he gained some crucial insight into the significance of TIME in the resolution of 'the problem'. Instead, he appears to suggest that it was only later that particular evening, at home, that the role of this important factor suddenly materialised, probably based in part on his previous discussions with Besso, and the subsequent storm broke loose in his mind. He does point out however that Einstein later gave various, not always consistent, accounts of that important day. The imagined tram journey certainly fits well into the account of him gaining his vital insight about then (seemingly after his acceptance that the answer to his problem also lay through previously considering the principle of relativity). [Ie Information about time must be by signals (eg of light) which necessarily take time themselves; they're not instantaneous. And any two reference frames involved are symmetric with neither having precedence.]
270. In any case, while Kaku discusses the problem confronting Einstein primarily in terms of the imagined race with a light beam first considered by Einstein in 1895, aged 16, and the contradictory predictions this threw up as between Newton's and Maxwell's theories (ie this was essentially 'the problem' (then), Folsing approaches the matter in terms of a wider and more detailed background. In this, we see that Einstein had considered the matter (the problem?) from all points of view during the period 1900-1905 when he had vascillated from a pro-ether, pro-variability for light's speed orientation to the very opposite, especially latterly - ie in his quest to resolve 'the problem'. Many discussions ensued with his friends in Berne about this difficulty around 1903-05 including latterly that evening with Michel Besso. And then, that next morning, he greeted Besso with his famous "Thank you, I've completely solved the problem: An analysis of the concept of TIME was the solution. Time cannot be absolutely defined (ie as if there was some ultimate 'real' time that we try to discover and which applies everywhere equally)..."..and there is an inseparable relation between (various different local) time(s) and signal velocity". Such 'different' times would presumably be those on symmetrically* opposed frames of reference both of which are deemed to move equally in relation to the other rather than one being taken as absolutely still and only the other considered to be moving in some absolute sense in relation to it (with only one time or the other deemed to be the 'real', true time as a consequence). [*See discussion of the 'asymmetry' problem below and its apparent relevance to both the principle of relativity and to the variability of Time (and Space).]
271. One immediately thinks of the soon to be ever-present ratio of v^{2}/c^{2} in the new transformation equations as confirmation of this. There is thus no 'real' time to which we can always adjust our local time - by simply being aware that such as light signals from same always entail some (generally slight) 'lag' in our knowing any such 'real' time. We can however make adjustments vis a vis any local times of others based on differently moving frames of reference by considering what proportion such differences in speed are in relation to the speed of the signals concerned, as, typically, the conveniently constant/reliable speed of light. Just as speed per se of any moving body must adapt proportionally to the upper limit of possible speed, so its constituents, including time, must do so also. Thus, time (and distance) must be relative to (dependent on) that ratio of v to c.
272. [Note: Einstein equally gave his (related) analysis of simultaneity credit for revealing the answer! Did Einstein's "Thank you..." imply that Besso had contributed that vital 'symmetry' aspect and its crucial implication of the principle of relativity that there was only relative motion and no absolute rest as a reference criterion? Both frames were equally pertinent and neither can take precedence. Time was relative (to difference in one's speed etc.) In any case, replacing absolute, unvaring values of time and space (in some calculation pertaining to that elusive 'problem') with relative ones, solved it. [Also vital seemingly was the recognition of the mutual equivalence and non-precedence of the two or more systems involved in any assessment of motion.] He would later write it all up in terms of a different conception of the problem and thus a different exposition of the answer - one that addressed more specifically that convenient, more economic conception of the problem: that is, in terms of the apparent but incorrect incompatibility between the principle of relativity and the constancy of the speed if light - two principles of laws of nature that somehow had both to be upheld and proven to be mutually compatible (as he eventually appreciated). Something would have to give to accomplish this. Their seeming incompatibility may have been only implicitly recognised by Einstein initially as he sought to manipulate various elements of his problem - before this became more explicit and he could see what had to be done and to find a way to justify 'doing it'! 273. [We may also note here that even the fastest possible signal - light - must entail a 'lag' between an event and the registering of the time of its occurrence by an observer by means of a clock located at the event. There are no 'instantaneous' signals available. Einstein would later stress that there can be no model of the consistent laws by which we, as observers, understand nature that are not based upon the limitations of our actual experience and perceptions. There isn't a more 'real' nature - of superior validity - behind or inaccessible to our own experience. His theory would thus rest confidently upon the limitations of that experience as regards the reality and replicability of our perceptions of local time and space and their utility in understanding and predictably controlling real nature. It is this consistent predictability that justifies this confident stance; his ideas, thus based, proved always to work in practice; they represented our reality, not just a kind of inevitable illusion. But they seemingly did so in terms of 'cross-framework comparisons' only and presumably not within one's own local environment in which so much of our lives unfold. Or did Einstein's new conceptions now apply somehow to everything - ie universally?? This may eventually be discussed - if only ultimately.]
274. Next, Folsing reviews the concept of Time per se (as formerly conceived) as a preliminary to this latter discovery by Einstein (re its relativity) - ie that it was necessary to raise the deeply held assumption of its absolute nature from the unconscious/subconscious to the conscious (ie in order to get one to think about it and see its possible fallacy). Newton's view is first described: 'Time is absolute' - but there is nothing to measure it against; it is based on faith and is actually quite arbitrary. Everyday or 'common' time on the other hand is relative and is measured (not always accurately) by various methods based on regular motions in the universe. But the concept of true (absolute) time allowed of the idea of there being a 'now' that was somehow the same throughout the universe. But Einstein was influenced by reading Mach who said absolute time was only a metaphysical concept of no scientific value. And from his study of Poincare, he saw that while Poincare thought it a convenient convention, in 'La Science et L'Hypothesis' (much read by Einstein and his Berne colleagues apparently), Poincare was more definite that neither absolute time nor simultaneity made sense. There was no real physical basis for assuming that there could be just one (simultaneous) 'now' at different places. [It seems to me that there can be such a simultaneous instant - everywhere (in the 'life' of the universe - but that there is simply no way this can ever be be confirmed.]
275. While it might well sound reasonable, it was based as much on hope as anything. [But to say there was no 'now' everywhere' seems just as arbitrary and hopeful??] Poincare had earlier written this up in an essay entitled 'The Measure of Time' which was published in 1898 in a philosophical journal nor read by most physicists. In it, importantly, he concluded that before time could be objectively measured, it would have to be more rationally defined - and not in some arbitrary or optimistic manner but in a way that accorded with the simplest forms of the relevant laws of nature in which time played a part. An operational definition based upon our actual experience. It is suggested by Folsing that Einstein very likely obtained a copy of that essay. But because Lorentz had introduced the concept of a 'local time' (t'= (t-vx/c2)) - but as a mathematical device only - the idea of Newton's absolute, true time still remained in his unconscious and so still informed his final theory.
276. Poincare provided a physical interpretation of Lorentz's local time in his outline of a method of synchronizing clocks by light signals (the same procedure that Einstein would use later as a vital element in constructing his relativity theory). Such time was that which would apply in referential systems moving at a velocity v relative to the fixed ether. The clocks so regulated were thus believed by Poincare, like Lorentz, to not represent true time but only this/these local time(s) (as a device) to make the maths come out right. They thus remained within the conceptual framework of the mechanical (ether based) model of Lorentz's theory despite Poincare's ideas being so close to those that Einstein would perfect within the year - after realizing that this 'local time' was in fact the appropriate (and only) time available; there was no other, more 'real', absolute (non-local) time - 'out there'. The synchronizing method revealed the limitations of attempted true (absolute) simultaneity. The two local times - differing according (relative) to differences in the velocity of separated inertial systems - were the real times for each! These earlier readings of Poincare seemed to have prepared Einstein to question the subconscious, unspoken assumptions generally held about an erroneously accepted absolute concept of Time. Thus, he may have said at some point "Let's assume that time may in fact not be absolute...but...?relative...but to what; in what circumstances?". He had had many such 'circumstances' and 'arrangements' moving about in his mind for years in which all the variables pertaining to his problem, including the speed of light, frames of reference, principles of relativity (quite recently), asymmetry problems and moving bodies were being manipulated in various ways. Somewhere in this morass, the ideas of some kind of non-absolute time and space must eventually have fallen into place as the only solution.
277. While the source or origin of Einstein's 'stroke of genius' regarding Time was not revealed in his famous 1905 paper on Electrodynamics (nor in his 1916 book for the layman), he apparently did touch upon it fairly directly in late 1907 in a publication called the 'Jahrbuch der Radioaktivitat'. "It turned out, surprisingly, he said, that it was only necessary to (re-)define the time concept precisely enough to overcome 'the difficulty' (ie to which the supposed absolute nature of time had contributed) - ie the incompatibility issue centred on the constancy of the speed of light as a law of nature which should therefore comply with the universal dictates of the (later elaborated) principle of relativity. For that concept of absolute time had allowed light's constancy to remain incompatible with the principle of relativity. All it needed was the realization (based on what!?) that the auxiliary term introduced by Lorentz called 'local time' should be defined as the Time, pure and simple". From this firmer base, it was possible to calculate any distant, moving time by appropriate equations. This would somehow overcome (at least mathematically) the problem of the requirements of the principle of relativity not being otherwise met in Lorentz's theory. Ironically, was it not Lorentz's method also (and earlier) simply to choose the length and time values 'precisely enough' to account just for Michelson's failure' ?? For Einstein, 'precisely enough' were the values calculated for time and space that just allowed the velocities of any moving body (including light) effectively to fit proportionally into the available space of possible velocities. [Note: the qualification 'effectively' is significant here; the exact 'mechanism' by which this process comes about is actually more precisely described in terms of the necessary time delay in conveying the information required. See later.]
278. But rather than just 'using' this device to this end, Einstein realized that the quantitative adjustment it made related precisely to the effect that the difference in velocity of the reference systems concerned (as a proportion of the speed of light) had on the magnitude of time's actual real measurement from a given frame of reference. By holding the speed of light to its one allowed constant speed even if perceived/measured from elsewhere when it had the supposed (but actually non-existent) boost from a moving origin/frame, the value of time (and space) had to adjust proportionally. If that adjusted time was applied, then the true constancy of light's speed would equally be concluded. [This may have been based on the variables used by Michelson in 1887, including the velocity of the Earth and the distances the light covered, etc.] Time was thus defined to always be of the (?malleable) magnitudes needed to maintain the one possible value of c whatever were the magnitude of v (and thus of the variables x, y, z and t (or x', y', z', t') involved - thus serving to overcome the problem exactly in any and all situations - ie generally. He could later derive a similar adjustment regarding the equally 'malleable' contraction of space from this new conception of Time. Both would contribute (or be made malleable) equally. [But these new conceptions simply state what the new values of time and space must be (on this basis); they don't explain why or how they come to be such !! What is the mechanism ? Symmetric perceptual anomalies or...??]
279. With no absolute time (or space), the problem regarding measures of light's speed proving inconsistent with our traditional measures of the (much slower) speed of all other bodies when observed from differently-moving platforms - ie via Galilean transformations associated with the original principle of relativity - disappears when we realize that those latter speeds were wrongly premised on absolute time and distance. They should (apparently) have been based instead on variable such measures - as implicit in the new transformations (as Lorentz advanced on different premises) in which Einstein realized time and space must be variable in order for light's speed to accord with the proper principle of relativity - one based on the variable time and space that was always the case (ie when viewing 'faster'-moving frames - as we can't assume their times and distances are the same as our local ones). We must 'adjust' them in terms of the constraints imposed by the constancy of light's speed and on the time that information of the other's time takes to arrive at the other reference frame.]
280. Einstein thus 'saw' that the time as seen locally was the only valid, reliable time available and that, because of the symmetry of relative motion, any other local times as perceived by those in other frames of reference were just as valid there - there being no other 'ultimate, absolute or 'real' time - or space - (that better governed some kind of 'real' absolute motion and speed) - 'somewhere out there' - in terms of which those local times and distances might somehow be more appropriately interpreted. It was always 'only relative'. By 1916, he offered another view: "...the difficulty was due to the arbitrary nature of the basic kinematic concepts..." (ie time and space). That is, it was due to simply assuming (arbitrarily) that they were absolute in all circumstances - based on some kind of unquestioned 'face validity' or even faith. As in the preceding explanation, he replaced this with a non-arbitrary, defined concept - one which he could justify on more rational grounds. He once also said that he found his solution (ie on that May evening in 1905) "..by means of an analysis of the time concept" (and ideas re symmetry - each coordinate system being of equal merit and validity regarding the operations of all natural laws). And finally, in 1924, he added that "By means of a revision of the concept of simultaneity (and thus of time) into a 'shapeable' (or 'malleable') form (by stipulating that the speed of light must be taken as a constant), I arrived at the theory of relativity".
281. The terms 'shapable' and 'malleable' are of course not very scientific or mathematical but one may assume that they signify that the time on the clocks, so defined, were in effect dependent on the relative speeds of their coordinate systems as perceived/measured/recorded from outside themselves while the time taken for light to travel any given distance must always remain the same. With that set time having to 'hold fast' (at its extreme end of the motion continuum), all other times for the lesser speeds of all other moving bodies would necessarily be perceived as of proportionally varying magnitudes; that is, between that fastest possible speed (or shortest time per distance) for light and, at the other extreme, the slowest speeds (and longest times per distance) for all other moving (or non-moving?) bodies (as proportions of the speed of light) as percived in their own frames of reference. They were thus 'variable' and relative - to those speed differences, if any, and so depended on or from which 'base' one was perceiving one's own or others' speeds of moving bodies.
282. The magnitudes of time and of space are thus assumed to vary according to the respective velocity differences of the inertial systems (bases) concerned (vis a vis that of light); they may be thought of as being thus 'moulded' into either very short or long magnitudes, say, so depending - ie as being 'malleable' - at least as viewed by those in the other reference system, not in one's own. Equally, as he has said himself 'all he had to do was select the 'local time' of Lorentz which best fitted (was 'shaped to') the needs of the transformation equations - for they somehow were appropriate for both of the contrasting purposes for which they had been derived. I believe, Einstein would later say that this was because they both reflected the same underlying theory of electrodynamics arising out of Maxwell theory. He already had such a local time of course - as arose from his 'insight'. The extent to which time and space would be affected quantitatively were, it seems, suggested by the values previously concluded by Lorentz. But anything more precise and revealing was never stated by Einstein. Thus time and space become 'malleable' to allow the speed of light to remain 'rigid'. (Or was it to keep the moving bodies 'rigid' and so not 'really' contract and something comparable re time not 'really' dilating - only appearing (to some to do so?)
283. [Note: In his 1920 book (see below), Einstein makes a case that by putting the question as to what answer would one need in order that the incompatibility is exactly neutralised (ie such that the extent of the velocity difference between the two relevant frames no longer provides an invalid result when new transformations so determined are applied) and implies that the answer is readily derived thereby. I think that the path to this conclusion was, at least initially, less straightforward than this might suggest. Apparently, he found that the Lorentz transformations provided him with the correct answers (?ca 1903) and all he had to do was find a more reasonable basis for the new variable forms of time and space that must apparently result! He thus had the quantitative answer he needed but not yet the qualitative basis for it (eg the lag in signals as concluded from his tram journey, etc)? The (?perceived) quantities based on that would presumably depend on the difference in the speeds of the two reference systems involved as a proportion of the maximum possible speed (that of light). Lorentz would have arrived at values that fitted his analysis of Michelson's results but would have generalized these for all situations presumably and Einstein would have had this as a model to arrive at the same values but from his more realistic (?valid) conception of what was really the basis of such transformations.
284. We can thus only surmise that the crucial breakthrough came after his discussions with Besso about symmetry coupled with his imagined tram journey later that same evening. Folsing suggests the following may have been relevant to his discussions with Besso: "They likely had before them one of Poincare's papers about the synchronization of clocks producing the equivalent of Lorentz's 'local time'. Both his 1904 paper on' The State and Future of Mathematical Physics' as read in St Louis, and his related contribution to Lorentz's 'Festschrift', were known by Einstein. This procedure may well have included some aspect not utilised by Poincare that Einstein saw as relevant to his analysis (lacking an ether and accepting c). Being by this date sceptical about absolute time, they may thus have focused on Poincare's (?dilated) 'local time' as derived from his procedure in synchronizing separated clocks (and being virtually the same idea as advanced by Lorentz) and examined it for any validity as a replacement for the so called absolute, true time that increasingly they couldn't accept. While this would give a different time for every inertial system, such a result would mean that the constancy of the velocity of light for any observer (however moving) would be inherent in Poincare's definition of simultaneity rather than be more 'forcibly' accounted for or brought about by the use of Lorentz's unreal mathematical device or adjustment. To consider it as such a replacement, they (or Einstein on his own that evening) must have become more convinced about the validity of each observer's perception of the world being as meaningful and valid as that of those moving relatively to the other(s) - with both (all) being equally valid. That is, no one had primacy, nor therefore did any one frame of reference. Local time was the only time available on which to base a science of nature. Valid conclusions about the laws of nature and its outcomes could only be based upon such defined local times - varied as needed whenever unknown times elsewhere needed to be calculated (as seemingly they would often have to be).
285. Lorentz had of course also introduced the idea of a contraction in the length of any body travelling (in that direction) through the ether. This too was required to make the results of such as Michelson's experiment account of the seeming constancy of light (when, I believe, they still believed it should vary in that arrangement). [If they didn't still believe this, I need another means to account for why they justified such contraction.] The fruitfulness of what Folsing calls Einstein's 'exceedingly daring idea' about Time (it was after all, still an hypothesis, as were other aspects of his theory) seems to have occurred to him later that evening when he 'easily derived Lorentz's assumed contraction effect from his newly revised concept of time without any further assumptions' (ie about the influence of charged particles, etc). [Clearly, as with the modification of Time, Einstein appreciated that there was always something about the constancy of the speed of light that, to meet the dictates of the principle of relativity, required both Time and Space to 'adjust' (be relative to....) - but not in the ways which Lorentz had suggested - as dictated by his insistence on a fixed ether, absolute concepts of time and space and apparently some variability in light's speed vis a vis an ether. But in Einstein's conception, the bodies were taken as being 'rigid' and didn't 'shrink in any physical sense. Length and distance, like time, were relative concepts and their perceived (and 'real') magnitudes would be dependent on other factors (of perception?).]
286. He was thus now able to provide the exact transformations for both Time and Space needed to account for the different results arising in different inertial systems, with light's speed proving constant and an upper limit of any velocity. Of more technical relevance, he would (says Folsing) have probably then examined the behaviour of the Maxwell-Lorentz equations under his new transformations. Would they be invariant despite light's constancy? Yes! And finally, he found that another independent hypothesis introduced by Lorentz to meet certain mathematical outcomes - the Lorentz force' - also resulted when his new transformations were applied. His new theory thus accounted for a more comprehensive principle of relativity - that accommodated the constancy of light as a valid law of nature, for Maxwellian theory and for Lorentz's otherwise poorly-founded (if arithmetically accurate) transformations. Everything came together perfectly. Einstein told colleagues that "my joy was indescribable". A truly new electrodynamics (indeed physics) had been born that night in May 1905. He wrote it all up during June that year and, according to his son Hans later, he then went to bed for a fortnight! It was published in September 1905 and gradually filtered through the international physics community over the next two to three years.
287. Folsing continues by introducing some of the factors that were associated with Einstein's sudden insight into the importance of a better definition of time to replace the long unquestioned and rather arbitrary one of the absolute time (and space) asserted by Newton). After this fairly general introduction to these factors (which continues below), he later goes into greater detail as to their apparent timing and inter-connections. Having (1) discounted any role for the ether around 1903/4, and (2) accepting about that same time that the speed of light was after all probably an invariable constant, Einstein seems to have then focused on (3) the role that some sought after general principle (possibly with its (4) symmetry implications) may play in (5) his new ideas about simultaneity and time (following "one simply had to re-define time to equate to Lorentz's local time"). It may also have helped overcome the need for a fixed reference system as provided for other theorists by the assumed ether, which he had abandoned. In any case, any new conception of time would resolve 'the problem' to the extent that he appreciated that it was a problem because such considerations had always used absolute, unvarying values for time and space. These must have entered into his calculations regarding the seeming incompatibilty as between a constant speed for light and the more general application of the principle of relativity to encompass that apparent truth.
288. His new ideas on Time appear to have arisen in part out of Poincare addressing the problem of Lorentz having no rational explanation about the need to 'adjust' a local time measure in his earlier analysis. Poincare approached this matter in terms of an analysis of simultaneity of an event as perceived by those in different frames of reference (I believe). [But why?] The demands of the principle of relativity may have been a consideration as well. For he and Besso had been discussing a related problem regarding the asymmetry of current induction which related in turn to ideas about a fixed reference system (as the ether). That principle proclaimed there was no such special, fixed platform; the view from each was of equivalent validity. He must have soon realised that the view he would have of the 'frozen' town clock time on his imagined tram journey would be the same as any ground-based observer would have of Einstein's own clock time as he sped away at such a speed. For two such observers to establish some agreed (or defined) common time at which some event occurred - ie 'simultaneously' (as just when the clock hands showed them both that it was 11.00 o'clock or how long some event took), they would have to exchange information as to their own respective times and could only do so by sending signals which informed each other about these. As can be appreciated however, if they are not only at different locations but are moving relative to one another, this becomes very difficult. It must have been at this point that Einstein considered Lorentz's idea of a 'local time' as well as Poincare ideas on simultaneity and his reference to a principle of relativity; those on differently moving platforms would have equivalent or symmetrical perceptions of each others position. Several such inter-connected ideas were thus coming into play at about the same time. As Einstein said later "a torrent of ideas were unleashed in my mind" after that imagined tram journey had triggered the ideas about the apparent relativity of time (and space) and so broke the 'log jam'. His step 5 conclusion then had a crucial 'mechanism' - as a step 6 - by which such a necessary new conception of the magnitudes of time (as so concluded) could actually come about (be accounted for) - by some reasonable mechanism.
289. From Galileo's original conception of the principle of relativity (by whatever name it was then called), Einstein knew that it was not possible by observing natural phenomena (ie pre-light speed considerations) in a closed-off moving frame of reference to determine whether one was (apparently) moving or not (ie relative to something apparently stationary) or moving the slower or faster. Hence, in theory, the observer on the tram and the one on the ground could be equally viewed as the one moving away from the other at some enormous speed. Moreover, even if one was moving at near the speed of light relative to the other, all natural phenomena should continue to appear as normal since the speed, if smooth and uniform, has according to the principle of relativity (and its inertial basis) no effects on outcomes of such activities occuring in either environment. [But would this apply also to such activities if they concerned the one thing in the universe that didn't manifest a change in its measured effects in response to any such activities? That is, the speed of light. If one had an hypothesis that said that this one motion would not be affected, how would the principle of relativity still hold and should it necessarily always hold? In cross-system comparisons, the velocity of the 'others' system wouldn't - just for light - be added to that of light, to calculate the final outcomes. 'Neutralising' the extent of this 'failure' - ie to match the traditional expectations of all other such motions - required something 'to give'; ie for time (and distance (velocity) to be 'shaped' to fit the dictates of that unyielding speed of light.]
290. In order that the timing of any event in either frame of reference make sense in their respective environments, it was thus necessary to accept the 'local time' of each which may well not be the same (one just doesn't know and we can't, said Einstein, assume they are, as Newton claimed). Both are equally valid and neither can be viewed as more of an 'artifact' than the other. The concept of simultaneity (and rime) must therefore be 'malleable' enough to accommodate the reality of both situations with equal validity. There is thus no one 'absolute' time - only the respective and real 'local times'. So time must be defined accordingly - as 'relative' to the speeds each reference frame is moving in relation to (and observed by?) any others. Local, relative, malleable times are thus the real times, not any absolute time (which doesn't exist) and this 'adjustment' would turn out to be one of the the variables which 'must give'. The local time of each coordinate system is an isolated reality to itself and can not be communicated to 'elsewhere' except by time-requiring signals - albeit very fast and reliable ones such as light.
291. Time (as we have already reiterated above) is not absolute and constant in nature - always the same everywhere as formerly believed - but only exists as a human conception for one's needs - whether locally or at a distance and is thus relative (to one's speed) and so variable. It was this variability that allowed the awkward constancy of light's speed to prove compatible with the dictates of the principle of relativity [these should be set out here or nearby again]; ie by always interpreting the latter in terms of the variability, not constancy (absoluteness), of time and space. This would (and always had) apply to all moving bodies, and the more so, the faster they moved relative to any comparison frame. [We could ask, therefore, what time Newton would say it was in Berne if he had accompanied Einstein on his tram journey but hadn't known what the time was when they left there? Or, vice versa. Would he say that time was or was not dependent on signal velocity when comparing times on differently moving frames of reference? And how would he establish if two events in different locations, possibly moving (or measured by those) at different speeds, occurred at the same instant (ie be simultaneous)?
292. Einstein's conclusions were still in the realm of theory and were assumed to be revealed only through the amplification possible in an imagined speed near that of light. At slower speeds, such effects (in 'cross-system' comparisons) would still be the case (thus the term 'traditionally' above) but proportionally very much less; they would never have become apparent with normal measuring equipment and a lack of concern about the anomalies of the behaviour of light. Other tests of the theory would nevertheless still have to be applied, later. [This seems to anticipate contraction, time dilation, etc.]
293. The relevance of the matter of simultaneity is shown in a discussion about the problem of establishing that two events that occur at two different locations do so at the same instant (or not) - such 'sameness' being judged/determined by ...? [And/or that one event occurs at a mutually agreed common time.] To establish this, it is necessary to synchronise the local clocks at the two places. When properly understood, this is a problem that must lead to a profound change in a practical definition of time (which in turn is basic to appreciating how light can meet the requirements of the principle of relativity without the need for a fixed ether medium or reference system). To effect such synchrony, it is useful to use the fastest possible signals and, to be reliable and valid, they should also be of constant speed between the two locations. To Einstein, light alone fulfilled these criteria. [To see details about this, it may be necessary to read that part of Einstein's paper, as Folsing doesn't seem to cover this; although see later. One wishes to see more clearly how analysing the simultaneity issue contributes to understanding the relativity of time per se and why it was thought best (or only) approached from that particular point of view.]
294. It may be useful to switch about here from Folsing's analysis of Einstein's ideas on Time to an exposition by Einstein himself of similar material - as written for the lay reader in his 1920 book - in its Sections 8 and 9. [This is covered as well in paragraphs 434-440 below and may thus be removed unless it contains anything considered more useful.] In this, his procedure to synchronize clocks in a frame of reference 'at rest' again entails the use of light signals between locations A and B, but instead of stipulating that the time for same will be the same as it takes for a return journey back to B, he does so for two halves of the distance - from A to a mid-point C and from B back to that same mid-point. His frame of reference in this latter case is a straight railway embankment 'at rest' on the (albeit moving) Earth. By this means, as in the earlier version, he thereby 'sets' the clocks at B (and C) to be synchronous with that at A. The time of any event is then given reliably by the reading of the clock in its locality near any of these points. Local time (on the Earth) has thus been so defined. Our interpretation of the events of our experience must rely on such defined times; there are no other, more valid, 'real', universal times out there to utilize.
295. He continues this setting but will include a train travelling at a constant velocity v along the rails parallel and immediately next to that embankment (as though they're virtually the same line). Events that occur on the train will be in relation to the moving train as its reference frame. The definition of simultaneity and of time can be established just as for the embankment. He then asks whether two events which occur simultaneously at points A and B as viewed on the embankment (as defined above) are also simultaneous when viewed relative to the train (ie by those moving with it)? We imagine that when the events at A and B occur, the mid-point between them at C on the embankment corresponds with the mid-point C' on the moving train (for just that instant), as do those of A and B on the train and embankment. While the light from A and B travel towards the stationary observer at C at the same speed and thus reach him at the same moment (do we ignore the Earth's speed vis a vis the Sun (say) for this reasoning?), observer at C' is moving with the train towards B and away from A. Thus the light from B reaches him before that from A does. The events at A and B are thus seen as simultaneous by observer at C on the embankment but are not seen as such by the observer at C' on the train. To him, the event at B occurred before that at A. [But can he not make due allowance for these factors (assuming he knows them)? Possibly, but is that not what is now taken care of by the new transformation equations?]
296. Thus events that are simultaneous in relation to one reference system may not be (or appear?) simultaneous with respect to a different reference system with which it is in relative motion and vice versa. This is called the relativity of simultaneity. Given that there is no absolutely stationary reference frame (as a still ether), there can be no absolute simultaneity - only that due to equal reference systems in relative motion - with neither having any means of taking precedence over the other. The reality and primacy of each is symmetrical and equal. This is a very important point. Either can be equally considered as moving or as 'still' with respect to all other meaningful physical processes - neither is in fact really still; both are moving relatively to one another and to all other frames of reference in the universe (eg the Sun, the Earth, the stars, etc.) and they to ours. We have to build up our model of reality - of the nature of the universe - from our only available, differently-moving reference systems - with laws that apply validly and equally in all (ie symmetrically). Thus, every reference frame has its own particular time and without knowing the frame to which physical events refer and its relative motion, there is no meaning to statements about the time of that event - there being no universal time of one constant magnitude. It is always of some variable magnitude depending on the relative motion of its reference. The same applies to dimensions over space in the direction of the motion concerned. [Does this mean that even on one's own frame, time and space can not be considered effectively constant or 'absolute'...or maybe 'almost'?]
297. The significance of the foregoing conclusion is that it explains why discarding the concept of an absolute time and accepting instead the idea of a natural definition of relative simultaneity (and of relative time), provided the means by which the seeming conflict between the principle of relativity and of the constancy of the speed of light disappears. They are, by means of a relative time (and space) now mutually compatible and thus both valid, given this reality of there being only relative time and space - ?always dependent on differences in the speed of the reference frame of the observer/measurer and that in or from which the body concerned moves.
298. This completes our summary of Kaku's and Folsing's interpretations of Einstein's ideas on Time (with some comment plus Einstein's simultaneity discussion). A more general account of the 1905 paper by Folsing, which may provide a different slant on certain aspects of the Kinematic Part in particular can be found at paragraphs 350 to 390 below. We now address the 2nd Part of Einstein's 1905 paper - on Electrodynamics (which Folsing may or may not also cover in that general account later).
299. In Part II of the main 1905 paper, Einstein essentially shows how the kinematic laws of his theory as deduced in Part I above (some say the major contribution) apply - not only to the mechano-dynamics of moving bodies but in particular to their electro-dynamics - in Sections 6 to 10. Where classical mechanics had in the past arrived at certain conclusions with respect to the motion of fast moving bodies such as electrons, ions, electromagnetic forces and even light itself, it did so under the tacit assumption that their respective velocities were of potentially unlimited magnitudes, as the time and space measures underlying same were assumed to be essentially absolute. But the theory of special relativity has shown that such conclusions should now be re-calculated under conditions in which the velocities (v) concerned must be recognised as being proportionally constrained in their potential magnitudes and more correctly represented in terms of the ratio v/c. This would reveal various new conclusions in those aspects of physics in which the theory would prove relevant. Part II on Electrodynamics would thereby lead us (via considerations of such as light's Aberration, the Doppler effect and Electron motion), to the concepts of momentum, mass and energy as now seen in relativistic rather than classical terms. Such considerations would in turn also lead to the famous equation E = mc^{2}. In this regard, we shall examine Einstein's subsequent, shorter paper of 1905 (on Inertia and Energy) - which follows directly from the latter sections on Electrodynamics. (His 1911 paper (on Gravitation and Light) will also be considered - as an apparent preliminary to his more General theory.) We shall then consider Einstein's 1920 book on Relativity - both Special and General - written essentially for the lay reader. That covering special relativity will, amongst other things, provide a less technical overview of the same Electrodynamic aspects which underlie the derivation of that important equation concerning Energy and Mass.
Section 6 On the Relativity of Electromagnetic Field Equations.
300. In this first section of Part II, two related and fundamental topics concerning the dependence of the effects on electric and magnetic forces of the state of motion of the relevant system of coordinates are covered - viz: The Transformation of the Maxwell-Hertz Equations for Empty Space. and The Nature of the Electromotive Forces Occuring in a Magnetic Field During Motion.
The phenomena of Doppler's principle, of Bradley's abberation and the dynamics of the slowly accelerated electron are then subsequently covered in the following sections (7 to 10) of this Part - presumably because the effects of special relativity on these phenomena relate ultimately to such effects on the more fundamental conditions of the electromagnetic fields and forces concerned - as covered in this present section. [We may note that while Einstein's paper essentially concerns 'moving bodies', this section deals essentailly with the wave-like 'field forces' which are involved in the velocity of light which may be considered as one such 'body' (ie as any moving phenomena). For light was eventually conclude to have both wave and particle-like properties in that it was emitted in discrete if minute packets of waves (photons) - which are almost instantly accelerated up to the speed c. While considered as a body of present concern, other forces (and associated energies) would be similarly involved and required in accelerating other very small bodies (as electrons and ions) of various mass values up to slightly lesser speeds.
Einstein first considers in this present section how the six electrodynamic field equations of Maxwell-Hertz (which pertain to the inter-actions of the independent electric and magnetic forces which constitute such fields) may be interpreted in terms now of the special theory. They are considered in the first instance to hold true in their original form for a stationary system K (when relativity considerations are less germane) - where X, Y, Z denote the vector of the electric force and L, M, N the same for the magnetic force; ie as shown as:
1/c x dM/dt = dZ/dx - dX/dz ; 1/c x dZ/dt = dM/dx - dL/dy ; and 1/c x dN/dt = dX/dy - dY/dx
301. But, if and because we may now wish to refer the electromagnetic processes considered to a different system K' of coordinates moving relative to K, with a velocity v, we must then apply the transformations of special relativity (which recognised the role of the ratio of v/c rather than v alone). The six equations then become (rather more complex...but at least simplified by replacing the term 1 / sq rt 1 - v^{2}/c^{2} (by which v is effectively so transformed) with the symbol B - as follows:
302. The principle of relativity requires that if the Maxwell equations for empty space (ie a vacuum in which an electrodynamic field is assumed to have been established) hold good in any system K, then, by the principle of relativity, they must also hold good in systems K', K'', etc that are moving uniformly relative to system K (or indeed vice versa). This is apparently tantamount to saying that 'the vectors of the electrical and magnetic forces - x', y', z' and l', m', n' - of the moving system K' [which are defined by their potential ponderomotive effects on electric charge or magnetic masses, respectively (just as they were themselves instigated/engendered by relevant sources in stars, flames, incandescent filaments, etc)], satisfy the following equations:
303. It is then concluded that the two systems of equations found for moving system K' must express exactly the same things since both systems are equivalent to the Maxwell ones for system K. By further algebraic manipulations and certain assumptions, Einstein concludes that the various vectors of one system (eg K) relate to those of the other system if not identically (as in the cases of x' = x and l' = l where velocity considerations are apparently not relevant) then as functions of the other in terms of the usual ratio v/c of the transformations (rather than just v), where such a consideration is relevant).
304. How to interpret the meaning of these 'force-vector' equations involved in the electric and magnetic fields which reflect the measurement and perception of actual electromagnetic effects (as light?) from the point of view of a moving reference frame ? To answer this, Einstein suggests that (theoretically) we 'let a point charge of electricity (as an electron?) have the magnitude 'one' when measured in stationary system K. As such, it would exert a force of one dyne upon an equal quantity of electricity at a distance of one cm therein. By the principle of relativity this charge would/should also be of the magniitude 'one' when measured in (from the viewpoint of) the moving system K'. If this quantity of electricity is at rest relative to system K then by definition the vector x, y, z is equal to the force acting upon it. If the quantity of electricity is at rest relative to moving system K' at the relevant instant, then the force then acting upon it, measured in the moving system, would be equal to the vector x', y', z'. The first 3 equations above would thus represent the following: 'If a 'point-body' of unit charge is in motion in an electromagnetic field, there will act upon it both its own electric force and an 'electromotive force' which is equal to the vector product of the velocity of the charge and the magnetic force, divided by the velocity of light.'
305. Einstein also re-states this interpretation of the 3 equations in a seemingly more contemporary (ca 1905) form thus: 'If a point-body' of unit charge is in motion in an electromagnetic field, the force acting upon it is equal to the electric force which is present in the locality of the charge which force we ascertain by transformation of the field to a system of co-ordinates at rest relative to the electric charge.' He notes that 'the analogy' holds equally with respect to the 'magnetomotive forces' (which are presumably always associated with the electromotive ones). [The electric and magnetic forces engender (and re-engender) one another after being instigated at some source but whether either must 'come first' (thus implying a kind of precedence for that one and thus a kind of dependent consequence (albeit inevitable associate) status for the other, I'm unaware.] Einstein notes that the electromotive (and magnetomotive?) force(s) play(s) 'merely' the part(s) of (an) auxilliary concept(s) in the developed theory which owes its (their) introduction to the circumstance that electric and magnetic forces are dependent upon (and do not exist independently of) the state of motion of the system of coordinates. Finally, he refers back to the opening statement of his 1905 paper here when noting that the asymmetry questioned there in regard to the current produced by the relative motion of a magnet and/or a conductor is now further clarified and questions as to the 'seat' of electrodynamic motive forces do not now arise.'
Section 7. On the theories of Doppler's Effect and Bradley's Aberration.
306. Einstein is now in a position to re-consider the actions of electromagnetic waves on certain known if not always fully understood phenomena, if and where relativity transformations now be appreciated as relevant. Thus, if we imagine that in the stationary system K as described in 6 above there is a distant source of electromagnetic waves (eg as star light) and that the system's coordinates' origin is itself in an isolated part of space, the force vectors (X, Y, Z and L, M, N) (which seem to act as a form of momentum on some object - to which its energy may be transferred) define the amplitude of those waves and may be represented by the following 6 equations:
where: theta = velocity(w x [t-1/c(lx + my + nz] and l, m, n represent the direction cosines of the associated wave-normals).
307. What then is their constitution (form) as viewed instead by an observer at rest in a moving system K' ? By applying the transformation equations concluded in 6 above for the electric and magnetic forces, and those found in Section 3 earlier (see paragraphs ..... which should be reviewed in detail), we would obtain directly the six transformed equations:
where theta' = w'[t - 1/c(l'x' + m'y' + n'x')] and w' = wB(1 - lv/c), l' = l-v/c/1-lv/c, m' = m/B(1=lv/c and n' = n/B(1-lv/c).
308. From the above equation for the K' velocity component w', it follows that if an observer is moving with velocity v relative to an infinitely distant source of light (as a star) of frequency f:v, that frequency - as perceived by the observer (f:v') - is given by the equation:
(if, that is, we assume that that the observer is moving such that the line connecting the source to the observer makes an angle theta with the direction of the moving observer referred to a system of coordinates which is at rest relative to the source of light). This equation then represents the Doppler principle (effect) for any velocities whatever. When the value of theta is 0, the equation simplifies to:
309. [The Doppler effect is of course experienced when any wave form (as due to sound or light) approaches an observer (or vice versa) at a given velocity such that the time taken for the successive waves to reach thr observer (or measuring instrument) is increasingly reduced accordingly. This results in the perceived/measured frequency of the waves continuing to increase from the level emitted at its source. If the source moves away from the observer (or vice versa) the converse effect on the frequency is experienced.] The precise extent of these effects when so viewed/measured will depend upon the v to c ratio (as shown above) rather than just on v alone (as assumed in the past presumably) - if we accept that the observer is in a differently moving reference frame than the source of the waves. We may note, says Einstein, that when f:v = -c, f:v' = infinity. (This is apparently in contrast with the more customary view).
Bradley's Aberration.
310. If we call the angle between the wave-normal (direction of the light ray) in the moving system and the connecting line between source and observer 'theta', the equation for l' assumes the form
This equation expresses 'the law of aberration' in its most general form. If theta = 1/2 pi, the equation becomes simply:
[Aberration occurs when the (?transverse) motion of an observer on Earth seeks to establish the position of a distant star by means of a telescope when that position is effectively 'offset' by virtue of the starlight concerned striking the approaching wall of the telescope before it reaches the observer's retina and is thus 'misperceived'. This may be corrected by adjusting the angle of the telescope as required.]
311. In any case, we still have to find the amplitude of the waves, as they appear in the moving system. If we call the amplitude of the electric or magnetic force A or A', respectively (depending on whether it is measured in the stationary or in the moving system), we obtain:
If theta = 0, this simplifies to:
Einstein adds that we may deduce from the above that 'to an observer approaching the source of light at the speed of light (c), that source must appear of infinite intensity'. Why he points this out here (and has considered both the Doppler effect and Aberration above, I trust will be revealed subsequently.
Section 8. a. Transformation of the Energy of Light Rays. and b. Theory of the Pressure of Radiation Exerted on Perfect Reflectors.
a. 312. If the wave amplitude is divided by 8 pi (ie by about 25) it will equal the energy of the light per unit of its volume in the stationary system. By the principle of relativity, the amplitude as viewed in the moving system would thus be A'^{2}/8 pi. The ratio of these two amplitudes A'^{2}/A^{2} would reflect that of the energy of light in the two systems - moving and stationary - for a given volume of light. But, is the volume the same in the two situations ? Apparently not. Light from a central source spreads out in all directions equally such that at its periphery it effectively forms an expanding spherical surface - like a fast swelling balloon. In the stationary system K, no light energy passes through that outer boundary (surface) - since it expands at the speed of light (c). Such a surface (R^{2})would thus permanently enclose the same light complex, with its original energy. We may enquire as to the situation when viewed in the moving system K' - when the shape of the expanding 'sphere' would in fact become an ellipsoid. The equation for its surface (R'^{2}) now becomes much more complex than that for the true sphere - with full representation of the ratio of v/c now incorporated. If S is the volume of the sphere, and S' that of the ellipsoid, then by a simple calculation we find that the ratio of these two volumes (and surface areas?)
We then consider the light energies E and E' enclosed within the two expanding surfaces (as viewed/measured within the stationary and moving systems, respectively. Their ratio (E'/E) would equal A'^{2}S'/A^{2} and this in turn equals:
E'/E = sq rt 1 - v/c / 1 + v/c (if theta = 0 ).
We may note that, remarkably, the frequency (see above) and the energy of a light complex thus vary according to the motion of the observer - in accordance with the same quantitative law.
b. 313. In this sub-section, light waves, as those described in Section 7 above, are considered to be reflected from a perfect reflecting surface (mirror) in order to determine the 'pressure of such light' on that surface and the direction, frequency and intensity of same. Why we seek this, at this point, is not addressed seemingly. The light to be reflected has the characteristics (or properties) of amplitude (A), frequency (f) and incidence theta - as referred to system K. When viewed from system K' the corresponding quantities are:
For the reflected light, referring the process to system K', we obtain
Finally, by transforming back to the stationary system K, we obtain for the reflected light 3 rather complex equations for these basic characteristics (variables) of our light rays: ie their Amplitude, Incidence and Frequency:
[these to come if required]
The energy (as measured in the stationary system K) which is incident upon a unit area of the mirror in unit time is thereby calculated to be:
the energy leaving per unit of mirror surface per unit time is then:
314. The difference between these two levels of energy represents, by the principle of relativity, the work done by the pressure of light per unit time. If we take this work as being equal to the product of the pressure of the light (P) and its frequency (f) - ie work = P.f - then the pressure of light is:
To a first approximation and in agreement with experiment, this equates to
Einstein notes that 'all problems in the optics of moving bodies can be solved by the method shown above. What is essential is that the electric and magnetic 'force' of the light (its ?pressure...or ?momentum) which is influenced by a moving body (this needs clarifying; how influenced?) be transformed into a system of coordinates at rest relative to the moving body (ie moving with it). By this means, all problems in the optics of moving bodies (including Doppler and Aberration seemingly) will be reduced to a series of problems in the optics of stationary bodies'.
Section 9. Transformation of the Maxwell-Hertz Equations when Convection Currents are taken into account.
. 315. This section begins with another set of the famous 6 field equations - again in calculus format - pertaining to the force vectors X, Y, Z and L, M, N - with the time (t), speed of light (c) and coordinates x, y and z all represented, as are the density of the electricity (p) and the velocity (u) of the charges concerned. Einstein notes that if we imagine the electric charges to be invariably coupled to small rigid bodies (as ions and electrons), then these 6 equations represent the electromagnetic basis of the Lorentzian electrodynamics and optics of moving bodies. If such equations are valid in a stationary system K, they may be transformed with the assistance of the transformation equations given in sections 3 and 6 to the moving system K'. They are then represented by another set of 6 field equations (in calculus form) with ample representation of our usual relativity ratio v^{2}c^{2} and its cousin u.v/c^{2} where, from the theorem of the addition of velocities, the vectors u = the velocity of the electric charge (and p, which is also relevant here, = the density of the electricity), as shown above. With the vectors u representing the velocities of the electric charge when measured in system K', we have the proof, states Einstein, that on the basis of our kinematic principles (of relativity) the electrodynamic foundations of Lorentz's theory of the electrodynamics of moving bodies is in agreement with the principle of relativity [and thus, I imagine, that theory is equally (and probably better) accounted for (ie with fewer ad hoc hypotheses) by Einstein's theory of special relativity (and thus probably better represents reality].
316. Finally, Einstein adds that an important law may be deduced from the foregoing equations. This is that 'when an electrically charged particle in motion in space has a given charge when regarded (viewed/measured) from its (own) moving system K', that charge does not change if and when regarded from a stationary system K but remains constant.
Section 10. Dynamics of the Slowly Accelerated Electron. (#1)
317. In this section, we consider the motion of an electrically charged particle (an electron) within an electromagnetic field. Such an electron may be assumed to move, states Einstein, according to a law of motion based on the following 3 equations:
where x, y, z denote the spatial coordinates (position) of the electron, m equals its mass - as long as its motion is slow (but how defined?) - and where eX, eY, eZ are the components of the ponderomotive force (e) acting on the electron when viewed from within its own coordinate system (K). [We may note that these equations prove comparable to that for the momentum (p) of any body of mass m moving at velocity v in relation to such axes. In the present case, the body concerned (the electron) is first subjected to such forces and would thereby have the equivalent energy and momentum transferred to it.] Secondly, we assume that its velocity during a given (brief) period is v. What then is the law of motion that governs its movement in the immediately following instants of time ? We may assume that the electron's position at the moment we first condider its motion to be at the origin of its x, y, z coordinates and that it then moves at velocity v along the X axis of stationary system K. At the moment when t = 0, the electron may (?equally) be considered to be at rest at x, y, z relative to a moving system K' which is in parallel motion with the same velocity v, along that same axis X. With these assumptions, in combination with the principle of relativity, the electron would move over the immediately ensuing moments (of small values of t) - as viewed from system K' - according now to the 3 equations
in which the symbols x', y', z' and X', Y', Z' refer to moving system K'. If, further, we decide that when t = x = y = z = 0 then t' = x' = y' = z' = 0, so that the transformation equations of sections 3 and 6 above hold good (and where B is shown to equal 1 x sq rt [1 - v^{2}/c^{2}], we would then have
[Note: We seem to have obtained somewhere the value here c^{2} isolated from its usual companion of v^{2} when considering the transformed time (t') element for moving system K'. We may note that the ratio v/c is relevant to the transformation of the velocity values of all moving bodies which pertain to their parameters of present interest and that the elements of same (v and c) are normally only squared and immediately square rooted in order to eliminate negative values - and bringing this ratio back towards a vaue of 1/1 or a large fraction of same (as 0.8/1, say). In what circumstances in our discussion of the velocity (and later the mass and energy parameters) of a moving body such as an electron might the value of c become relevant where it is not in ratio with the body's velocity (as v/c - after the squaring is necessarily removed) but becomes utilised instead on its own and still in its squared form - as the denominator of the value of the time (t) the electron takes to move along the x dimension - when calculating the transformed value of that time (t') - ie by means of the fraction: t - v.x / c^{2} ?? After all, this would produce an exceedingly minute value (for this particular ratio) in so far as the denominator has a value of 90,000,000,000 !! It would appear that v^{2} over c^{2}] is somehow replaced here by t - v.x / c^{2}], with t - v.x somehow representing or effectively replacing v^{2}. Is it large enough to roughly balance that enormity of c^{2} ? But, in any case, what happened to the square root sign normally associated with v^{2}(or even t - v.x) / c^{2} ? Or was it never involved earlier in this particular algebraic process ??]
318. With the help of these equations, we may in any case, states Einstein, transform the equations of motion from moving system K' (?back) to unmoving system K [do we not usually do this the other way round?] and thereby obtain a set of 3 equations of electron motion for unmoving system K - as:
which may be referred to as set A. We now inquire as to the mass of the moving electron - with respect to both its longitudinal and transverse aspects. To this end, we may re-write the equations of motion of set A in the following form
where eX', eY', eZ' are the transferred components of the ponderomotive force (e) acting on the electron when viewed in the system moving with the electron at its speed. Such a force could, notes Einstein, be (?theoretically) measured by a spring balance at rest in that moving system. If we call this force 'the force acting on the electron' and use the formula force = mass x acceleration (in system K), we derive the two equations for its mass:
320. [One wonders why this particular body (an electron) requires two equations to represent its mass ?] In any case, with a different definition of force and acceleration, we would obtain other values for such masses. This, notes Einstein, shows that 'when comparing different 'theories of the motion of the electron', we must proceed cautiously'. [One wonders if this is implicitly directed to those who may have previously adopted Lorentz's electron theory in this regard ? However, he mentions in a footnote that this definition of force is not as advantageous as defining it in such a way (as Planck noted), that the laws of momentum and energy assume their simplest forms.] He also mentions that the above conclusions about the electron's mass would also apply validly to a ponderable material point if the latter acquires an electric charge, no matter how small (as it would then effectively constitute an electron?).
321. He next considers the kinetic energy of the moving electron. If an electron can be considered as initially at rest at the origin of its coordinates in system K, and is subjectd to an electrostatic force (X) along the x axis, it will acquire kinetic energy of motion to this value of S eXdx (where S = the sum of the integrals). This amount (as withdrawn from the electrostatic force/field) will exactly equal the subsequent kinetic energy of motion (W) gained by (and now held within the mass of) the moving electron since, by being only slowly accelerated to some velocity v from its initial state of rest, it would not give off any energy as radiation (heat/light), as it apparently would otherwise. During the whole process of the electron's motion being considered here, the first of the equations of its motion as shown in set A apply (when calculating its acquired kinetic energy). [This appears to show the velocity of such motion along the x coordinate (ie d^{2}x/dt^{2}), by which it has gained that energy, as being a function of the electron's mass (m) (ie E = f m) in ratio with its electric charge (-e), the particular function by which these two measures produce the resulting Energy concerned being rather complex - as: -e/m.B^{3}.X where B = the transform term for velocity (1 - sq rt 1-v^{2}/c^{2}) which, amazingly, is somehow cubed !). We would then obtain an equation for that kinetic energy gained by the electron after being so 'boosted' along the x axis of its coordinate system (?K'):
Thus, as v approaches the value of c, the kinetic energy W would approach infinity in value (which it could never reach). This further supports the earlier conclusions that nothing other than light can move as fast as that (c).
[Note that in the above equation we see our first example, I believe, of the ingredients of the famous equation E = m.c^{2} - where c appears in a squared form (possibly derived from its role as shown above for the first time (in transforming the time factor) that is not simultaneously 'square rooted'). As mentioned, E = f(m) and the particular function that it is of same, is that of that mass (m) multipled by the enormous value that is c^{2} which is then multipled by a variation of the transform term B (possibly because this is a formula for kinetic, not total, energy (as hopefully will be elaborated later). Just prior to this, we have also now finally introduced the concepts of both mass and energy into this, the electrodynamic part, of the special theory of relativity (which had previously focused, in the kinematic part, on the velocity only (with the variabilty of the underlying time and space thereof) of the moving body per se; its energy and mass weren't then relevant or considered. This is further developed in his subsequent paper of 1905 - on the Mass - Energy relationship (where that elaboration will be found) and for which this present section seems to serve as an introduction. We shall later consider more specifically the derivation of Einstein's famous equation (essentially for the latent energy E_{latent} of any and all unmoving bodies - by means of a number of different approaches of which this present section may conveniently be considered as #1 - and subsequent ones as #2 (the later 1905 paper), #3, 4...#7), somewhat later.]
322. Finally in this section, Einstein lists the 3 properties of the motion of the electron which are revealed by the set of equations A - which are accessible to experimental verification. Firstly, he notes that from the 2nd equation of set A, it follows that an electric force Y and a magnetic force N have an equally strong deflective action on an electron moving at its velocity v, when Y = N(v/c). Thus, when v is a large part of c, Y and N are almost of equal value and in any case have the same effect on deflecting the path of the electron concerned. If v is rather slower than that, both the electric force and the magnetic one are weaker but their defective powers, though less, are still equal. By this means, it is possible to determine the velocity of the electron; for it would utilise the ratio of the magnetic power of deflection (A_{m}) to the electric power of deflection (A_{e}) for any velocity in that this ratio is equal to that of v/c. That is,
This relationship can be measured directly - as by means of rapidly oscillating electric and magnetic fields. [One would assume that when v = c and the ratio v/c = 1, the ratio of the electric force to the magnetic force would result in the constant c - the speed of light itself - ie a little faster than that of the electron.
323. Secondly, from the deduction concerning the kinetic energy of the electron, it follows that between the potential difference (P) traversed and the acquired velocity v of the electron there must be the relationship (with S again = the integration sign):
324. Finally, we can calculate the radius of the curve of the electron's path when a magnetic force N is present (as the only deflective force) which acts perpendicularly to the direction of the electron moving at its velocity v. From the 2nd equation again, we may obtain the equation:
These latter 3 relationships are, notes Einstein, a complete expression for the laws by which the electron must move, according to the theory (of special relativity) here advanced (and presumably contrasts with that advanced by Lorentz the year before).
325. He then concludes his main 1905 paper at this point by expressing his indebtedness to his colleague Michel Besso for his several valuable suggestions "in working at the problem here dealt with". We may now appreciate that 'the problem' so worked on is quite complex in its many ramifications but should nevertheless be capable of being analysed down to its essential core. We thus conclude that the problem worked on (and resolved) in his 1905 paper was in fact.........(to be completed)...... .
Summary and Comment on Part II.
326. Within both Parts I and II, Einstein has been concerned with 'moving bodies' - either in regard to their kinematics (Part I) or their electrodynamics (Part II). In part I, the form and type of such generally tangible bodies, albeit envisaged within a theoretical framework, was quite obvious. But in Part II, this was much less the case. We know that the relativistic effects arising from appreciation and application of the discovered relevance of the ratio of v/c, rather than using v alone, are barely measurable at slow velocities and only become significant when v is a large proportion of of c. Nevertheless, in the consideration of the motion of any moving body, the most precise analysis would now require the application of that crucial ratio. It is a pity that there seems to be no moving body of a tangible type (as a very small single piece of buckshot, say, of about 0.5 mm) whose velocity in certain circumstances might in the past have been the subject of analyses which threw up some anomaly which, when re-analysed now in terms of v/c rather than just v, produced an explanation and possibly some new discovery. This would be even better if the velocity of same was some value considerably beyond those of 'everyday' levels (ie beyond about 5000 mph, say) but yet wasn't one of those moving at something like 90% of the speed of light (or even just 10 % of same) but, rather, was nearer to, say, 1 or 2% (ie about 2 or 3 miles per second, say). But there seems to be no such intermediately-moving bodies to so consider and hence we must shift from various 'everyday speeds', pass by all those conceivably in-between (but lacking), and only consider those at quite unimaginavble ones (at say 90%+ of c) - as characterise such as various intangible electromagnetic waves, cosmic 'rays', alpha particles, electrons, ions and, of course, light itself. The concept 'body' is thereby pushed to its very limit. We thus find Einstein applying his relativistic conclusions to Maxwell's field equations in which certain field constructs may well move at velocities v approaching but not quite equaling that of c and hence the ratio v/c should now be utilised rather than just velocity v - as used by Maxwell and Hertz. Einstein thereby arrives at new conclusions and deductions which would have escaped Maxwell and his contemporaries and near contemporaries.
327. Now, it appears that Lorentz, in addition to addressing the matter of Michelson's nil result of 1887 regarding light and ether (and coming up with his explanation for same), was soon to focus his attentions onto new ideas then in circulation regarding the electron. His theory in this regard soon integrated his ideas on light and ether with those on the electron - by about 1904. Einstein, on the other hand, became concerned about the constancy of the velocity of light and its seeming incompatability with a principle of relativity for all moving 'bodies' (of which light was considered to be one). His special theory of relativity accounted for certain anomalies noted in the past in this sphere and he appears to have searched out other phenomena (minute 'bodies') that might be better (and more accurately) understood now under his new relativity conceptions. Some of these have been considered in Part II of his paper above. And the last section of that Part (section 10) thus focuses on the motion of the electron and, amongst other things, brings into consideration those elements of moving bodies purposely not considered in the kinematics of Part I, namely the energy and mass involved. This is a topic that he would pursue further in a 2nd paper which he clearly wrote up almost as soon as he had submitted his main paper and was published soon after that same year. It could be argued I believe that it was Lorentz's interests in the electron that guided Einstein in re-intepreting those new ideas in terms of his special theory (especially as the latter 'body' was one of those few whose velocity made it a prime candidate for such attentions) and in addition to simplifying and rationalising that part of his (Lorentz's) theory, seems to have led Einstein into further studies into the area of mass and energy per se - albeit for any other 'bodies' that travel nearer to that of light than those (generally more 'massive') ones that travel at more 'everyday' speeds - which will, I believe, be virtually ignored henceforth.
As Einstein's 2nd 1905 paper - on the Inertia-Energy relationship - seems to follow on fairly directly from the foregoing Sections (8 to 10) of the slightly earlier main paper, we shall therefore address its contents (as version #2) next. {We may subsequently incorporate a review of both Section 10 above (version #1), and of the following account on this topic (# 2) with later analyses of the derivation of the famous equation E = m.c^{2} (as versions #3 to 7) to which they lead. Version #3 will be based on relevant material from Section 15 of his 1920 Book on this same subject.]
328. From Einstein's later 1905 paper 'Does the Inertia of a Body Depend on its Energy Content ?' (version #2)
On the basis of the same two principles on which he developed his special theory, Einstein then deduces* an associated 'result' regarding not just the velocity of bodies, including light, but the relativity of their mass and energy - two associated concepts of any moving body not addressed in depth previously in his KInematics(although section 10 above did introduce some of these ideas). That is, where the velocity of all such 'motion' should be considered in terms of v/c and not just v, as previously, this factor should also bear upon the mass and energy features - of all such moving bodies. [One wonders why this slightly later paper wasn't titled 'Does the Mass of a Body Depend on its Energy Content?' - as inertia is I believe only ever a function of any body's mass, and both, plus energy, in turn, was generally considered to be a function of the body's velocity. However, when this question was further analysed, it appears to have lead to a conception of the magnitudes of such mass and energy being not only inter-dependent for a body in motion, but for any one that was 'at rest' (in a relative sense) as well. If Inertia, which effectively underlies the principle of relativity, depends on a body's energy, does that mean that this fundamental principle itself also so depends - that is, where a system's uniform velocity is effectively irrelevant vis a vis the different velocity of any other such system - in regard to all laws of nature regarding moving bodies considered with respect to either system ? How might the latters' energy content affect this requirement ?] [* 'Deduced' (in this first instance )as opposed to the derivations in Sections 6 to 10, or established through experiment, say.]
While a conclusion (deduction) concerning the relevance of a non-moving body was eventually arrived at, it was so by virtue of analysing initially only the usual moving bodies as perceived and measured in terms of the usual two frames of coordinates (K and K') moving relative to one another. But in this case, it would be with regard not to an electron but to 'bundled' waves of light (photons). We thus imagine a quantity of such light waves (as discrete photons) which move at an angle theta with respect to the x axis of the coordinate system x, y, z (ie K) and which possess both mass (m_{k}) (in theory at least?) and, as the latter is moving, kinetic energy (E_{k}) - as measured from that system. [We may note that when describing in our earlier considerations 'actual' bodies moving in this respect, the angle of their motion, relative to the x axis, was generally unmentioned and were assumed, one imagines, to travel parallel with (and along) that x axis - although this may or may not not have been the case with regard to the forces and waves (vs bodies) considered in the later sections of part II. But, in any case, why isn't there some explanation as to why, 'suddenly', this aspect is now a consideration ?] The mass and/or energy possessed by the body (light particles) concerned (moving relative to (ie within) its own 'unmoving' system K) when measured from a second coordinate system x',y',z' (K') movimg parallel and uniformly at velocity v relative to the first system (now symbolized as E_{k}') does not, states Einstein, equal the values m_{k} and/or E_{k} as measured relative to (within) the first system. Rather, they are particular functions of E_{k} - ones which recognizes that both the mass and the kinetic energy of such masses (including light in theory) moving at velocity v are (as with time and space) not functions of v alone but rather functions of v as a proportion of the ultimate possible velocity c (ie effectively of v/c) whenever perceived from such a differently moving viewpoint. (And hence are characterised by a delayed reception of the perceptual information regarding it/them.) That is, (in the case of Energy):
[(And, in the case of mass, it should (presumably) be, similarly, m_{k} = ........?.......).
329. These are the 'apparent' results - of the lag in receipt of the relevant information (itself conveyed at the velocity of light (c)). [But this would be learning something about properties of light (its mass and/or energy) by means of the 'information conveying role' of light itself ? Could this not become confusing when we are also considering its energy conveying (or 'mass' (?momentum') possessing) roles simultaneously ?] In any case, we recall that it is by the application of such a proper translation equation (incorporating v/c) that we can see that the principle of relativity again applies and so any effects of different uniform motion of a 2nd reference system from which any measurements (of activitiy and/or properties on the 1st system) are taken should have no net effects on the perceived outcomes - be it of velocity, mass, energy or, indeed, momentum) - after applying the requisite transformation equations which then results in calculated values for all such variables which differ from those that might have been calculated (and assumed to be correct) if the role of special relativity (via v/c) hadn't been appreciated and applied.
[Note: We should probably review Tao's step 1. about here as it would seem to apply here as much as in the earlier account about the derivaton of the transformation equations initially. Or, steps 2, 3, etc ?]
330. The above conclusion regarding the quantity of mass and/or energy as perceived and measured being dependent on one's viewpoint (which Einstein appears to refer to here as a 'principle of energy' in the latter case; one wonders why it isn't referred to similarly as a 'principle of mass' in the former case also) is then utilized when he proceeds to apply it to the extent to which mass, velocity and energy of moving bodies generally are functions of one another (in terms of v/c) - in each case recognizing that the attendant velocity (and pertinent mass and energy?) must always be considered in relation to the limitation imposed upon it by the ultimate possible velocity - that of light (c) - and presumably the relevant informational lags pertaining thereto. Thus, M = f(E,v) and E = f(m,v) [and V = f(m,c) or ?] such that these variables of a moving body are also subject to the crucial ratio of v/c. When considering such measurements from differently moving frames of reference, one can't use Galileian transformations to verify validly the classical principle of relativity with its assumptions about an unlimted velocity; only the new ('Lorentzian') equations (with their recognition of the significance of the ratio v/c)) - are suitable to verify the more general (if still 'special') principle of relativity which requires no net differences (between results obtained initially on the differently moving systems (and observation 'stations) to be so concluded. Ie diferent reference frame velocities, if uniform, 'don't matter' when measuring such parameters of a mass's motion - providing they are appropriately transformed.
331. If one then considers the energy E(s) of a stationary body* at rest in the 1st system x,y,z (when referred to that system), then the same body's energy when referred (relative) to the 2nd system x',y',z' moving at velocity v relative to the 1st, may be symbolized as E(m). We imagine the body concerned to emit light (at angle theta to the x axis of that 1st system) with an energy content = 1/2 E(e) (also called 1/2L) when measured relative to that system, and simultaneously produce an equal quantity of light (ie 1/2 E(e)) sent in the opposte direction, the body itself meanwhile continuing at rest relative to the 1st system. The above 'principle of energy' must apply to this process, says Einstein, and, because of the principle of relativity, do so with respect to both systems of coordinates - their differing uniform velocities having, as agreed, no (net) effect on any such law of nature (by definition), after the relevant transformations are applied. If we symbolize the reduced energy of the body after the emission of light as E(s-r) and E(m-r) - when measured relative to the 1st and 2nd systems, respectively - then by applying the relation given above, we obtain:
E(m) = E(m-r) + 1/2 E(e).(1-v/c x cos theta / sq rt 1-v^{2}/c^{2}) + 1/2 E(e).(1-v/c x cos theta / sq rt 1-v^{2}/c^{2}
which, by adding the two 1/2 values of the body's emitted light energy (E(e)), equates to:
By subtraction, we then obtain from the above equations:
[*Note: The non-descript 'body' referred to in the above 1905 analysis may, I believe, be likened to the free-floating box described below in account #4 by Adams (said to be based on a thought experiment by Einstein possibly presented after 1915 in which a 'photon' of light (this term not yet used in 1905, I believe) is emmitted and absorbed by opposite walls of the box/body and the energy (and mass) so transferred is similarly analysed, although no difference in uniform velocities of any coordinate systems K and K' are apparently involved, while seemingly no components of momentum feature in the present analysis but do in the case of the floating box. One will have to determine just how the essential elements of these two analyses can, presumably, arrive at the same conclusions regarding the equivalence of mass and energy (with the factor of c squated involved) but by rather different means.] The two Energy differences in our present analysis - ie E(m)-E(s) and E(m-r)-E(s-r), which occur in the last equation above, have says Einstein, simple physical meanings: They are energy values of the same one body referred respectively to the two coordinate systems which are in motion relative to each other - with the body in system x,y,z taken to be at rest (stationary) relative to it. It is clear, says Einstein, that these differences can differ from the kinetic energy Ek of the body with respect to the other moving system (x',y',z') only by an additive constant C, which depends on the choice of the arbitrary additive constants of these two sets of energies (as synbolized above). [One again wonders if all this could be equally applied to the masses concerned ?]
333. We may thus calculate the two (energy) diffences respectively to be equal to the following values:
since C does not change during the emission (or absorption) of light. The difference between the kinetic energies of the body as measured from the differently moving systems is thus seen as:
334. The kinetic energy of the body with respect to the 2nd (moving) system (x',y',z') is thus reduced as a result of its emission of light and the amount of this reduction is independent of the properties of the body. Moreover, the difference in the body's kinetic energy as so measured (Ek(m) - Ek(s)) depends on the velocity difference of the systems concerned (and associated informational lags) - which is the same as in the case of the kinetic energy of the electron. [But are the 'net' results not effectively equalized by an appropriate transformtion thus maintaining the role of the principle of relativity ?]
By neglecting magnitudes of 4th and higher orders we may, says Einstein, then accept that
and from which it follows that:
If a body emits energy E(e) in the form of radiation, its mass is reduced by the amount E(e)/c^{2} and thus we may say that 'the mass of a(n unmoving) body is a measure of its energy content'. That is, E = f(mass (and/or inertia?)) and the particular function that it is, is (somewhere and somehow)) shown to be a multiple of c^{2} which, we may again note, appears to have been separated (in a sense) from its typical accompaniment v^{2}. One should 'work through' just what this step signifies in the underlying physics (vs maths) (as to some extent we did in #1 above and intend to do with respect to the later derivations (#3, #4, etc) which, we assume, entail the same essential parameters). [Again, if one could logically approach this type of conclusion by analysing instead the masses concerned similarly, one should arrive at the same result but in terms now of 'the energy of a body being a measure of its mass (content)' - with any increase in a moving body's kinetic energy apparently reflecting an increase in its mass. But, need we differentiate the source of that energy - as between kinetic (from its velocity) and 'latent' (from its non-velocity-indluenced mass) ?? But - latent energy per se hasn't been introduced to us as yet ! And the famous equation doesn't seem to appear in this Mass-Energy paper itself although its subject is paramount to same and that paper is cited as the origin of its derivation in both #5 and 6, I believe.]
335. It is in any case, concludes Einstein, that radiation energy must convey inertia (or ?mass) between emitting and absorbing bodies. {And that the mass and inertia from such a body must likewise convey energy between bodies, if losing and gaining same). The former conclusion at least could be tested, he says, in terms of bodies with very variable energy contents such as radium salts. If they emit energy of E(e) ergs, their mass in grammes would be reduced by E_{s}(ie L)/9 x 10 (raised to the power of 20!) and any absorbing body's inertia would, amazingly, be increased to that same extent. The magnitude of the number 10 when so raised seems to me so astronomical as to have required some explanation or comment at this point - but oddly none appears. {Possibly the power should actually have read 'just' 10?] One might have assumed that energy (in ergs) in a given amount of mass (with its associated inertia) may have amounted to such an enormity (as seems implied in E = m c^{2} - where, in kms per hour, that squaring produces a remarkable velocity value of 90 billion (ie 10 zeros) kps!), but not the other way around (in respect of such an enormous reduction); although, maybe it does?). [We will return to the subject of this later 1905 paper (in the version (?re-vision) of same in section 15 of Einstein's 1920 book on Relativity, written for the layman, in which, additionally, he may approach the matter differently - after further consideration. (Other approaches to the derivation of E = m.c^{2} will also be considered (see parags 597-660+). We would also hope to explain how the magnitude of energy said to be held in a latent form in any moving or, more impressively, 'unmoving' body is not a direct function of its possible kinetic energy even though the above analysis seems to centre mainly on kinetic energy - as a route into these non-kinetic mysteries !
In any case, it appears to be the case that one would never deduce that an apparently inert block of unmoving mass 'just lying there' as it were, might contain such an enormity of energy (as represented by the equation E = m c squared) without involving in the reasoning behind same some reference to the activity of radiation or photons of light somehow pertaining to (ie being absorbed by or emmitted from) that particular mass. To approach the theoretical derivation by some such means (and later its verification empirically) was presumabbly only ever considered after 'clues' about same were suggested in terms, say, of the apparent equivalence of the laws of the conservation of energy and mass (by Einstein) based on ...(see earlier)... . Or did such clues first emerge only when he considered the relevance of relativity to such as the motion of the slow moving electron, or other topics as we have recently considered above ? Certainly, without any motion itself (in its entirety), the possible energy attributes of such an inert mass (at an immensely greater extent than through its simple combustion) - (and whether or not it was ever releasable) - would not entail consideration of its own velocity (v) per se. To discover any relationship between its mass (m) and any suspected (enormity of) 'latent' energy (E) it may conceivably possess, and the particular function between them, did necessitate a theoretical role for the addition or removal of energy of a particular form - one in which the basis of that function (a form of velocity) could be derived.
If there was a strong suspicion that there was some relationship or equivalence between mass and energy, we might consider that for a typical amount of mass - say 1 kg - that the energy equivalent for same would be some reasonable function of that value, one that would indicate an equivalence that fell somewhere within our everyday range of practical energy values - as 1 kg of mass held, say, 5 times that number of energy units (as ergs or whatever), or even 50 times. If the multiple seemed much beyond that, we could consider another, larger energy unit and suggest that 1 kg mass held just 1 such 'super-erg'. As it turned out however, that multilple proved to be 90,000,000,000 times the kilogram equivalent (!) What a unitary 'super-unit' that would be! But it didn't so 'turn out' to be such on the basis of blind empiricism - simply seeking to get some idea what it approximated by trial and error experiments. Rather, it was deduced by the theoretical and algebraic manipulation of the particular variables involved which (necessarily) included velocity values pertaining to photons (radiation) considered either to leave a mass, initially as minute 'bits' of same but which, on leaving, transformed instantly' into light energy of motion at a velocity (c) - which velocity provided the basis of that required function...and/or as radiation packets, the energy and momentum of which impacted into such a mass and similarly instantly transformed thereby into equivalent 'bits' of mass. Such theoretical predictions would only then be verified empirically by weighing relevant masses before and after such arranged transformations. But, while photons of light may well have a velocity of c, nothing has one of c squared - which turns out to be the actual function concerned, not just c itself. That is, while in a sense, m = E (and E = M) - the lstter is the case only if m is first multiplied by 90 billion! (ie by c^{2}). It is how that enormity was derived as the fucntion that we seek a straightforward answer.
The empirical fact that the 'rate of exchange' between units of mass and those of their latent Energy is as enormous as it is, indicates that the particular function between them (c^{2}) as derived theoretically is, with its equally immense magnitude, quite understandable. One should not then be perplexed therefore that the physical entity which accounts for this has the magnitude that it has. It is entirely fitting for the actuality. We simply wish to 'see' just how it emerges as that physical reality - albeit by way of algebraic operations applied to the interacting variables. If the 'bit' of mass that has the potential to transform into a photon of energy when emmitted with a velocity of c, we might assume that it then moves out in a straight line at that velocity (and certainly not at that value squared). However, we might consider for a moment that light actually spreads out in all direction so that a value of c^{2} might more reasonably represent its unavoidable dispersal over a squared surface area instead. But this would be wrong in that it would actually spreads out as an expanding sphere - ie over 3 dimensions - and thus (on this shakey basis) one would expect the multiple to be such as c^{3} ! Moreover, we may recall that the kinetic (if not latent) energy of a moving body is a function of its mass and its velocity - but with the latter again squared ! Was this determined only empirically or, as with the mass-energy relation, was it or could it be derived on theoretical terms also (even if not done firstly) - thereby allowing a physical explanation as to why, apparently, it must be a squared velocity - as, we assume, this feature can be so explained with respect to E = m.c^{2} as well ?
We do not of course 'utilize' the value c^{2} simply because it happens conveniently to be of the amazing magnitude empirically required...or because it just happens conveniently to be there at all and available - but rather because it is only in terms of this particular function that the conversion actions underlying this equivalence must and do proceed. This is how, physically, they are so related and thus why one finds the empirical result one does. And the energy so represented within mass is not there in the form of particles moving at the velocity of c (never mind at c^{2}, but as part of the binding forces within the mass's atoms and molecules which may be released as tremendous light/energy. How, physically, the maximum possible velocity in nature proves a relevant factor in such binding energies and forces, I am unaware. But we must accept that the source of the value c^{2} in the derivation can only come from the velocity shown by the photons of light which apparently must enter into any search for deriving that equation which represents the relationship between mass and its latent Energy; there is no other source for this amazing function in the theoretical analysis. Mass is a store for the energy utilised by photons of light and radiation generally - providing them with their enormous velocity, momentum and energy (for their size) - but with no mass per se. In the opposite direction, it is just the converse - with the impacting photons of energy somehow transformed (back) into those powerful binding energies/forces holding the molecules densely together in the form of mass. How many photons per quantum and/or how many quanta per atom or molecule are required, I'm unaware. We continue with further interpretations of the derivation of E = m.c^{2} at paragraph 597 (with version #4 by Adams).
336. But, in the meantime, we show next a different version of the Einstein's main 1905 paper (by Folsing) - also written for the layman - although it too might be better seen after Einstein's own 1920 version, which likely contributed to same. [Does it touch on the contents of section 10 above - with respect to Energy, mass and the beginnings of a role for the component m.c^{2} or indeed, subsequently, to the later 1905 paper on this important matter (version #3) ? We shall see:
A more General Analysis of the Main 1905 Paper: - by/as per Albrecht Folsing.
350. (new numbering (from 337.) needed) The title of Einstein's paper - 'On the Electrodynamics of Moving Bodies' - is, says Folsing, in line with the orientation of several other physics papers around that time (ca 1900-1905) concerning similar matters. That is, with investigations and/or ideas about the new electrodynamics as evolved from Faraday and Maxwell, through Michelson, Hertz and Lorentz, to Poincare and others. The 'dynamics' element in this would deal essentially with the forces that determined the motions concerned. But Einstein's paper had, despite its name, actually placed the 'kinematics' of such moving bodies as the more fundamental contribution - from which his new dynamics (and indeed a new physics) would follow as a consequence. His kinematics would be that of rigid bodies (moving in relation to reference systems of coordinates over measured space and time), as well as with that of the velocity of both weighable bodies and more tenuous electromagnetic processes therein. [Why the bodies are called 'rigid' is uncertain to me; possible he was under-lining the fact that these bodies wouldn't contract physically in any absolute sense as, in theory, did Lorentz's ? However, reference to the physics of 'rigid bodies' (in contrast to those of some elesticity?) has been noted elsewhere and seems to be a more general and recognised category; so I may be rather 'off key' here.] The title of Einstein's paper may also have relevance to Lorentz's ideas about the electrodynamic effects on bodies' lengths as they move through an alleged ether (unmoving) - something of which Einstein's electrodynamics of moving bodies explicitly denies having any need; his is thus a quite different kind of 'electrodynamics' but clearly pertains to that same general domain. The title in this respect may have been even more appropriate therefore! Folsing also refers to Einstein 'concretizing(?) the measurement of time and space'. I would have thought 'plasticizing' would have been more appropriate ?
351. Thus Kinematics in this context concerns the spatial and temporal geometry in which moving bodies move (although it is more than that) without consideration of the forces and energy needed to move bodies of given mass; thus mass itself was not yet a consideration. One of the fundamental elements of his theory as it evolved was based upon the conclusion he came to when considering the faulty asymmetry in the interpretations of Faraday's current induction methodology that had somehow become accepted during the late 19th century. He saw this as a fundamental error in the electrodynamic theory of moving bodies (his very subject) that had since evolved through Maxwell and Lorentz. For the theory to be valid and robust, it should, he felt, have reflected the actual basic symmetry - and its fundamental implications about the absence of any absolute motion or resting place - which would imply an asymmetry. It thus had an in-built error that would come down to causing difficulties in later conclusions. For it implied that in one case (if not both), there is a necessary fixed (effectively ?absolute) reference point or system. The proper symmetrical interpretation, on the other hand, reveals that no such concept is involved or required; that it is only the mutual 'relative motion' in the relevant relationship that is necessary. That is, as between the mutually and equally affecting magnetic and electric fields on which, significantly, electrodynamics is actually founded. Both elements in any comparative motion thus have equal significance, with neither having primacy or being effectively more absolute or whatever. [NB Is this latter point an answer to concerns about how general is Einstein's theory and its many implications ? All frames are in motion and are so (only) in relation to some other (equally non-fixed) reference system; there is no ultimate 'fixed' system...somewhere 'out there'. So, is there not always an implication of some 'cross-over comparison ? No! We can have motion within a single given frame as eg entirely on the moving train or on the platform, or on the Earth itself - in all of which measures may presumably be taken with no reference to other frames! Are time and space not constant therein therefore - both in appearance and actuality ? Or is the 'potential' perception from somewhere else always implicit ? This may have to be be considered further below.]
352. Einstein points out an unexpected link between this matter of the seeming asymmetry he refers to in the first sentence of his paper and that of confirming that the Earth moves relative to a fixed 'light medium' (ie the ether). But such attempts had all failed. He thus concludes, as an assumption or hypothesis, not that some 'correction adjustment' is therefore needed to overcome some kind of built-in 'masking' of the expected influence of such a still ether (eg on light), but that whether considering electrodynamics and mechanics, there is no 'still' reference system that is at absolute rest. Einstein saw that the crucial concept was that of relative motion which, as it was so fundamental, applied to all moving phenomena - whether of tangible bodies or those of light (quanta of electromagnetic waves). He thus states early in his paper that there is no need for a concept of an ether medium for light - seemingly one that is fixed and could thus provided (also) an absolute reference system. Neither existed nor were needed. They would prove superfluous. [His two postulate alone, therefore, would prove sufficient, he stressed.]
353. From this platform, Einstein moves quickly to its natural implication. He had read Poincare's reference to Galileo's principle of relativity and knew that it concerned the stability/reliability of all laws of mechanics in what were later called different inertial systems - ie ones moving with uniform, non-accelerating velocity. The valid behaviour of such laws was appropriately referred to the associated reference markers (coordinate systems) moving uniformly relative to one or more other such systems. While Newton claimed that these in turn could be adjudged ultimately against some absolute reference system (as the ether), he accepted that his laws functioned quite validly within the former systems alone. It would thus seem to follow that the constancy of the speed of light, as another law of 'nature' (if not of mechanics per se) which would also have no fixed reference system by which to operate, should therefore also prove to operate according to the dictates of this same 'principle of relativity'. Uniform speed of its reference marker should have no affect on its normal operation therein (or therefrom). Thus, not only would it not be possible to establish whether it was Galileo's ship that was moving or the dockside nearby by the results of mechanical activities (motion) in a closed room on the ship, it would also not be possible if one used experiments entailing the motion of light either (despite its unique constancy - something that did not apply to other (ie mechanical) motions). Quite simply, uniform speed of the reference system should have no affect even on laws (as electrodynamic ones) which required unchanged outcomes whatever that uniform speed might be - just as it had no effect on mechnical laws which did manifest and require changed outcomes !
354. This was referred to initially as a 'conjecture' by Einstein which he would soon raise to that of a 'postulate' which he described as a (more generalised) principle of relativity. It would be more comprehensive then Galileo's or Newton's versions - but based on the same underlying principle. [One might assume this would be in terms of the concept of inertia but he refers also (unclearly) to 'our experience' as its more general rationale; see earlier discussion; also Bondi's view.] Crucially, however, its valid application would have fundamental implications concerning the basic components of which all motion is composed - that is, on time and space (as would be explained only later). In the same sentence in which he states this first postulate, he introduces a second one regarding the constancy of the velocity of light in a vacuum - which is independent of the state of motion of the source of that light. He points out that this idea may appear incompatible with the principle of relativity (as that was originally understood for all of mechanics in Newtonian terms) but implies that he will show that it is in fact compatible - at least with an adjusted (ie generalised) basis of that principle. [NB It would be useful to show here just how and why they did appear incompatible before his analysis and thus what would have to be amended (as discovered by that analysis) to counter this. (It was, I believe, thought incompatible because it didn't allow the added (or subtracted) velocity of the reference system of its source to leave it as its law (alone) required (that is, unchanged). It had always been assumed that this was not the case with laws of mechanics whereby the altered overall velocity of any body put into motion in such a system would have its true net value revealed by the addition (or subtraction) of the velocity of that system. The problem lay in the values of the components of the velocities added (or subtracted). They had to be more 'flexible' - not only for the case of light but, to a lesser extent, for mechanics as well.)
355. It seems quite possible that after studying Maxwell's equations in university (ca 1898) and concluding that the velocity of light in a vacuum was a constant, but later having some doubts about this when influenced for a time by Lorentz's ideas, including a role for the ether (ca 1903), he finally settled upon his earlier conclusion. This second postulate was therefore likely his belief before he managed to 'fit it into' the rest of his scheme (ie only after considering the implications of the principle of relativity and his discovery of how to overcome this - by relativizing time and space; a symmetric 'lag' in times gave the same values, according to speed differenes, as Lorentz had concluded quantitatively - to allow his view to 'work out'!). Trying to 'fit it in' before this led to problems trying to overcome its inevitable conclusions when following the usual methods of empirical science, so he decided he must find a more general principle under which such empirical facts are most probably subsumed. It appears that it was Poincare's reference to the principle of relativity (but in what context?) around 1903/4 that provided him with this sought after general approach. This in turn recalled (with Besso) about 1904/5 the problem of the asymmetry of induction and the related matter of the failure of establishing the fixed ether - both of relevance to electrodynamics and thus to light, as well as to the idea of any asymmetric fixed reference systems.
356. This principle was therefore seen as increasingly likely to be appropriate, relevant and what he needed to make possible his belief in the constancy of the velocity of light. When combined with (1) his growing disbelief in absolute time and space (as per Mach and Poincare), (2) the latter's description of local time which arises when tying to synchronize clocks/establish simultaneity and (3) his own ideas arising from his imagined tram journey re Time, he was drawn inexorably towards his realization that...only relative time and space could be the case (ie relative to/dependent upon differences in the speeds of reference frames and their observers). This was what was necessary in order that the principle of relativity could allow (be compatible with) the constancy of light's velocity and its non-additivity with the velocity of its source. That is, time and space had to adjust (be variable) to fit in with this unyielding component of any final velocity. Time and space would be measured with clocks and measuring rods whose units were relevant to the inertial system in which they were perceived. Such a fundamental revision of the measurement of these basic factors had not been considered necessary by anyone else. As Folsing states: " Einstein's decisive step was the relativization of space and time". But he elaborates on this by pointing out that what he really did was correct the way these were measured - ie in terms of his re-definitions of time and space as dictated by his two postulates (beliefs). His theory can then be derived from these (4) premises. [But how does the information lag fit into all this ?]
357. His two prime postulates were not theoretical 'guesses' but related to the experiences of earlier physics. That regarding relativity was in effect inherent in the 'old mechanics' but, significantly, not in Lorentz's new 'electrodynamic mechanics (of the ether). It was effectively forced into fitting the latter by major efforts (via Lorentz's 'devices') as a generalization up to first order magnitudes of v/c. [Elaborate on this!] But Einstein nevertheless elevated it to a more general postulate (seen as a principle of nature which should co-exist with all othe principles) beyond such empirical evidence - based on his almost philosophical belief in its role as a general principle to explain uncertainties in this general area. [Possibly following his thoughts on the asymmetry and fixed ether problems and Poincare's mention of it?] The light postulate was (as I believed) inherent in Maxwell's equations and theory (at least as concluded by Einstein but possibly also Lorentz) but Folsing suggests that it was a purely empirical ?expectation and that later observation of same could as easily turn out the other way. Einstein himself admitted as much although felt it was at least probable (ironically) by virtue of the confirmations which Lorentz's theory (albeit based on the assumption of an ether at rest) received indirectly from experiment!? Direct experimental evidence was, however, not available at the beginning of the century and had to await confirmation by later predictions/deductions proving successful (which they did, invariably). In the meantime, Einstein asserted that the theory was at least supported by the fact that "..the two postulates suffice for arriving at a simple and consistent theory of the electrodynamics of moving bodies on the basis of Maxwell's theory for bodies at rest". Using the model of thermodynamics, he felt that his two principles also "could lead to reliable results" and so justified his faith in his theory. It would stand or fall as a whole rather than any of its components being directly testable.
358. The actual formulation (ie sequence of ideas) in which his theory was born is, says Folsing, scarcely known today". That is, what the exact order and timing of his logical steps were is uncertain. The various elements in the process are well known but just what led to what amongst them, and when, is a puzzle that even Einstein wasn't sure about some years later. There seemed to be an eventual 'fusing' of several ideas that he had manipulated in his mind for some years - at least after he hit upon the last pieces in the puzzle on that May evening - when everything fell into place and he 'saw' the central/general concept of it all. However, there appeared to be no clear primacy amongst the underlying elements. Analysis of his 1905 paper is probably the best way to understand the theory - if not the precise cause-effect sequence/reasoning of its genesis nor that final general determinant from which the more specific elements follow. After the Introduction (which I've analysed elsewhere), it is the early sections on Kinematics, says Folsing, that provide the best understanding of its evolution. This begins with a discussion and explanation of 'Simultaneity' in the present context. [We may see earlier just when this seems to have been first considered. This chronological placement can also be attempted with respect to each of the other major 'elements' in the theory, as also discussed above or elsewhere - ca 1902-1905.]
359. Definition of Simultaneity. [See also earlier coverage of this topic.] The relevance of simultaneity to the solution of his problem seems to have occurred to Einstein after he read Poincare's views about it in 1904/05 - as relating to the matter of synchronizing clocks and the 'local times' this produced. He then had his own crucial thoughts about time via his 'tram journey' which, with Besso's recent ideas, must have called up similar thoughts. It would appear that both his sudden insight about Time and 'then' his elaborating this in regard to Simultaneity (or vice versa?), were necessary steps which he had to utilise to resolve his compatibilty problem - ie in terms of the relativity of time and space. He knew that Lorentz's transformations proved compatible with Maxwell's equations about the constancy of light's speed (advanced to explain something else) and that they incorporated variations in the measures of time and space, but could not accept Lorentz's explanation of such vatiations (or relativities). It was his insight into the concept of time that began to dislodge the log-jam in his mind.
360. In his 1905 paper, he sets out the theoretical setting in which he can describe more precisely the geometry/kinematics entailed when the motion of a body occurs over space and time and in which the exact time at two different locations in such spaces can be determined. Any motion of a body can then be described in terms of the 3 dimensions of Cartesian coordinates in such a system considered initially 'at rest' (relative to some other reference system) as a function of time. But then he asks the reader to consider just what is meant by 'time'. (Considering what is meant by 'at rest' will come later.) We may recall that just as there is no fixed reference point in the universe by which any absolute (non-relative) spatial measures can be taken so there is no fixed starting point or measures of duration for time from which any absolute times could be concluded. We must define both length and time relative to criteria in our own experience. All that was necessary, said Einstein, was to find the values of time and space that proved consistent with the effects of the constancy of light's speed - that being the one constant fact of nature that was a crucial, determinant part of our experience. This was what different from Lorentz's reasoning.
361. Here he seems to use Poincare's critique about time when he notes that our use of the concept of time implies an almost unconscious assumption about certain 'simultaneous events'. Thus, if some event occurs at 7 o'clock, that really means that the event occurs simultaneously with a clock which shows its relevant hands at 7 and 12. But which clock? If it's a near clock, is that same time also shown on all clocks in the universe? This could be the case - but only if the information about the time as shown on any such selected clock could be transmitted instantaneously (ie at infinite velocity) to any and all clocks elsewhere - ie by having first synchronized them all by this 'magical', instantaneous way. (Their prior synchronization can't be assumed; it would be an unverifiable assumption or hope.) [Recall about here the case of the 'flying tram' (which thus also entails simultaneity) - as well as the 'flying room' with its centred light bulb and fellow traveller; the light reaches the fore and aft wall simultaneously (as he, who is moving with the walls, sees it) but an external 'stationary' observer would see the advancing rear wall reach the light first (actually vice versa) and the retreating fore wall receive it later. Think out why.]
362. But there is of course no infinitely fast method of sending such signals. There is light of course, which is at least exceedingly fast, although still finite and, vitally, reliable. Thus, simultaneity can be valid at the location of the clock - ie as between an event nearby and the time shown on the clock; the time the light reflected off the clock takes to travel from the clock to any observer of a nearby event would be insignificant in the present context and even less at the clock itself. But when events at different locations some distance apart have to be evaluated temporally, an agreed method of assigning times at these locations is, suggests Einstein, required (in contrast to Newton who would assume that time at one location was somehow already the same as at the other). For this Einstein uses Poincare's method for synchronizing clocks: Two identical clocks at separated locations A and B are taken to indicate, respectively, 'A time' (tA) and 'B time' (tB). These two times cannot be adjudged as synchronous, says Einstein, by means of somehow discovering this condition (of an identical 'common' time between them - ie as a physical reality we assume may exist) but only by validly defining such a communality by some agreed and appropriate physical method which can be checked as required - to serve our purposes.
363. In Poincare's method, the time so established at one of these locations (say A) is apparently used to establish that same time at B (and any other locations - as C or D, etc to be so synchronized). [It thus appears that the initially unknown or uncertain time at B can not be assumed to be synchronous with that at A and this method seeks not to discover some such prior common time between that may have met this criterion but rather to define and set this synchrony by making time at B equal to that at A. It begins by noting that the 'time' needed for light to travel from A to B may be defined or stipulated as the 'time' needed for it to travel back from B to A and that, hence, the two clocks would be synchronous by definition if: tB - tA = t'B - t'A (where t'A and t'B are the times at which the light leaves A and B, respectively) AND (as it appears to me) other clocks in location B are then set to agree with the A time in accordance with: tB = 1/2(t'A + tA). Such a time at A is apparently identical with Lorentz's 'local time' there. [ Seemingly, if it was the time at B that was to be used as the defined time, the formula would be: tA = 1/2(t'B + tB) - giving a similar 'local time' there.] This definition of simultaneity (and of time based on it) thus relies upon the stipulation that the speed of light is always the same. By insisting upon this, one can proceed to perform physical experiments which ,if fulfilling their various predictions correctly, provided support or justification for this initial 'stipulation' or definition. If it were deemed to be just an hypothesis, it would be tested more directly and immediately, one assumes.
364. [Note: In Folsing's account, the synchrony appears to be established by virtue of the formula shown before the capitalised word AND alone; the following formula appears as the means of setting other clocks to the same as both the clocks (at A and B) already so synchronised. At the moment, I can't see how the first formula achieves synchrony - simply by saying that the time signals take to travel between them is the same in both directions. [Answer: it does so by taking the speed of light as always being the same. With this 'proviso', clocks so synchronized must allow an agreement as to the simultaneity of two events at distant locations - when both are not in relative motion one to the other. The word 'if' before the 2nd equation (by which the distant clock is so set to the first one), is of course required as its validity depends on the light signals used to always be of that same one speed.]
365. Einstein's choice of light as the signal used is not (presumably) because it happens to be the fastest such means (by far) but possibly because he has in mind a unique feature about light (its reliable constancy) which when applied to the similar situation in which these measurements are not taking place within one coordinate system 'at rest' but in two inertial systems moving at velocity v from each other (or from that assumed one at rest). To be applied to both situations, however, he first provides a reasonable basis - ie Maxwell theory - for positing that the velocity of light in a vacuum is a constant. Thus, his new definition of time is so based - in terms specifically of the constant speed of light. It would apparently have been useful even for Lorentz's theory, with its ether at rest, as ultimately equivalent to his 'local time'. Poincare had noted this but failed to draw the significant implications that such time, so defined, was the only 'real' time available. Before examining the implications of this new definition of time in the moving situation, Einstein first sets out his geometric 'platform' even more precisely so that he can establish the implications of light's constancy not only on time but on its inevitable concomitant in motion - space (and/or the length dimensions of bodies).
366. In the next section addressed by Folsing, he focuses on Einstein's approach to establishing the relativity of time and of space (ie the physical dimensions of bodies in motion) by considering the situation when a resting frame of reference is compared with one that is moving. He begins by describing the length of a rod in these two situations. {See also earlier account.] This is preliminary to discussing the apparent contraction of the rod when it moves (as viewed by an observer in the resting frame). But, he then quickly describes how a pair of clocks (synchronised as shown above) are attached to the ends of the rod and by this method seeks essentially to show that time (not length) is relative. He then indicates that he will next establish by similar means that length (or 'dimension') is similarly relative (having first announced that the two lengths being compared would differ whereas they would normally be expected to be the same) - but no comparable proof of its comparable relativity (and any resultant spatial contraction) appears! It is, says Folsing, as though that part of the paper went missing at the final proof stage before publication!
367. With relatively simple mathematics, however, this similar proof apparently can be so derived, even if not shown in the paper! It is as if he first announced that time dilates but doesn't show just how nor interprets this as a reflection of it being relative. [Does either one not imply the other - motion of a body being a function of both components equally? [Note: There have been countless discussions over the years regarding just how and why time and space are affected when a moving body is measured from a relatively fixed position. While Einstein provides clues as to how he arrived at the fact that these are so sffected, he doesn't analyse these in any tight, cause and effect way, relying instead on his more general reasoning that Maxwell's theory and equations 'force' such conclusions but did so in his mind only after he had gained some insight into the apparent or probable basis or reasons for same. It was as though he 'saw' the necessary reality of it but couldn't be bothered to analyse it with clarity at the detailed level.
368. The actual details by which he does at least seek to show that time is relative (as well as announcing that length of the associated rod is also affected by the same factors - in that it contracts) is somewhat tedious in its verbal exposition. A more graphic/kinematic representation would certainly help (me); to show (?state/assert) that such relativities are the case is not quite the same as explaining why this must be so. However, his account will hopefully provide some such insight. Thus - after again formulating his two key principles as precisely as possible, he begins by asking the reader to..."suppose (ie imagine) a rod at rest, measured with a measuring rod likewise at rest, and let it have the length l". Folsing continues without quoting Einstein so exactly with; 'Now suppose r the rod is put into a state of motion with a constant velocity v. Einstein asks about the length of this now moving rod and indicates that this may be established by two fundamentally different operations'. [Continues - see p 185 and on, although we have covered some of this previously as well. Does Folsing also cover the Electrodynamic sections (of Part II) of the 1905 paper...and what about the 'Inertia-Energy' paper that same year ?]
369 - ca 389. [Yet to do.]
ca 390. While one may not have understood the mathematics of every step in the development of Einstein's theory (as reviewed above thus far), we may ask ourselves whether we can at least now set out more clearly just what was the exact problem he was seeking so to answer and what was his basic answer to same (ie his 'result') - even if every step is not fully comprehended. It would be useful if this included some insight into the actual causes and mechanisms by which the conclusions of Part I could in theory come about. For, it may follow that if one insists that the generalised (vs classical) principle of relativity must accord with the motion of all bodies, including one that had a maximum and especially constant velocity, and thereby conclude that for that to be the case, the magnitudes of time and space must (appear to) be variable/relative, as described, it would certainly further such a case if some reasonable (?electrodynamic; ?perceptual) mechanism could be more clearly advanced and justified to account for such albeit logically deduced effects.
Einstein's 1920 Book on Relativity.
391. We shall instead look next at more of Einstein's own writings on the matters pertaining to both his Special and General Theory - as compiled about 10-15 years after his 1905 papers - specifically for those without mathematical expertise. Hopefully, we may gain better insight thereby into just how he addressed the matter of the non-symmetry in the unknown distances and/or times involved. Was it the actuality (not just perception) of time and space (length) that were found to be variable or do the limitations of our mutual, symmetric perceptions (our 'experience') equate to the only 'reality' we can ever know ? As noted in the article by...............in 'My Einstein', Einstein felt that his theory should be understandable by anyone even without higher mathematical expertise. So, what do we find ? Does he describe matters again in terms of perceptual 'appearance' but then drop those concepts and replace them with those implying same form of actuality ?
392. [We may note here that around 1910-15, Einstein was engaged in a lengthy correspondence with Max Abraham, a brilliant physicist who had trained with Max Planck in Berlin and held chairs at Gottingen, Milan and Aachen (in the fields of electrodynamics and electron theory) and who was quite unconvinced about the theory of relativity (of 1905-10). He said that it went against 'common sense' and did not validly describe the real physical world. If such a mind had difficulty accepting or understanding it, what chance is there for the rest of us? No wonder one keeps 'thinking aloud' and writing out such thoughts - often repeatedly - trying desperately to find a logical path through the seeming morass. I'm not myself trying to find a chink in its armour (if it had a chink, I would be most unlikely to find it, for a start); I accept that it must be correct. It is just that I would like to really understand its apparent irrefutable logic.] Thus:
Einstein wrote the Preface to his book for the lay reader: 'Relativity - The Special and the General Theory' in December 1916, having published the first version of his General theory in the scientific press just the year before (1915). The book's authorised translation was later published in English in 1920. [For simplicity, it will be referred to here mainly as 'the 1920 book'.] We analyse its contents in two parts which we have entitled thus:
1. On the Kinematics, Mechanics and Electrodynamics of Moving Bodies - (i) for the Special case of Uniform motion only: ie - The Theory of Special Relativity and
2. On the Kinematics, Mechanics and Electrodynamics of Moving Bodies - (ii) for the General case of Unrestricted forms of motion: The Theory of General Relativity. Thus:
Part I - The Theory of Special Relativity. - as per the 1920 book.
393. Unlike his original and ground-breaking papers of 1905 and 1915, the 1920 book, says Einstein, was indeed written for those without the mathematical knowledge underlying theoretical physics. Moreover, he may have felt that certain of his ideas could in retrospect be more clearly presented than previously (when the earlier paper at least was written in a kind of brain storm in 3 weeks in June 1905) - including those already touched on elsewhere concerning time and simultaneity. This may help in further analysing the kinematics of the 1905 paper so that some of its more technical aspects will hopefully become better understood. Moreover, we would hope that more of its Electrodynamics will be further explained - including the derivation of E = mc^{2} by about Sections 15 (thus providing the 3rd coverage of this aspect - after those Electrodynamic Sections 6 to 10 of Part II of the main 1905 paper and, secondly, that implied within the later (?Dec) paper of that same year - on Inertia and Energy). We begin our review of the special theory (as per the 1920 book) with the very basics of kinematics:
Section 1 - The Physical Meaning of Geometric Propositions and Section 2 - The System of Coordinates
394. In these first two sections, Einstein begins by defining the use of Cartesian coordinates to specify the position of any body. For in any discussion concerning the motion of a body (with which mechanics, optics and our related discussions of electrodynamics and relativity will essentially be concerned) consideration of how to specify its position, before, during or after such motion, is (as has been emphasised before) clearly basic; it can only move, be measured and analysed in relation to 'somewhere' and 'something' that can be reliably specified and agreed upon. These early Sections 1 to 12 are thus another, more considered approach to what he described in his 1905 article as Kinematics [The later Sections (13 to 16) consider certain implications concerning Electrodynamics I believe while Sections 17 to 30 (of Part II) will cover a similar overview of The Theory of General Relativity - also for the lay reader primarily.]
395. Thus, the coordinates of this 'somewhere' may be usefully determined in theory at least in terms of imaginary perpendicular plane surfaces but attached to a relevant objective body of reference (such as the Earth) and not to formless 'space' per se. Any event involving a moving body can then have its position(s), dimensions) and speed specified by measuring rods and clocks or their equivalent in relation to such agreed coordinates. The purpose of Mechanics is to describe how bodies subject to various forces change their positions (move) in space over time (in terms of such defined reference systems), given that terms such as space, position and time are properly understood. Empty Space has no positional reference utility itself and so some defined reference system somewhere within that 'space' is first required to establish any initial position and subsequent motion of bodies, however caused - relative to that reference.
396. Thus, if we specify a particular point on the Earth - as a given railway station and its associated embankment - we may describe the motion of a passing train (the 'body' in this case) relative to that adequately defined reference system. Because of the size and character of this particular moving 'body', we are also able to consider a smaller body moving within the train. The motion of such an 'internal' body moving within the train (as a person walking along the aisle of one of its carriages, say, or a stone dropped therein by such a passenger) can then be considered in relation to either the train itself (now a reference system itself whether moving or not rather than a body per se) or to the outside embankment typically considered as a stationary reference system. We could however consider that the embankment, relative to which the train is moving, is itself moving, afixed as it is to our fast moving Earth - with the Earth and its attachments themselves moving in relation to the Sun. And so on. For reasons which will become clear, the conclusions which Einstein will be arriving at in his exposition on relativity will require frequent reference to situations in which the moving body concerned will be one which can be considered in relation to both its own immediate local environment (as the person walking inside the train, say, or the stone dropped therein) and to the reference system specified initially for assessing the motion of the larger body itself (here the train) - ie the embankment or station.
397. We may thus consider the Earth as a body moving relative to the Sun, or the train as a body moving relative to the embankment, or the walking man as a body moving relative to the train. In each such case we have a body moving in relation to its most reasonably pertinent local reference system (even if each is itself moving in relation to something else). And while it may be convenient to assume a particular perceptual stance in each case as to which is viewed as the moving body and which its pertinent ('stationary') reference system (often viewed as 'still' or relatively so), in all such cases there is no 'absolute' motion or stillness of any of these reference systems. In each case, the motion concerned can be conceived to be as much the reference system itself (as a 'body' of interest) moving relative to the body seen previously as the moving one (but now as the relatively still reference), as vice versa; the mathematics involved would be identical in the two cases. To both the bodies and reference systems concerned, the motions involved are totally relative and mutual and they are in a sense quite neutral as to which is deemed to be moving relative to the other. The three examples described would generally represent bodies seen as moving relative to their most pertinent 'local' environments.
398. [We can usefully refer to such as 'direct' body-to-reference motions (or vice versa). Thus, if we label the person walking on the train as 'A', the train as 'B' and the embankment as'C', we can describe the motion of the train relative to the embankment as 'B re C', that of the person relative to the train as 'A re B', and that of the person relative to the embankment as 'A re C', although the latter motion actually entails 'A+B re C'. The motions of 'A re B' and 'B re C' are thus of the 'direct' (or 'simple') type while that of 'A re C' (ie 'A+B re C') is a more 'indirect' (or 'complex') type. It will become increasingly apparent that in his exposition on relativity theory, it will be these latter, indirect types of motion of the bodies concerned which will often prove most relevant. (In all examples, the motions described are deemed to be smooth (uniform) and straight, as will be emphasised later).]
Section 3. - Space and Time in Classical Mechanics
399. As implied, rather than a person walking on the train, the moving body concerned could be a stone dropped on or from the moving train, say; this may be perceived by someone travelling on the train as dropping straight down - ie in relation to the train at least - but when perceived by someone on the nearby embankment, it would appear, relative to that local embankment, to drop in a parabolic curve. The walking person on the train could equally be viewed from the perspective of the embankment - that is, as an indirect or complex motion in our shorthand terminology. In each case, the precise mathematical description of these motions would assume a set of coordinates rigidly attached to the train or to the embankment, respectively, as required. As stated above, the path of the stone or person can not be meaningfully described in terms of empty space alone, with no defined frames of reference, be it the train or, as in this latter case, the embankment. One or the other (or even both) have to be 'nominated' or specified for any meaningful analysis.
400. But such motions for a complete description also require a measure of the times at which, for example, the stone is located at each succeeding point in its fall (or person during their walk) since such positional status of any moving body can be further differentiated in terms of its speed. This may be achieved in terms of the times noted for these successive positions on synchronized identical clocks observed by the relevant observer(s) - in relation to the agreed reference frame. For the moment, the exceedingly slight times taken for the light signals (with their information) to travel from the stone (or the person) and from the clock - to the observer's eyes may be ignored as the distances and times concerned are very short. But, later in our account, we shall point out that such 'light-conveyed information' will play an exceedingly important role in this theory, and in understanding it.
Section 4. - The Galileian System of Coordinates
401. Einstein further defines such Cartesian coordinate reference systems - as Galilean - when their state of motion is such (ie uniform) that the law of inertia, the fundamental law of mechanics, applies to them. The laws of Mechanics are valid only for Galilean systems of coordinates premised on that law. In such a theoretical Cartesian system of coordinates, one clock is usually assumed to be positioned at the origin of the 3 coordinates and the other (ideally synchronised with another at the location of some event) - in order to establish the motion and velocities of such bodies - as viewed within that system. [More here?]
Section 5. - The Principle of Relativity
402. This important principle is then defined, firstly in what he describes as its 'restricted' sense (ie as it had always applied to mechanics in the past), without providing any particular rationale for its introduction at this particular point. Thus, any body (as the person) which is travelling uniformly in a straight line with respect to a given coordinate system (K) - ie the embankment - is also so moving relative to any other coordinate system (K') (as the train) that may itself be considered moving uniformly in a straight line relative to that first given system K. By this definition, all such reference systems are described as Galilean and so all mechanical laws should work identically in all of them, their different relative uniform motions having (because of inertia) no influence on their valid operations (just as they all work consistently on our fast-moving Earth). The results of mechanical experiments in any such environment cannot of themselves therefore inform one in which such environment they have transpired. As neither has any 'absolute' motion associated with them (which might be thought to somehow allow a different or more valid set of laws to apply) than any of the others; they are effectively equivalent.
403. This is the classical principle of relativity for mechanics. Mechanical laws work exactly the same whatever the different relative velocities may be of systems of reference on which they occur or from which they are are measured. It was first expounded by Galileo in terms of comparing activities like pouring water from a bottle into a container (or bouncing a ball up and down) while on a smoothly moving ship with the same things done on the relatively 'still' dockside nearby. While their respective relative motions (if uniform) have no effect on the mechanics applied therein - ie when assessed in relation to their respective own immediate environments (eg 'A re B'), if assessed in relation to the other ('B')'s reference point (on 'C'), the results remain the same - 'once the extent of the other's motion (which was always assumed to be totally additive with the motion of the body concerned) is subtracted (ie 'A-B') (or in some cicumstances added as the direction of any comparison is totally symmetric); such corrections were effected by means of what were later termed 'Galilean transformation equations'. In terms of our foregoing descriptions of determining the extent of a body's 'direct' and 'indirect' motions (vis a vis their relevant reference systems), such transformations are only required and of relevance for the complex/indirect cases.
404. The principle of relativity originated within Galileo's early analysis of mechanics (ca 1625) partly in an attempt to convince the Church in Rome that the Earth wasn't the still centre of the universe. If the Earth actually moved at many miles per hour (around the Sun) as Copernicus and Galileo claimed, all our activities on Earth should be virtually impossible, argued the Church, as everything would be thrown all over the place. However, Galileo showed that, on the contrary, on a moving ship (representing the moving Earth) all normal activities can and do proceed as per normal (if the motion is smooth and uniform) and that this is exactly what happens on the Earth. The principle concerned wasn't given a specific name at that time, nor even later when it formed an equally important part of Newton's improved explanation of mechanics by 1690. And during the 18th century, it was probably a fully accepted principle somewhere in the background but wasn't particularly quoted or invoked to help explain currently studied mechanical phenomena. This seems to have remained the case even after Lorentz and Einstein first began analyzing their own respective interests in the motion of light (but would soon re-appear when these studies required deeper analysis).
405. Thus, in his present exposition, Einstein asks whether this principle should not apply to the motion of all natural phenomena (eg including the motion of light)? It apparently was implicitly assumed to do so - as long as 'all natural phenomena' involving motion (as of bodies, objects, particles, waves, etc), were deemed to be a part of classical mechanics, which they traditionally were, and obey its various known laws including that of inertia and the relativity principle. It seems to have taken its place in later accounts of his theory by Einstein (as above) only in retrospect - once its possible role throughout nature was more fully appreciated. It was however not a part of his thinking before about 1903 - apparently. And when it was finally (re-)considered, possibly after being mentioned by Poincare, it was likely assumed to apply to all natural phenomena deemed to be a part of mechanics. At first, this included light and optics generally. But, at some point, Einstein, accepting that light was probably not a part of mechanics as it was generally understood, considered that this important and fundamental principle should nevertheless somehow still apply to light (despite one very awkward property of it) - whether deemed to be a part of mechanics or not - on the basis that it was nevertheless a part of nature and of the laws of nature - of which mechanics were after all only a part. That is, that this principle was even more fundamantal than realized and thus should apply to all motion - including any that might be more validly described as being outside mechanics (for some reason).
406. This conclusion seems to have coincided with his decision to seek some general principle in terms of which that odd property of light (the constancy of its velocity in a vacuum) could be explained within the constraints of a less restricted principle of relativity. Once he had analysed an adjusted interpretation of the principle of relativity, he must have decided that in any later explanation of his theory, this important principle was itself the more general principle he was seeking. It should therefore be introduced early on in his explanation (partly to provide the historical context) but as though its relevance was recognized and its more general conception had actually applied 'seamlessly' to his developing theory from the start; indeed that it was the very foundation of it, rather than that it could facilitate a more logical post hoc exposition of it (once so appreciated). To this end, Einstein then introduces the idea that "...in view of more recent developments in electrodynamics and optics (eg Maxwell's advances in understanding the basis and velocity of light), it became more and more evident that classical mechanics [with its associated, if restricted, principle of relativity] did not in fact afford a sufficient foundation for the physical description of all natural phenomena" (ie even if a 'neccessary' one). [Specifically, such mechanics (with its still unqualified relativity principle) wasn't consistent with one aspect in particular of natural phenomena about which our understanding was still evolving - namely, that apparent constancy of the speed of one (and only one) 'motional' phenomenon in nature - namely, that of light (and all other electro-magnetic waves). The laws governing the motion of bodies before this were based on the consistent fact that the velocity of all such bodies would be varied in predictable ways if boosted or retarded by a faster or slower moving reference system when measured thereon from a neutral point.]
407. The 'recent developments' referred to would presumably be those reported during the 1880s and '90s mainly - after which it must have been gradually concluded (by ca 1903, say) - that the usual mechanics afforded an insufficient basis as an acceptable description for the behaviour noted and/or predicted for light. He continues: "At this juncture, the question of the validity [and ?relevance] of the existing principle of relativity (for all motion) became ripe for discussion..." [and, one might add, 'ripe for rather deep thought and solitary consideration, although one wonders just when Einstein at least first asked about the validity (or genealisation) of this particular principle and if so, its relevance to his long-considered problem pertaining to the constancy of the speed of light. One also wonders just what 'mental action' (and by whom) underlay that rather passive term 'became' (...'ripe for discussion...')?? Was it Poincare or just Einstein ? It seems that the validity of light's constancy was, on the other hand, not deemed to be equally ripe for discussion - at least by Einstein who had already accepted this .
408. He seems therefore to have decided that the rather dormant relativity principle may be the more general principle he sought to help resolve the problem as he saw it; secondly, he would soon see that if it was to apply* to light's speed (as to everything else), some adjustment (of the assumptions underlying it) would be necessary (to ensure its validity in this wider context; and thirdly, he would have to find a realistic basis, logic or justification for any such generalisation of its validity by such adjustments - both theoretical and empirical.] The possibility that this principle, if it was to apply to phenomena 'beyond mechanics' - as to light in particular (if this had yet been mooted - eg by Poincare), may not do so validly - without either it or that (light) phenomenon itself having first to undergo some such adjustment in its interpretation in view of the apparent incompatibility* between them. Such 'discussion' certainly transpired between Einstein and his good friend Besso at that time (ca 1904/05); did Lorentz and/or Poincare not also discuss/contribute to this matter about then, we may ask ? [* 'Incompatible' in that any attempted alteration in its velocity by means of a faster or slower reference system would be compromised when applying the usual transformations as these would yield a conclusion that gave a faulty net velocity for light - ie one that concluded that its speed differed from its one and only true value - and thus not in accord with the principle of relativity.
409. [*Importantly, Einstein then provides a needed justification for maintaining a crucial role for a fully valid, more generalised principle of relativity - ie to so apply (implicitly for all phenomena involving motion). Clearly, this justification (in Einstein's mind) would logically precede his confidence that it would (should) so apply. For he points out firstly that its accepted validity in the domain of mechanics, being based, amongst many things, upon such precision in predicting the actual motions of the heavenly bodies, strongly argues a priori for its validity also in any other, seemingly non-mechanical domains (such as that of light and electrodynamics generally). [Or, as Bondi said, to maintain the integrity and unity of physics generally.] Secondly, he points out that if the principle of relativity for mechanics did not also hold more generally for all such motional phenomena (including light once its inclusion was considered) then the Galilean coordinate systems K, K', K'', etc, moving uniformly relative to each other, would not be equivalent for the description of all such natural phenomena. For adding or subtracting a frame's velocity to that of light at least would not allow that 'law' to hold - something the principle of relativity apparently required. From this basis, he then reasons very carefully to show that this second 'powerful' rationale for the principle's wider validity, very probably also holds therefore for (ie applies to) all natural phenomena), including the 'law' of light's one and constant speed. Indeed, although he came up with justifications for the generalization of the principle of relativity, we may assume that it was his acceptance/belief that the velocity of light was a constant for all observers which actually took precedence and motivated his search for same (as the latter was necessarry to allow the compatability that was required].
410. By the foregoing arguments, he thus shows why it was not acceptable to seek some kind of 'special case' status (or some other explanation) for a seeming non-mechanical law of physics - that of the constancy of the speed of light - on the basis that there may be no justification to assume that any such non-mechanical law need necessarily prove compatible with the original principle of relativity - as it applied to mechanics. He accepts instead that, as argued above, all laws of physics, including that pertaining to the constancy of the speed of light, should/must prove compatible with an appropriate (fully generalised) principle of relativity. It is thus effectively a part of his theory that in fact they do. Thus, a more appropriate transformation equation was required, one which resulted in a net velocity for light which was consistent with its law. On a more practical level, Einstein also justifies a role for the principle of relativity on the grounds (relating to the asymmetry problem) with which he begins his famous 1905 paper - namely by providing examples from both mechanics and electrodynamics which indicate that there is no absolute rest or motion in our universe (in either sphere), only relative motion, and that therefore the principle of relativity must be a factor in the analysis of the motion of all phenomena, including light. This too, he points out, "..is a very powerful argument in favour of [the ?universality of] the principle of relativity..." per se. Moreover, he suggests that it is unlikely that a principle that holds with such exactness in one domain of nature (mechanics) would not do so as well in another (electrodynamics). [I would like to see a more detailed breakdown into what should be a more direct basis for the necessity/logic of this fundamental principle. Or must it remain so general and abstract/indirect ?]
411. We might consider here this problem of asymmetry further: According to Faraday (based on Oersted's discovery), when a magnet moves relative to a copper wire (conductor), an electric current is induced in the latter. It makes no difference whether it is the magnet or the conductor that moves relative to their immediate surroundings; Only their relative motion, one to the other, is of relevance to induce the effect. No fixed reference point (as an apparently fixed conductor, or magnet) is needed - if either was or could be so interpreted or arranged. However, in the Maxwell/Lorentz theory of electrodynamics (and that of others), the theoretical interpretation of such induction appears to be quite different for the two situations. In the first, if the magnet with its surrounding magnetic field is deemed to be the element that moves (relative to the other), a time-variable (?moving magnetic) field is said to exist in space which, according to Maxwell, gives rise to 'closed electric lines of force' - that is, to a real field of such force lines in the copper conductor. It is this induced ?moving field which sets in motion the movable electric masses (electrons), as electric current, within the conductor. If however the magnet (and its permanent field) is deemed to remain at rest while the conductor (with its electrons) moves across or through it, the Maxwell/Lorentz interpretation says that no electric field is created. Instead, it is assumed that a current is induced in the conductor by the fact that the electrons so moving with the conductor are subject, through that physical movement relative to the still magnetic field (like a still ether?), to an electromagnetic force (emf) that was assumed (hypothesised) by Lorentz. This asymmetrical interpretation apparently gives scope to the idea that one such element (here the magnet and its field) may be viewed as a (?necessary) fixed reference point almost in an absolute sense. But Einstein saw no need to introduce such a 'cause' of the induction in just one of the two equivalent cases - nor any fixed (?absolute) reference point - for either component. No absolute motion was required; only relative motion, he said, for either outcome (which were in any case identical) was required for this outcome to transpire reliably.
412. Both Einstein and Besso (an electrical engineer) recall being troubled by this asymmetric interpretation as generally given at their university and it apparently came up in one of their many discussions around the time of their meetings in late 1904/early 1905. [It may have been when they were (finally) considering the theory that Lorentz had been working on ca 1895-1904 in which he accounted for the real physical contraction of moving bodies in terms of the effect of their electric activity at the atomic level being affected by their (allegedly 'real') motion through an assumed fixed ether. This seemed analogous to the induction of a current where either the magnet or a coil of wire were unnecessarily deemed to be still and only the other component moving. This was in error said Einstein; neither needed to be 'still' as the motion of either, relative to theother, provided all the stimulus needed to produce the effect.] The electric industry and associated academics had apparently long assumed that if the magnet moved, an electric (not electromagnetic) force and associated current ensued in a conductor, but if it was the coils of wire in the conductor that moved around a still magnet, it was instead an emf that was produced (as the precursor to any current?).
413. Einstein found this awkward asymmetry unnecessary and could see that in all relative motion, which of two elements was deemed to move or be at rest was, effectively, merely a convention of "the choice of perceptual attitude by the observer". What mattered to the unthinking elements concerned was the relative motion as between the (potential) magnetic and electric fields regardless of the state of motion of the observer or of one or other of the two elements relative to that observer. The meaningful reality was the relative motion between the two elements jointly. No fixed reference system was needed (nor existed). [The same principle applies when a match strikes a friction surface; either or both can move to cause the ignition.]
414. It was, said Einstein, "this actual symmetry in the relative motion of induction that compelled me to postulate the theory of relativity" [seemingly vis a vis his consideration of a principle underlying the principle of relativity, as well as the equivalence of which time was the 'actual' time in different reference frames after his tram journey and simultaneity considerations)! The effects of the relative motion of the two travellers (like the magnet and copper coil) was equivalent; neither determined a more valid time or distance perception (nor current!]. The mathematics were identical for the perceptions from mutually relative frames of reference. It was as though the single fixed ether was replaced by two equivalent if differently moving reference systems and an always constant speed of light. With no ether to provide a fixed reference system, neither time nor space had a means of establishing their alleged absolutism or reality; they must be relative - ie to an observer's speed. But the actual mechanism underlying this seeming logic was presumably still to be clarified.
Section 6. - The Theorem of the Addition of Velocities in Classical Mechanics
415. Einstein then considers the situation in which the motion of a body (eg a man walking (self-propelling himself by his own force) along the aisle of a moving train - as a moving reference frame or environment - can be measured in relation to either that local system (ie our 'simple, direct' motion: 'A re B') or to another Galilean system - as the nearby embankment (our 'complex, indirect' case: 'A+B re C', as described in more general terms above). Unlike the simple/direct local situation ('A re C'), establishing that the motion of 'A' (in and of itself) in relation to 'C' (ie 'complex' case) is no different to what it is in relation to its local reference 'B', entails applying the Galilean transformations - based on the theorem of the addition (or subtraction) of the respective velocities involved. Thus, as one walks along a moving train carriage aisle at a certain pace (in the same direction as the train moves), one is walking at a faster pace (ie and over a greater distance for a given time) - at least in (the more complex) relation to the embankment outside (due to the addition of the train's (B's) velocity). The differences are calculable by means of the said transformations: The train's velocity (v) is, in this case, added to his own walking velocity (w) so that his final overall velocity (W) relative to the embankment becomes (W) = v+w . [But why does v times w enter the equation and not v + w ?] This example refers to the basic mechanical application of the principle of relativity to the motion of a body. [To repeat: 'the complex case always entails applying the Galilean transformations' if one wishes to determine the net magnitude of the body's velocity within the moving reference frame (here the train).]
416. Once adjusted by such transformations, it is found that the net results (the speed and thus distance per time at which the man walks in the two differently moving systems remain identical. The velocity of the train has not altered those net values. (Although, says Einstein, we shall see later that (surprisingly) the true values turn out to be not quite as indicated by that simple transformation. And while the true values are, at the speed of a train, only slightly different from those as calculated above (and those differences were certainly too small to have been apparent to anyone originally, nor to have been measured or verified directly if they were), the theoretical arguments to be presented will indicate that such small differences should be the result nevertheless and such conclusions may be verified if and when such subtle differences can be more precisely measured. Thus, just as the result with light would prove to be invalid, so it would even with slower moving bodies. [We may note that Einstein here anticipates the resolution of his problem in the use of the phrases 'turn out' and 'to be presented' without yet providing the reasoning as to why this will be the case. It may have been preferable if this was presented to us as it first came to him rather he already knowing it.
417. Moreover, it is interesting to consider what may have transpired had it been possible to measure empirically such subtle differences originally. By noting the successive effects as the speeds increased (and if there were, conveniently, several phenomena which moved at speeds between, say, 10,000 mph and 100,000 mp second, up to that of the constancy of light, it may have been the case that the empirical reality of an actual 'law' (vs theory) of relativity would have been necessarily concluded by this prior empitical approach - rather than having to proceed theoretically from the opposite direction, as it were, ending up by concluding theoretically that there would be those 'surprise' differences in speeds hidden there (as followed from the invalidity of summing velocities to a value greater than that of light (c) from which follows an inevitable proportionality for the perceived velocities of the moving systems for all slower moving bodies as well (as the effects are linear) - as compared to those calculated by the simpler (classical) transformations. We may again note that it would be the components of velocity - distance and time - in which these subtle variations reside (and hence in their associated velocity measures) and not directly in the latter per se.]
Section 7. - The Apparent Incompatability of the Law of the Propagation of Light with the Principle of Relativity
418. Einstein's explanation of his theory (and of that 'surprise' of the previous paragraph) then proceeds by introducing a 'consideration' that may not, he states, immediately appear to follow clearly from the foregoing. [It is quite possible that, at the start, it wasn't immediately clear to him either - ie that it so followed; but only latterly.] This concerns the apparent incompatibility of the 'law' of the constancy of the speed of light with the 'law' of the principle of relativity, at least as it had traditionally applied to mechanics (as we have touched on above). We have accepted that the former 'law' represents an (?the) important example of the major 'other natural phenomena' referred to above which, while possibly lying outside the domain of mechanics, should nevertheless be covered by that principle (as argued above) - lying as it does within the ambit of natural laws - although possibly not as it originally stood - with its implications about how the magnitudes of the components of motion had always been calculated. For the speed of light was shown by both Maxwell's equations and by the Dutch astronomer De Sitter (it seems) to not depend on the speed of reference system M with a source emmitting light - in relation to any independent point of observation. It should somehow always remain a constant, but this was incompatible with a principle of relativity which was verified by means of the original transformations.
419. Thus, when we consider a ray of light sent along the carriage aisle (rather than the man walking), its speed (assumed to be through a vacuum) can also be calculated relative to the moving train - when it would be = c, or to the stationary embankment - when we might assume (as argued in the case of the man walking along the train aisle) it would become = c+v. But this would be a value greater that c - its seeming 'boost' in speed due to the motion of its emmiting source (the moving train) relative to the bank being expected* by the principle of relativity somehow to not effect its final speed of c. But if the usual amount of M's velocity was subtracted from the value actually measured for light (ie at c only), the result (c-v) would be invalid as it would show light to be less than its constant.
[* ie 'expected' - on the basis that all laws of nature were 'expected' (and so acknowledged publicly as of some uncertain date - namely:...??) to accord with the principle of relativity. The confidence and authority for this assertion is clearly the fundamental bedrock underlying the reasoning that led to the theory of relativity. Where is this discussed in the literature?]
420. It was, essentially, in order to explain that 'somehow' that the theory of relativity was formulated (once the 'hypothesized pre-emminence' of the principle of that name was accepted, as argued earlier). [This was the 'consideration' that had to be faced - for this characteristic applied to no other known phenomenon and seemed to fly in the face of the 'demands' of the original principle of relativity - at least it had to be so faced from the moment when the relevance of those demands to the problem being addressed by Einstein first became apparent and/or acknowledged (whether by him or any others). This seems to have been around 1904. He seems not to have attacked his problem initially by being directed or guided to seek a way to resolve this incompatability (since it wasn't initially formulated by him in such terms I believe until later); rather, he kept trying (unsuccessfully) to resolve it by manipulating in his mind all the other elements concerned (light; ether, velocity, etc) until he became increasingly aware that the general principle (of relativity) that he eventually accepted must somehow be satisfied pointed more explicitly to specific factors constituting velocity in general which had to be 'adjusted', there being no others to so adjust and, in any case, they having been so adjusted similarly for rather converse reasons by Lorentz (after Fitzgerald) which quite likely gave Einstein certain clues to consider his comparable adjustments.
421. Now, the above basis for doubting that this would be the speed of light calculated in such circumstances (of the light sent along the train aisle) is not however that to be described by Einstein. Rather, he begins with the ray of light being sent not from within the train but from and along the embankment itself (again as if in a vacuum) - when the light's speed relative to same would = c (just as we found it would be when sent from a source within the moving train - relative to that moving reference). He then asks what would be the light's speed (sent along the embankment) relative to (ie as seen on) the moving train? In one sense, the latter light can be compared with the man walking in the train - ie the speeds of both can be considered relative to either the embankment or the train. In this example, the speed of the light along the embankment - but relative to the train (by applying the usual addition or subtraction of velocities) - becomes c-v, which is now less than the speed of light. This proves awkward in that the speed of light is assumed (as per Einstein's interpretation of Maxwell's and de Sitter's findings and reasoning) to remain constant - at c - in all circumstances. And as such a 'law', the principle of relativity would require it to apply equally to any two or more reference frames moving at different speeds. We should probably question the validity of such calculations therefore in that any speed accorded light - with respect to any frame of reference - that does not equate to the value of 'c' can be considered invalid virtually by definition. Indeed, rather than relying on this general acceptance regarding the expected speed of light, whatever the motion of the reference frames used (as Maxwell and De Sitter apparently indicated), Einstein approaches this matter by pointing out that this result must be wrong, rather, on the basis that it conflicts with the principle of relativity. We have already suggested this above but it may be further elaborated here. [One might enquire whether Einstein's logic here is more fundamental than that of Maxwell or de Sitter? Or are they different ways of saying the same thing??]
422. Thus, according to that principle (which as shown above, he places in a kind of sine qua non position) the speed of light, if accepted as a general law of nature (within the ambit of motion), must perform the same in relation to whatever reference body (whether that of the moving train or of the 'still' embankment) it is measured. This differs uniquely to all other natural laws pertaining to motion (eg as in mechanics) in which no constancy of basic velocity was ever assumed, thought possible or required; indeed, it was generally expected to reflect totally any such added or subtracted speed due to the velocity of the associated reference frames but, with this accounted for in any subsequent calculations, to otherwise perform as normal. That is, that while applying the transformation equations with their relevant addition or subtraction of velocities to the motion of the walking man (or to any mechanical law generally), in relation to either reference frame, did not produced any anomalous results (or at least certainly did not 'appear' to do so), applying them to the case of the 'law' of the constancy of the speed of light would (alone) do so. Adding or subtracting totally the (smooth, uniform) speed of the comparative frames of reference appeared to leave the measured results of normal mechanics of nature unaffected - as the principle of relativity required - but doing so with a feature of nature which demanded a resultant unchanged constancy of its velocity (something no other law in mechanics demanded) would lead immediately to an inconsistency with the demands of the principle of relativity - for the velocity concerned wouldn't be concluded to have remained constant as that law and, therefore, the principle of relativity, required - no matter what the velocity of its source (which shouldn't ever matter). Everything (law) should always 'work the same' - whatever the velocity of its 'platform' and from wherever observed or measured.
423. But in dealing with a rather similar problem arising in quite different circumstances, Lorentz found this could be overcome by the use of different transformation equations (in which the velocities to be added or subtracted were appropriately adjusted to exactly overcome this problem) - the values of time and space always inherent in such velocities being effectively (implicitly) altered from their otherwise constant or absolute ones (as generally assumed throughout prior history) to being in effect variable ('malleable', as Einstein would refer to them) instead. And it was seemingly this same awkward constancy of light's speed (even if unrecognized as such) to which Lorentz decided he had to adjust his results in order to overcome 'his' problem - and by exactly the same magnitudes (of space and time). But Einstein could not accept the uncertain bases/reasoning for Lorentz's adjustments - chosen essentially to simply fit the 'facts' as it were. In Lorentz's case, light's speed wasn't initially recognized as constant; rather, he felt it should be variable (when affected by the still ether) but that this had been 'masked' by effects of the ether on the space-measuring apparatus and on using the wrong time-measuring parameters. But at least he had specified the exact values that were needed in essentially the same sphere ('ball park') even if based on different premises and problems. [One should be able to account for such adjustments having the same magnitudes in the two albeit contrary approaches when deriving the same quantitative transformations.] And, as Poincare pointed out, these adjustments of velocity necessarily concerned adjustments of the time as well as the space variables inherent in all velocities. That there is only relative motion would apply equally importantly to the matter of the equivalent validilty of the local Times necessarily accepted by those in differently moving environments, each having equivalent primacy (as discussed later). These two facets of velocity would seem to be part of the same logic.
424. Being convinced that the principle of relativity must apply to all laws of nature pertaining to motion/velocity, Einstein could probably see that if Lorentz's equations were applied to his problem, in place of Galileo's, the requirements of both that principle and that concerning light's speed could be met - that is, whenever determining measures pertaining to the motion of any and all bodies (including light) in one frame of reference from another moving in relation to it (our 'complex' case). For this would allow the speed of light to not benefit from the speed of its source - which it shouldn't. All he needed was to find a better explanation for the variability in the measures of time and space inherent in Lorentz's equations that must somehow arise when the comparative frames of reference are some distance apart and/or moving at different speeds. He would then have to derive these conclusions in a more formal way using the appropriate geometry. A more general set of equations are needed - ones that apply validly and accurately with respect to both such normal mechanics and to any other (much faster) phenomena such as light with its awkward constancy (and all speeds between). And they shouldn't be chosen just to 'fit the facts' post hoc.
425. A single general 'rule' is therefore needed that can ('somehow') accommodate all such moving phenomena - and do so proportionally according to the speeds involved; the universe of moving bodies must be conceived as part of a single conception, govererned by the same, one set of rules. It would then prove to be the case that the original, Galilean transformation equations applied in the normal case of mechanics, involving the motion of a body (as in the train), was in fact not quite accurate - when measured from a differently moving platform (as foreshadowed - theoretically - by Einstein and discussed above; presumably, this was later confirmed empirically). But pointing out such an implication inherent in the term 'appear' above and thus the slightly inaccurate result regarding the speed of the walking man do not of themselves clarify or explain the matter of concern (it being in any case much too slight to have ever been measured or noted at that time). How was it ever recognized therefore? For such recognition only followed after the 'somehow' was eventually explained - at least in theory - if not 'shown'. For this, we shall have to continue with Einstein's account which firstly re-states some of the foregoing:
426. Firstly, as mentioned, he points out that the above result concerning the motion of light could well suggest that either the law of the speed of light or that of the principle of relativity may be wrong. But, as already indicated, he is inclined to accept that both must somehow be correct. But, as explained, the latter relativity law doesn't allow for different speeds of light in different reference frames. It is accepted as appling only to general laws of nature. And that principle has the security of its seeming inviolability in being so simple, natural, powerful and elegant (as argued above) that it should hold univerally - ie generally - for all (general) laws of nature (entaling motion). Possibly the law of light was really more complicated therefore - so that it could somehow prove conformable to the rigid latter principle (by accommodating some variation in its speed in various special circumstances)? But Einstein also argues against this - quoting as evidence (in addition to the work of Maxwell and de Sitter mentioned above) "the epoch-making theoretical investigations of Hendrick Lorentz on electrodynamic and optical phenomena concerned with moving bodies which showed that experience in this domain leads conclusively to an electomagnetic theory of which the law of the constancy of the speed of light is a necessary consequence".
427. The latter sentence must be read very carefully for it doesn't quite (or necessarily) imply that Lorentz himself arrived at that conclusion about light nor that his ideas agreed with those of Einstein in all aspects of their differing theories. For example, it could still allow Lorentz to have maintained his earlier view that the speed of light could vary under certain circumstances pertaining to his acceptance of the ether and be explicable in terms of his new transformations...but that the theory underlying this was however capable of being so (re-)interpreted that the speed of light should in fact have been concluded actually to be constant and thereby explain such as Michelson's results (from which Lorent'z theory arose) without the new transformations. That he derived them nevertheless proved a most happy contribution to Einstein. It was as though Lorentz's theory was a kind of reciprocal of Einstein's (in this particular sphere at least) and the new equations it required and produced fitted Einstein's needs if applied to his converse interpretation about light - providing he could come up with a more rational basis for their constituent variables.
428. Thus Lorentz's 'adjustments' to space and time by which both his and Michelson's expected variation in the speed of light was (wrongly) salvaged, led to his famous transformation equations whose quantitative values would prove to be correct in another sphere - even though the qualitative basis (an action of a still ether on the structure of material bodies moving through it in some kind of absolute sense; and on the passage of time) of their earlier needs turned out not to have been so. In fact, Einstein's later explanation would actually have predicted Michelson's result as it stood - with its confusing constancy of light's speed, but without positing any action by an ether or the need for any adjustments in the times or lengths involved in Michelson's results (at least if viewed from the Sun, say). While he later concluded that such adjustments of exactly the magnitudes suggested by Lorentz in his transformations (for his faulty premises) would nevertheless appear to be exactly as required - but not to account for (or be based on) an unexpected constancy of light's speed but rather, and paradoxically, in order actually to 'maintain'* that expected constancy - whatever may be the speed of the moving source (reference frame) in which it originates - as viewed from any other seeming stationary reference body (and vice versa). By the time Einstein became aware of Lorentz's equations, they had probably acquired a degree of autonomy from their origins with Michelson of whose results Einstein thus claimed he wasn't aware, before publishing his own theory.
429. [*'Maintain' is not quite the correct concept here in that light does not of course require any such external help to always have its one and only speed - relative to any reference body (whatever the latter's own relative speed) - as this is inherent in light's nature itself; but this concept does refer to the need to use the correct values of time and space for any event considered when, to use Einstein's phrase, "changing over from one body of reference to another" (moving at different speeds) which thereby does at least 'maintain' a correctly calculated value for the speed of light. That is, holding validly to this value for c effectively forces recognition of (results in) actual variations in the values of time and space when deriving the appropriate transformation equations. (See the earlier discussion in which "changing over from one reference to another" is described in terms of 'A re B' vs 'A re C', etc as concerns the 'direct' and 'indirect' assessments of motion. This was designed to help answer the question of whether Einstein's 'new physics' applied universally - ie to the many situations 'in life' where one is not necessarily 'changing over from one moving body of reference to another' but rather continues to be concerned only with one's usual immediate environment ?? If so, then length and time presumably don't vary in that much more typical latter case - the one most of us seem to inhabit most of the time. Or have I missed something here? Just how universal are his soon to be discovered realities about time and space - ie at whatever velocities are involved ? Of course every environment is really moving - relative to some other such platform - with neither seemingly more stationary or mobile than the other (except in reference ro themselves (as syatem K, say).]
430. [We may note about here that while Einstein concerns himself mostly with the effects of viewing the motion of a body in one (moving) reference system (as K'), from a more stationary one (K), with regard to the components of that body's perceived velocity through space - ie that the distance moved therein may appear shorter (contracted) and the time taken to be longer (which may be 'accounted for' by applying the appropriate transfomation equations) - in Lorentz's examples, it is typically the contraction of the body itself that is focused on (eg as the measuring arm moving through space on the Earth). With Einstein, some contraction of the moving body (as a bullet or thrown ball, say, or even the length of a moving train) may well appear to be the case as well, but this element is not usually referred to as it is not the primary focus of concern. It is the space (distance) through which any such body on same moves is so perceived that seems the focus of attention. This difference may have to be considered at times and not overlooked. (Possibly the space through which the body moves is the equivalent of the 'train's space' - ie it is shortened only because the train itself is necessarily perceived to be shorter ?) And while Einstein has explanations primarily for the effects on Time (see below) and Lorentz on Space (a body's length), the two men appear to have less to say about the respective complements of these respective differing components of velocity.]
431. If the law of light couldn't readily be adapted therefore, it appeared to some prominent physicists at the time that possibly the basic principle of relativity itself might have to be either rejected (which would thus oppose the conclusions described earlier about its ?seeming universality) or, more likely, 'adapted' (ie more fully explained or adjusted within the framework of mechanics, or even by expanding that framework). It was at this juncture (ca 1904/5), says Einstein, that the theory of relativity 'entered' the arena to resolve this seeming dilemma. ['Entered' is of course a very passive term here; it massively begs a very profound question.] Through it (and the new analysis of the basis of the variations in time and space on which it is based), both of the above 'laws' could in fact continue to apply fully (via these new transformations) and so prove to be mutually compatible and not inconsistent after all. Einstein would have noted that Lorentz had found a need to adjust the magnitudes of both time and space in his attempt to account for certain findings in which an ether was assumed as light's medium. Seemingly before Einstein saw his problem so efficiently in terms of the incompatibility between the speed of light and the principle of relativity (post hoc, as it were), he appears to have realized before this that he too needed to find a basis for a similar adjustment in the values of these two fundamental features of motion (ie as a basis for more comprehensive/coherent transformation equations) - one that did not however involve any role for an ether.
432. [Possibly because Lorentz's theory included a plausible basis for such an adjustment for space at least, but less so for time, he was particularly sensitive to any analysis that might focus more immediately on that latter, unexplained factor - 'to break the logjam' (especially if he had any doubts about the absoluteness of time (or space) beforehand - as possibly suggested by the writings of Poincare). Happily, this was apparently sparked by that imagined tram journey in Berne one evening in May 1905 (as described below). [I feel we shall never know if Einstein saw the problem in terms of the incompatibilty issue (with its convenient structure to set out his line of reasoning as though it proceeded essentially in those terms) before the tram journey described below or whether he struggled with it otherwise and could then see it in terms of that more easily communicated analysis only latterly.]
Section 8. - On the Idea of Time in Physics
433. In establishing what is meant by the concept of Time, Einstein first considers the concept of simultaneity per se (as discussed previously). In both cases, he points out that it is vital to be certain that they are defined in ways that allow of confident verification and measurement. [We may note here that Einstein is now giving us the benefit of his (eventual) insight on time as though he realized that such a prior definition of time was obviously necessary 'from the start' in an analysis such as his when in fact he is actually speaking with the benefit of hindight (albeit his own); it likely didn't slot into his initial thinking in this straightforward, efficient manner - at the time - but was eventually crucial.]
434. He begins by considering how to establish that two events that occur some distance apart, that appear to be simultaneous, actually are so. Just why he begins from this basis is not immediately apparent. He uses the method touched on elsewhere in which there is a central position C, mid-way between two distant locations A and B where the assumed simultaneous events occurred. He begins by noting that assumptions about two events being simultaneous in some absolute sense are in reality not assessable. We can only proceed on the basis of defining what we mean by this concept in terms of an agreed and reliable method of measurement. In this method, two locations A and B are separated by an exactly known distance at the mid-point (C) at which an observer can, by angled mirrors, see light flashes sent from the two equidistant points at the same moment arrive at his eyes simultaneously. But a problem arises in that this assumes that the light travels the two distances A - C and B - C at exactly the same speed. But to assess this would mean we already had a means of accurately measuring time. It is thus circular. We break this circularity not by assuming, supposing or guessing something about the speed of light but, says Einstein, by stipulating that, of all signals, light does always take the same time to travel these two distances. A stipulation seems to differ from an hypothesis in that one can proceed logically and confidently - on more 'solid ground'; it is presented as a pre-requisite, as it were.
435. We have thus defined simultaneity not discovered it and did so with a stipulation which is itself based ultimately on a postulate - that the speed of light in a vacuum is indeed a constant. This of course is a hypothesis based on conclusions from at least three important pieces of research as already mentioned. [He thus provides here some help in differentiating the subtle difference mentioned elsewhere concerning definitions, stipulations and hypotheses. If the outcomes based on this prove true, it provides support for these underlying elements of the theory regarding both the speed of light and the principle of relativity and their compatibility so explained.] [Note: the above example of establishing simultaneity seems to assume that the light travels the two distances on some absolutely still frame - which the Earth isn't! Light may travel at the same speed over the two legs but are those two distances equal if the mid position and observer are moving with the Earth?]
436. This definition of simultaneity can, in turn, lead to a definition of our greater interest - 'time'. {Again, one must assume that he already had reason (based on his imagined tram journey) to doubt any alternative, pre-existing concept of (absolute) time whether defined or not.] This is then done in a more formalised way by imagining clocks placed at points A, B and C along a straight line which are set by the foregoing method such that all hands point to exactly the same time as on the clock at point A. Under such conditions we may now define the 'time' of an event at any of these points as 'the time shown on the synchronized clock in the locality of that event'. That is, the 'local time'. As mentioned elsewhere, such synchronization assumes that the rates such clocks pass time also remain identical for them all. The duration (in time) of any event at any of these points/localities is thus also the same. This 'local time' (eg of Lorentz or at Berne, etc) was what he seems to have concluded in his 'eureka' moment was the only time we have! He could see that it provided the (sought after qualitative) basis of the variability of time (and space) he needed (to justify acceptance of their quantitative values as suggested in Lorentz's equations) and once recognized as such, he could derive it more formally in terms of the foregoing analysis of simultaneity.
Section 9. - The Relativity of Simultaneity
437. Einstein then continues by comparing the simultaneity of two events at A and B as viewed from a stationary position along that line with that of the same events as viewed from a moving position (as on a fast train or plane). He asks if they are simultaneous (occur at the same time) with respect to those viewing same from both the embankment and from the train. That is, can we assume that 'time' remains always the same no matter from what frame of reference one is viewing/measuring/timing such events? He shows that the answer is in the negative. For the mid-way observers may be at the same position along the line (ie at C) when the events occur at A and B simultaneously (in the stationary frame) but when the light from same reaches the stationary observer - who thus sees them as simultaneous, the moving observer finds that light from point B towards which he is moving, will have (necessarily) reached him a little before it does the stationary observer while the light from point A will reach him a little later. They will not appear to him as simultaneous therefore [even though they may be? No! This is the same point as arises in the case of the time in Berne. There may well be an understood 'lag' in conveying the information over distance by even light signals but simply 'knowing' this apparently doesn't prove a sufficient explanation; as we may, more typically, not know this!? (I need to re-assured myself further on this aspect!)] The light from point B will be seen as occurring before that from A. Thus, events that are seen as simultaneous with respect to the ('stationary') embankment are not so as seen with respect to (?from) the moving train, and vice versa.
438. This is summarized as 'the relativity of simultaneity'. Every reference system has its own particular, local time. Just as with the motion (distance per time) of any body, unless we are told the reference system or body to which a statement about same refers, there is no valid meaning in a statement about the extent, speed or, in the present case, the time of an event - either regarding when it has occurred or its duration. It is relative - to that particular reference point. It is very difficult I believe to over-emphasise the crucial point made here: The difference in the times taken for the relevant information to reach the (seemingly) moving observer is the total, ultimate, objective reality in his perception of time (and similarly of space). There is not only no other way for the information to reach him, there is no other more knowable, ultimately more real, time that could be conveyed to him by any other means - for exactly the same limitation applies to the observer on the seemingly still view point; both may be equally deemed to be the one moving and neither can have any access to a more valid perception of their respective local times (or space) than the other. [Maybe this answers my above point; knowing of any 'lag' in receiving the information due to the albeit brief time light takes to convey it doesn't inform one as to whether the time that could be calculated after accounting for that lag is anymore valid than any other time. They are equally valid. (Well, something like that). One should re-check to see if Einstein uses the term 'appears' in any of his analysis of these aspects in Chapters 8 or 9.]
439. Thus, events that are simultaneous with respect to a (relatively) stationary reference system will not (?necessarily) be (?appear) simultaneous with respect to a differently moving one - and vice versa. This latter point is crucial and implies that an arrangement whereby the moving observer sees two events as simultaneous would have the stationary observer reporting that the same events were (did) not now (appear) so. As mentioned, this is referred to as 'the relativity of simultaneity'; it 'depends' on the extent of difference in the speeds of the two moving reference systems. One may appreciate that this, in turn, somehow implies the 'relativity of time'. For the two observers, the situations are identical or symmetrical; both are equally legitimate therefore as to whose perception is the more valid. Neither in fact is 'more valid'. They are equally so, even if different. This is why the effects of the relativity of time and space is not a question of being real or only apparent; it is (validly) as it is - for any given observer - with any contradiction not being resolvable in terms that any given local time is more or less real for one or another; they are equally the only 'reality' available for each in their respective circumstances. There is no other actual 'reality' out there! Every reference frame (locality) has its own particular time (like 'starting moment') and, apparently, rate of time's passage/duration for a given event. Equally, the length of any 'rigid' object is not contracted in some physical sense. [Nor is Space??] Its actual length is shorter or longer than the same object relative to (observed from) a differently moving platform in the sense that nothing has any absolute, real length. There is thus 'local length' as well as 'local time' and no more objectively real measures of either. They are truly only relative, as is all motion, size, time, etc. [But is this the case for only the indirect 'A re C' case (and not for any direct 'A re B' cases?? See below (parag 206) for more on the relativity and contraction of Space (but of a body or of space per se??) in addition to the dilation of Time as discussed above.]
440. Before the publication of the theory of relativity in 1905 (to which these ideas on the relativity of time and space were fundamental), it had always been tacitly assumed that time was always 'absolute' and unvarying - and thus the same everywhere, at whatever speed one was travelling relative to anyone else. (Ditto for length/space/distance.) But the most natural definition of simultaneity just elaborated can, says Einstein, be seen to be incompatible with that former assumption. Without knowing to which reference frame the timing of an event refers (or the position/motion/speed of the bodies concerned), there is no sense in describing the time or duration (or position/speed) of that event, says Einstein. By replacing the concepts of an absolute time (and space), unaffected by motion of the associated frame of reference, with those of relative time (and space) - which are so affected - and for which the relevant reference system or body is specified - allows the constancy of the velocity of light to be compatible with the principle of relativity, but not otherwise. The reasoning behind this important statement is elaborated further below. It seems a little hopeful that the typical reader will see any basis for this statement just in terms of its limited content per se; why would anyone assume that the motion of another reference system might have any effect on the perceived distance or timing of a moving body thereon ? And if it is relevant, why would anyone (eg Einstein) look there first ? What is it about that situation that might make anyone suspicious ? There are most likely some other clues as to why one might so check. What were they ? It is however not clear why anyone in the past might have been concerned about how time or space in a moving environment (vis a vis one's own apparently stationary environment) might, conceivably, ever be affected by differences in their respective speeds. In what situations would concerns about that have ever come up, one wonders? We might consider how Einstein first became concerned about such - ie beyond his worries about racing with the light beams. [It is at about this point that we might also ask how these considerations might prove relevant to the apparent acceptability (to Einstein) of utilizing Lorentz's differently derived transformation equations with their implications of varying length (of a body not (?just) of the bodyless space through which it may move) and, somehow, varying time.]
441. In any case, just as with Galileo's original conception of the principle of relativity whereby those on a smoothly moving ship could perform all activities governed by the laws of mechanics in exactly the same way as on the relatively stationary dockside (and if windows were blacked out they would not know whether it was they or the others who were moving past whom (that is, by observing such activities) - and vice versa!; it makes no difference - they're equivalent and totally symmetrical as far as physics is concerned).....So too under Einstein's broader conception of this fundamental principle (with its now relative and so varying magnitudes for both time and space), all activities governed by all laws of nature, including both mechanics and that for the constancy of the velocity of light, proceed identically (ie fulfill the requirements of their respective laws) in whatever smoothly moving environment they occur. It is just that when these are performed in very fast moving environments and observed/measured from a much slower frame of reference, the relativity of time and space (as necessitated by the inclusion in this generalization of the law of light) becomes more and more obvious. [Time may appear slowed and length shortened as observed by those in either environment when reporting on the others where their respective speeds are very different. Again, it is equal and symmetrical; there is no preferred or provable reality. This is exactly that same at slower, normal speed differences (again relative to each other); they still occur at such speeds, as with the man walking on the train as measured from the embankment, but these very slight effects on such values are barely measurable then and have no practical effects on daily activities (and why they were never suspected).]
442. [The next three paragraphs reiterate some of the foregoing but should be retained if and where it makes anything clearer.]: Einstein states that because of the relativity of simultaneity as shown by his examples, we must conclude that Time can not be absolute and independent of any influences - such as a moving reference system from which information as to its actual time is delayed. (For if the moving situation produces s different Time (as Einstein claims, it apparently following from non-simultaneity), it does not agree with Newton's assumption - of being the same everywhere and in all different moving systems or whatever.) We thus accept that it differs because of that moving system and its effect on the method of synchronising clocks with light signals which necessarily don't reach the moving observer simultaneously (due to the differences in the distances that light must traverse to reach him). Einstein has stated that Time (via clocks) is always associated with such (light) signals and there is no other way to establish Time in a given reference system. At the moment, we shall just accept this. By discarding the assumption that time is absolute and 'unmalleable', he says, we may now accept that c and the principle of relativity are compatible. The conflict about that compatibility was thereby removed. This may well be true but there seems to be a couple of intervening steps in the logic missing here. This assumption of Time's absolutivity must be replaced by showing it is...(what?)... relative presumably, but, again, he doesn't make this apparent - yet. At least we see (or accept) that Time is different in the two reference systems and the type of difference will be explained (as relative) later, we assume.
443. Also in Section 9, Einstein portrays the above conflict so resolved by describing someone walking along the aisle of a moving train at the rate of, say, 1 metre per second relative to that moving system and pointing out that they would also be walking at that rate in relation to the stationary embankment (thus ignoring for the moment his speed relative to same due to the speed of the train). That is, he walks that same distance for the duration of each second of Time as measured for each of the two reference systems. But, we've seen (where?) that these two times (ie the durations of each second of time are different in the two systems - moving and stationary. (That is, the apparent duration of time (as viewed from the stationary frame) in the moving frame is slower than that seen within the stationary frame due to the time it takes to get the information to the stationary observer.) The two events (walking a metre as viewed in terms of the two reference frames) can not just be assumed to be equal in duration therefore. He also shows that not only are the two times different, but the two distances covered are as well. On this basis, he again claims that the conflict has been resolved.
Section 10. - On The Relativity of the Conception of Distance
444. This section deals with the relativity of space (ie distance or length). Thus, if the distance between two points is measured along the aisles of several carriages of a train travelling at a velocity v, we would normally consider the distance so measured relative to the train itself. But this distance on the train can also be considered with respect to the (relatively) stationary embankment. Thus, at a given point in time (t) (on the embankment), two points on the embankment (called A and B) are directly opposite those on the train (which we may now call A' and B' - which are moving together at the velocity v). The positions of A and B be determined exactly by means of the definition of time as given above (Section 8) and the distance between them measured off using the same measuring rods as used to measure the distance on the moving train. These two distances would be expected to be identical. But the question arises - is the distance along the embankment between A and B (as viewed and measured by reference to that embankment) the same (or appear to be the same?) as the distance between the points on the moving train (A' and B') - as also viewed and measured from the embankment ? Equally, we may ask if the two distances are (or appears to be?) the same when both are perceived and measured from the point of view of the moving train?
445. Without actually answering these questions, Einstein implies that the two distances may well not be equal when measured from these different points of view (which, by the way, have equal relevance and validity (neither can be justified as being the more valid) - just as the distance covered by the man walking along the moving train in a unit of time, as measured in the train, is not necessarily the same distance (per unit time) as it is when measured from the embankment. [Note these distances per unit time equate of course to velocities.] Thus, the assessment of both times and distances when considered from within a given reference system and from one moving relative to it may well not (appear to?) prove the same, implies Einstein, and he seeks in the following Chapters to verify this seeming likelihood further and show thereby that this discovery has relevance to the question of the compatability or otherwise of the laws of light's constant speed and of the principle of relativity. For the newly appreciated variabilities of time and space will help explain why light's speed manages to be unaffected by the speed of its source or observer.
Section 11. - On The Lorentz Transformations. [This is covered below]
Note: The foregoing, with some added interpretation and reference to related topics, was based mainly on Einstein's 1920 book - up to its Section 10. On a later occasion, I happened to re-read another copy of this same book (picked up while on holiday) and subsequently wrote out further if lengthy accounts about Einstein's crucial insight into 'the nature of Time' (and Space) as partially covered, but with less background, in the later Sections 8, 9 and 10 above. This additional analysis is shown here before final Summaries of the remaining Sections (11 to 16) further below.
The 1920 Book Further Considered.
446. As noted earlier, Einstein wrote in the Preface of his new book 'Relativity - the Special and the General Theory' that it was indeed intended for "..those readers who...are interested in the theory but who are not conversant with the mathematical apparatus of theoretical physics". In it, he thus guides the reader rather carefully and mostly by verbal rather than mathematical reasoning towards his derivation and interpretation of the Lorentz Transformations (in Section 11) with their implications of the variability (ie relativity) of time and space - the crux of his special theory of relativity. [As noted above, we have already considered and summarized the first 10 Sections of this book and may now repeat some of this (for Chapters 5 to 10 at least) before considering, as mentioned, the remaining Sections 11 to 16.]
447. Thus, in Sections 5 to 10 of his book, Einstein discussed the basis of the earlier ideas on the motion of objects in general with respect to the two fundamental components of velocity (time and distance), when their suggested relativity had yet to be proposed. They were, until 1905, if considered at all, not assumed to be relative or dependent upon anything in particular but rather to be essentially 'absolute' - in and of themselves - not affected by anything else, including in particular the relative motion/velocities of different relevant reference systems dependent on those constituent measures. However, after finding some difficulties in accounting for certain reports in the physics of light over the previous 10 years or so (in which the velocity of light appeared to differ from that expected (at least in terms of Maxwell's ideas), Einstein was gradually led (by considerations we should be able gradually to delineate) to giving thought to the role of these components of any body's motion - time and distance - and their assumed absolute magnitudes. For they seemed to be the appropriate (only remaining?) focus as to why the Galilean transformations did not cope correctly with calculations regarding light's speed when measured across differently moving reference systems. The classical mechanics (based on absolute time and space) through which the principle of relativity applied by means of the usual arithmetic of these transformations appeared to be inappropriate in this domain. The solutions concluded were not consistent with Maxwell's constancy of the speed of light. He thus looked out for any evidence that such long held assumptions about time and space (and their associated arithmetic) - which appeared to be the source of the inconsistency - might not be warranted within this electrodynamic sphere of the physics of motion.
[Our resume and analysis of this material will inevitably include some repetition of ideas covered above and in the writings of Seife, Kaku and Folsing. Hopefully, it will further address the basis of the term 'appears' so often used in the foregoing and how, why and when that term conveniently and quietly drops out of the typical descriptions - increasingly to be replaced, with little or no fuss, by the terms 'is', 'are', etc or their equivalents.]
448. In Section 5 of his book, Einstein begins by defining the Principle of Relativity as it applied firstly in the sphere of classical mechanics (as already described above) and provides evidence that supports its 'essential' validity. And yet, certain indications in the realm of the motion of light suggested that its application or suitability in this latter sphere at least might nevertheless be reasonably doubted. But before considering this further, he begins by defining this important principle very carefully in terms of the complete equivalence of all laws of mechanics in all Galilean coordinate reference systems (K, K' K'', etc). By 'Galilean' he means systems whose states of uniform relative motion and that of the bodies within them conform to the law of inertia - being the fundamental law of classical mechanics. Thus, all natural phenomena in such equivalent systems should run their course according to the laws governing them in exactly the same way whatever the differing but straight and uniform velocities may be - of the systems in which they so function and/or from which they are measured. This is the principle of relativity and it was generally accepted as applying to all laws of mechanics and indeed (implicitly) to all laws of nature generally. [This seems a most reasonable place for Einstein to begin the present analysis (to show how and why his theory was necessary) in that the motion of bodies (mechanics), and it important principle of relativity, is so fundamental to understanding nature.
449. But later developments in the domain of electrodynamics suggested (at least to Einstein) that classical mechanics alone did in fact not explain all natural phenomena (and in particular not the implications of the law of the constancy of the velocity of light). The validity of the principle of relativity was thereby placed in doubt since applying it to certain electrodynamic arrangements - at least as analysed in terms of classical mechanics and the associated Galilean transformations - appeared to give false results (as mentioned above) - ones that were not consistent with that latter law regarding light.
450. The relativity principle's validity (as then interpreted) would be increasingly questioned by any who addressed this matter seriously (as Einstein did around 1903-04). Was it not appropriate just in the domain of electrodynamics or was its application to that sphere (and even in any domain) by means of classical mechanics the essential problem . Might a different kind of mechanics, based on different premises regarding time and space (those basic components of all motion), be more appropriate when applying laws of both mechanics and those pertaining to electrodynamics ? Einstein reasoned that as this principle of relativity appeared normally to apply so well in one domain of nature (mechanics), there was a strong likelihood that it should, somehow, also hold good in this and any other domain (but possibly do so in terms of an arithmetically differently based mechanics - specifically in terms of the actual varying magnitudes of the components of velocity - time and space). [But, he doesn't show here how and when he decided this may be the most successful 'point of attack' (analysis)! The above sentence beginning 'Might a different mechanics...etc was not in fact posited so directly at about this point; we just don't know when (or even if) he so reasoned.
451. He then advances a second, more powerful argument in favour of the essential validity of the relativity principle - which should apply in all situations, whether mechanical or electrodynamic. If this fundamental principle was not valid (ie did not hold in all circumstances) then different uniformly moving systems (K, K', K'', etc) would not be equivalent for the operation of all natural phenomena - as required for Galilean reference systems based on the universality of the principle of inertia. This would suggest that there must be one such system (say K*) for which, if any, all laws of nature must hold true (as we find they are generally so reliable) and this would likely be one that was 'absolutely at rest' in relation to which all other systems would be in some varying and uncertain degree of real not relative motion. We would, said Einstein, expect less simple, less predictable laws to apply in. the latter. The motion of any such non-stationary system would then be 'absolute' in relation to the hypothetical fixed system but would be different from all other systems rather than all being equivalent and mutually relative.
452. The velocity of the various different systems would therefore not be effectively neutralized and placed on an equal footing as before; their differing motions would enter into various unpredictable outcomes for the bodies set to motion within them. But, when one considers activities on the Earth - whose motion is not uniform in that the direction of its velocity does vary through the course of a year, it would not be at rest (and motions of bodies on it would not be absolute) relative to this hypothetical system K* throughout the year and yet there have never been any (?significant) variation in the outcomes of the laws of nature according to this variation of the Earth's (direction of) velocity. This, says Einstein, is another powerful argument in favour of the essential validity of the principle of relativity which is fully consistent with this reality. Surely it applies validly to all phenomena in nature therefore - including light - and not just to that of mechanics? But, as concluded above, do so on the basis of a different kind of 'mechanics' - one we might call 'electrodynamic mechanics' (or, now, just 'electrodynamics' for short - with its new conceptions of time and space (as yet un-announced).
453. With two such arguments supporting the validity of this principle, Einstein thus felt there must be (such) a way by which it can be shown to hold for all domains of nature. He thus began to focus his attention on the possible application of this principle more generally - within both the mechanical and electrodynamic domains - including the demands of a constancy for the speed of light. Was there any scope for re-interpretation of the components of all motion more generally - ie both mechanical and electrodynamic ? He addresses this in the next Section (6) - as analysed below following the next few paragraphs (which cover some of my own thoughts at this point).
454. [I would imagine that the forgoing analysis of the basis of the validity of the principle of relativity would be consistent with the following: The more restricted principle of relativity (as it was assumed to operate within the sphere of mechanics alone) states basically that the differing velocities of different reference systems in relation to which laws of mechanics may operate and from which they may be measured 'make no difference' - to their essential outcomes, once those prior differing speeds are accounted for. The reason for this is not because there is some pre-existing principle with which such laws must somehow accord but because of the prior effect of inertia which acts on the bodies and systems concerned more directly to determine such equivalent outcomes (I would suggest).]
455. The uniform motion of any immediate or local reference system on which any laws of mechanics may be operating (relative to that of any other system) have no effect on those laws. Any effect such relative motion might conceivably have had on such laws is completely neutralized by virtue of the action of the inertia possessed by all bodies within and including that system. This neutralising influence may be termed the principle of relativity as applied to mechanics. It acts indirectly by means of the action of inertia. Similarly therefore, the influence of the more general principle of relativity isn't the direct reason why all laws of nature, including that concerning the constancy of the speed of light, are also not affected by such differing speeds of reference systems but, rather, because of the operation of some comparable prior and more direct determinant (through which such a general principle itself operates indirectly). Is this primary determinant again inertia, or...what ??
456. Thus, when the difference in the speeds of different reference systems are accounted for (as by subtraction, say), the net velocity of light in either system - if measured from the other - remains at its one constant value nevertheless (as required by its law) while the net velocity of any other moving body subject to its own laws of mechanics (entailing an application of force, say) will reflect the operation of the principle of inertia on the uniformly moving systems concerned and so prove to be identical - in, say, an increase in essential velocity) - whatever the associated speeds of the relevant systems - again accounted for by appropriate transformations - which will now reflect more accurately the extent of the particular difference in the systems speeds. The communality or generality of the determinants in the two (or more cases) would of course be what is conceptualized within the more abstract principle of relativity.
457. What is it, in the final analysis, that determines that the differing speeds of reference systems can have no effects on the operation of any and all laws of physics which take place within and as measured between them ? One assumes that it isn't the principle of relativity per se but a more directly acting determinant.
When a body is forced into motion within its own reference system, that effect when measured from a system moving at a different (say much slower) velocity will normally reflect both the action of the law concerned and the difference in the velocities of the two systems. By subtracting that latter difference from the result (of the addition of velocities) one would expect to be left with the effect due to the law itself. But if the body concerned is a pulse of light, the effect when measured from a slower moving system remains exactly the same as when measured from within its own local system. Subtracting the difference in the velocities of the two systems is unnecessary. This is because, unlike the mechanical law regarding the motion of most bodies, that law pertaining to light requires light to have the same velocity from wherever perceived and measured. The difference in the velocities of different systems may be directly additive with the effect of the force acting on a body (to increase its velocity) and thus would normally have to be accounted for (removed) in order to reveal the action of the law per se.
458. In the case of light, however, its propagation or propulsion is intrinsic and requires no external force once released/emitted. And it then has but one possible velocity, from wherever measured, and thus is not further boosted by the greater speed of the system from which it is released (relative to the slower system from where it is observed and measured) and so that difference in the speeds of the two systems need not be subtracted from the velocity of the light when measured from that slower system. The law of light has in effect already accounted for any such difference (as it is moving at the maximum velocity possible) whereas the laws of mechanics act on bodies at much slower speeds which are only some proportion of that maximum possible speed. The greater the velocity of the body (and system?) concerned (relative to the other system), the less would be the proportion of the difference that need be subtracted (say). At much slower speeds, the greater would be the proportion of the difference to be subtracted to reveal the actual effect of the law concerned. What is needed therefore, is a single transformation equation that adjusts for such a range of possible (?proportional) velocity differences for all bodies, including light, when measured from a different system - ie from 0% to 100%.]
[Note: For the present repeated consideration of Sections 5 to 10, I have also made briefer resumes (in blue) of what appears to be Einstein's essential 'lines of reasoning' over same. Thus, for the foregoing section (5), I would simply note that Einstein concludes mainly that:
459. The essential validity of the principle of relativity is indicated by certain strong evidence but in its original form this is only the case if one accepts that classical mechanics (with its unmentioned assumptions of time and space) validly represents all natural phenomena. But electrodynamics has (in the form of the implications of the law of the constancy of the velocity of light) shown that such mechanics provides an insufficient foundation for the explanation of all natural phenomena. That is, certain findings in electrodynamics of ca 1885-95 cast doubt on the sufficiency of mechanics (and of the principle of relativity of that domain) as fully accounting for all natural phenomena (and particularly for lack of any addition of velocity to that of light when the latter's source was previously expected to give it a boost - as it appeared always to do when the velocity of the body concerned had been boosted not only by some force therein but, relatively, by that of the system concerned as well). That is, there was not the expected addition of the relevant velocities. The more general validity of the principle of relativity so based was thus brought into question. But, as mentioned, its validity within the domain of mechanics at least seemed so strong (on the basis of the evidence mentioned) that it would appear to be most improbable, asserts Einstein, that it should not be valid also in this other domain - that of electrodynamics. Uniform motion of reference systems (all with identical acceleration status - of zero) meant that all activities, mechanical and electrodynamic, thereon would commence from equivalent starting points. Because of the universality of the principle of inertia, all their laws should therefore function identically and reliably; in brief, different inertial systems should make no difference. To do so would however have implications for our conceptions of time and space.
460. In Section 6, Einstein first describes the situation in which an 'addition of velocities' (ie of a moving body measured on a larger body that was itself also moving) had in the past appeared to transpire as expected - that is, within the realm of mechanics - as a man walking at a uniform pace, generating his own force to this end. Thus, depending on his velocity, he will walk a given distance (w) in a given time t - when judged in relation to his immediate reference frame. If he now walks at that same pace but does so on a railway carriage travelling itself at a velocity v in relation to the embankment, the distance W travelled by the man in a given time t relative to the embankment will apparently become W = w + v . This may be described as the theorem of the additions of velocities - but of classical mechanics (as touched on above). In the past, the net effect on the velocity of his walking in relation to the embankment was calculated by subtracting the velocity of the carriage (as per a Galilean transfomation) - since it has no effect on his net speed in the two situations (as per the principle of relativity and inertia). (It may be noted that the symbol W can refer here to both the velocity and the distance covered depending on whether the time is shown and that where velocity is concerned, there is always a reciprocal relationship involving both time and distance. Knowing any two elements will allow one to know the third and definitions of either can, by algebra, probably be adequately formulated in terms of any two of these three measures.)
461. In any case, Einstein anticipates later developments in his exposition in that he points out that this simple linear addition of velocities (and of the distances and times necessarily involved) and the subsequent subtraction of the train's velocity if appropriate by means of the usual transformation equations) to give his net velocity actually turns out not to be the case in reality (ie if measured extremely carefully but, more pertinent here, is also not the case on important theoretical grounds to be explained later). One might add here (also as an anticipation) that the difference between the simple addition and subtraction shown above and that which it will turn out actually to be is, for most normal velocities (and distances per time), so minute that the need and relevance for the latter, more precise values, was never appreciated because they were far below the scope of all but the most sensitive and precise methods of measurement. Thus the classical theorem of additions and any subsequent transformations had always been accepted as totally accurate and fully reliable for all practical applications and speeds but, as mentioned, will now be qualified below on theoretical grounds initially, and only later by actual confirming measurements. [I believe that he should really have given us some hint here as to why this apparently valid 'addition of velocities' is in fact not the case. It has to do with the problem with light; because it couldn't be 'added to' (other than by 0% of the light source's velocity), so would every other moving body (moving at much lesser speeds) have somehow to be added to 'proportionally less' than had otherwise been assumed to be the case. That is, by between about 99%, say, (at very slow velocities) and only 1 or 2 % (at very fast ones).
[The shorter blue summary of Section 6]:
462. The theorem of the addition of velocities was thus exemplified by a simple example of a man walking on a train and the principle of relativity of mechanics applied (via Galilean transformtions) to calculate his net speed or distance walked in a given time in relation to another reference (the embankment). However, Einstein then points out that this result (believed to be accurate in the past) will actually turn out to be not quite correct, the reason for which will be explained later when a similar practical example is given which entails elements of motion that are more amenable to very accurate measurement. Moreover, a theoretical argument will also be provided later on the basis of which a more accurate transformation equation will be applied in order to arrive at the correct result. For all laws of nature to accord with the same one principle of relativity, it would become necessary to recognize that time and space were not constant as previously assumed but were in fact variable - at least whenever events in one system were measured from any differently moving system.
463. NB: Thus,in Section 7, he follows up on such implications (of the constancy of the velocity of light) by considering the electrodynamic example referred to above which physics had found didn't fit an analysis by the usual classical mechanics. The relevance or validity of the traditional principle of relativity and of the Galilean transformations based on same (as had always appeared to be correct previously) were thus gradually placed in doubt (by whom and when??). That is, the natural phenomena of the constancy of light's motion didn't seem to run its course - as 'required' by the generally accepted principle of relativity - at least when analysed by classical mechanics and the usual transformations. ['Awkward; because the motion of all previously considered bodies were always variable; that was in fact how they were all seen to work and fit the expectations of Galileo and Newton. This constancy was inherent in Maxwell's well-accepted equations for electrodynamics (which however would, I believe, benefit from being made more explicable in situations in which observers or reference frames move at greatly differing speeds but light remains constant). But that of course was exactly what hadn't been thought through about it. It was eventually 'discovered' that it had to do with having comparably different perceptions (or measurements?) of time and space in these situations. Maintaining the old conceptions of these was what prevented this law from being seen to have fulfilled its own and the principle of relativity's requirements. Just when was this first realized (ie by Einstein) ?
464. [One wonders if Maxwell appreciate this when his equations indicated that light's speed would always be manifested at its one and only speed - by anyone, whatever their different relative speeds?? Or was it only Einstein who would appreciate this - and was it that conviction that spurred his search for an explanation (in terms of time and space relativity - possibly as there were the only available 'variables' left to so account for this?) or was the latter concluded first and the 'invariable' constancy accepted only after ?? (The former I would imagine.)]. Moreover, (the constancy of light's speed) was also consistent with the discovery by De Sitter observing double stars that the propagation of light cannot depend on the velocity of the body from which it is emitted. It remains constant regardless of the velocity of its source. [And, thirdly, it was, according to Einstein, also supported by Lorentz's analysis when seeking to account for the Michelson-Morley result by suggesting that light's expected variability was not found because it had been masked whereas Einstein asserted that it had in fact been correctly found to be constant. However, it is one thing to be able to state that light's speed is indeed a constant and another to explain why it must be so !!
465. The principle of relativity had been considered heretofore consistently in terms of the outcome of any law of mechanics on a moving body being the same whatever might be the velocity of the reference frame on which it is so applied - whether measured on that same frame or from any other frame moving at some other uniform velocity. In that latter case, one needed simply to calculate the difference in velocities of the two frames and by adjusting for this (by applying a Galilean transformation equation) one would obtain the correct and valid result for the law concerned - as required to accord with this principle. Such laws dealt with the expected effect of applying a force to a body of some known mass, say, and observing its expected new (different, non-constant) velocity or distance travelled in a given time. When measured from a different reference frame, the body's total resultant velocity or distance would be appropriately adjusted by subtracting the full difference in the velocities of the two frames thus leaving the full net value expected for the mechanics and laws concerned. The accuracy of making the measurements from a different system was assumed to be satisfactory. [Note that this approach would thus assume that it would be possible to add the full value of velocities upto and beyond that of light (ie to be infinitely fast if necessary), as though light's speed offered no upward limitation on velocity (and the additions of same) and so Galileo's transformations would require no proportionality (as though having to 'fit' proportionally into a limited available velocity 'space' - as determined by light's speed being the maximum possible speed. ('As though' is very pertinent here since the limitation concerned is not really one of space but of time - namely, the time required to receive the information regarding the motion of the body concerned.] But what should happen when the law concerned doesn't predict any increase in the body's subsequent velocity or distance - from whatever frame it is measure - whether from its own or from a differently moving one ? This was, after all, the basis of Galileo's original analysis of its action - in terms of inertia - to establish that the Earth moved around the Sun and not vice versa. [Even if, mathematically, they can be realistically conceived as moving equally and mutually relative to one another.]
466. [I don't believe Galileo (or even Newton?) formulated the principle of relativity (however then termed) in such general and abstract terms as did Einstein - with his reference to '..all phenomena should run their course..' etc and with respect to '..all laws of nature..' or even '..of mechanics', but rather just in terms of the specific kind of motion with which Galileo was concerned in establishing the effect of inertia to justify acceptance that the Earth revolved around the Sun (more obviously than the other way round). How to justify therefore that it should apply also to such as the 'law' of the constancy of the speed of light which doesn't entail any net increase (or decrease) in velocity of a body after applying Galileo's transformations to remove the otherwise masking effects of a differently moving reference frame (of that from which it is measured) ? On the contrary, it entails by definition no change in velocity whatever the difference in the speeds of the frames concerned. So the same principle is somehow so generalized that it can be invoked to account for both a net change in a body's velocity that may be (?partially) masked and for a complete lack of such change.
467. That these two differing outcomes can be subsumed by Einstein within the same general principle exemplifies the breadth of his conceptual powers. It wasn't something Galileo or Newton 'saw' but, as mentioned above, the principle wouldn't itself determine the outcome in any direct, mechanistic way but rather indirectly - by means of some more immediately direct determinant which we would assume possesses some communality with the determinant of inertia in the case of the laws of mechanics. What is this determinant ? [Answer: there is no such 'determinant'; 'inertial mass' is itself a kind of 'starting point' in this analysis.] Mechanics work the same whatever the uniform, straight speed of the system on which they apply or from whichever they are measured - because of the action/effect of inertia per se. All such reference systems are effectively identical, having identical levels of acceleration - namely, zero. Electrodynamics work the same whatever the uniform speed of the system on which they apply or from whichever they are measured - presumably because of the action/effect of inertia. (Otherwise, what?) In the former case, the additional velocity or distance due to the difference in the speeds of the two systems can be subtracted (or, now, some proportion of it - from 100% down to 0%?) to reveal the identical net effect of the law concerned. In the latter case, the additional velocity or distance due to the difference in the speeds of the two systems need not be subtracted (or only a zero proportion of it?) and thereby allow the same (vs net) effect of the law concerned (as that of the constancy of the velocity of light).]
468. So, we can ask whether the principle of relativity still applies in this 'new' electrodynamic situation? Einstein argues that it should in that it should be generalized to state that all laws of nature generally (not just of mechanics) should meet its requirement - that they should perform identically whatever the difference in the uniform, straight velocities of different reference frames on which they occur and from which they are perceived and measured - even if not examining the presence of an increase or decrease in velocity but an unchanging velocity whatever the velocity may be of the reference systems concerned. The outcome must adjust to not reflect that difference; it must be neutralized (or, as Lorentz decided, masked). This was the problem touched on by Einstein in Chapter 5 when he pointed out that this principle seemed not to apply when electrodynamic phenomena were analysed or dealt with - at least in terms of the old classical mechanics - implying that it may do so however if treated by different 'means' (than the mathematics of classical mechanics with its assumptions about the unchanging values of time and space (ie of velocity) in all situations) since the principle itself still appeared to be valid. [Note: those different 'means' entailed different bases of the components of the basic feature of mechanics - ie the motion or velocity of a moving body - comprised of just time and distance (space).] He then describes the basis of this doubt by showing what will happen when the principle of relativity (in its restricted, classical sense) is applied when the body concerned is a pulse of light (as we've already touched on above). For it turns out that both the walking man and the speeding light beam (and everything in between) require the application of a new 'time/space-adjusted' principle of relativity - rather than the original, restricted one.
469. [This repeats the earlier paragraph above and must be reduced and integrated: We might consider here whether the principle of relativity as first formulate by Galileo (and utilized by Newton similarly) was ever considered by them or others to have the potential generality eventually accorded it by Einstein. That is, did they not conceive of it entirely in terms of showing how the principle of inertia would account for the identity of all effects on the motion of bodies in differently if uniformly moving systems of reference where that difference in motion would be added to (or subtracted from) that of the body being measured - to confirm that identity or equivalence of the net effects ? It was not, I assume, conceived as a principle pertaining to all laws of nature that might one day be discovered in which, for example, no increase in a body's velocity (as that of light) had to be differentiated from that due to a difference in the speeds of different relevant frames of reference (or, equally, not the full increase of that shown by a body moving slower than light had to be so differentiated).
470. Einstein seems to suggest that it is because the principle of relativity should apply to all laws of nature (for reasons presented elsewhere) and not just to those of mechanics that we are justified in accounting for a need for the adjustments applied to the theorem of the addition of velocities (according to the velocity of the bodies concerned) by maintaining or concluding (as a (the only!) way out of a dilemma) that time and distance are not absolute but variable or relative (being dependent on - ie varying according to - the perceptions arising due to those velocity differences and the perceptual lag therein due to light not being instantaneous) - as shown quantitatively through Lorentz's transformations, even if the bases for these adjustments had quite different premises. That is, that they ultimately depend upon the extent of the difference in the velocities of the different reference frames involved where these exist and are relevant (and the perceptual concomitants thereof) - without involving the effects of an assumed ether. [Or, are they not ultimately so dependent but only correlated with this basis of these perceptual effects?? That is, is the actual 'cause' fully explicated in those terms alone ?? Yes, apparently so.] In any case, I can't imagine Galileo or Newton saying to Einstein: "Yes, cite 'our' principle of relativity (however then termed) as a justification for that solution to your problem for we can see in an instant that it must also apply to phenomena that show no net velocity changes (of the (only) kind that we were ever concerned about in our day) where differences in such reference system speeds are the case. Or, might they have said instead "One might well generalise our principle in that way but surely you could justify your adjustments of time and distance (to fit the apparent facts) without relying on that generalisation". Or does the success of that solution depend upon or serve to support the validity of that principle's generalisation ?
471. We may now show what would be calculated, according to Einstein, when the motion to be measured from a faster (or slower) coordinate system is a pulse of light emitted on the other system rather than a more normal body subjected to some force (as the walking man) and thus how a certain difficulty (a 'dilemma') arises when the results are assessed in terms of the expectations of the (restricted) principle of relativity - as based on the arithmetic of classical mechanics (ie on the same basis as Galileo or Newton would assume it should be applied as with any other moving 'body') and the transformation so associated - in which time and space were treated as always as non-variable; any other conception of them having never arisen. Thus, any such body must in the first instance be adjudged in its movement in relation to an agreed reference system. As before, this can be taken to be the railway embankment. A pulse of light emitted above the embankment would travel along it at the velocity of light (c). We then imagine a train travelling beside the embankment at velocity v in the same direction as the light. In terms of the usual mechanics of relative velocity, we would calculate that the velocity of the light relative to the train would thus be = c - v which is of course less than that of light relative to the embankment - ie by the value of v, the train's velocity. This is comparable to Einstein's original thought experiment as a 16 year old when he imagined he was travelling (like the train) but even faster - as near the speed of light itself so that, in Newton's terms, the relative velocity of light would appear to such an observer as being barely faster than he - with light's waves as almost 'frozen' still.
472. In our present example (as in that latter one), such a result, says Einstein, is in conflict with the principle of relativity (as applied in terms of the laws of mechanics) - as set forth in Section 5. But, says Einstein, like every other general law of nature, the law regarding light's velocity must, according to the principle of relativity per se (ignoring for the moment the use of mechanics in a particular application), be the same whatever may be the reference system in relation to which it is measured because that is what this particular law of nature (re light) and the relativity principle says must be the case. [We need here the reason why this almost magical outcome for light is the case] and that latter relativity principle (says Einstein - as argued elsewhere), 'must' apply equally to all such laws - including that for light (even though Galileo and Newton didn't claim this for it, I believe). That is, in this case, whether it be measured with reference to the 'stationary' embankment or to the train moving at a velocity v. But, as calculated here at least, this would appear to be impossible. [We should also review here on what basis or logic Einstein arrived at his conclusion that the principle of relativity warrants this 'over-arching' primacy and influence concerning the laws of all motion, as it is crucial to the logic of his argument as to what was the problem that needed a solution (his theory) - as well as why the law of light's speed per se so applies at all. In any case, he points out that the dilemma might be resolved by simply abandoning either this principle or the law concerning light. He suggests that one is unlikely to abandon the former as it appeals so convincingly to the intellect (presumably on the basis of the logic reviewed; see above) - being as it is, so natural and simple (and presumably also logical and reasonable and 'fundamental'). I believe he also presents elsewhere other cogent reasons why this principle has such primacy. Uniformly moving systems, whatever their relative velocities, are all equal and any law that works on one should work identically on them all - for they're all the same!!)
473. On the other hand, this would mean that one would then have to abandon instead Maxwell's (?generally accepted and ,again, 'simple') law about light - and replace it with one that was much more complicated and so less convincing. [But it is hardly 'convincing' as it is almost impossible to account for!?] But, besides the two reasons given above as to the validity of that law in its simple form, he also now quotes the work of Lorentz (seemingly done when seeking to resolve the problem of the Michelson-Morley experiment with light) from which the constancy of the velocity of light must, says Einstein, actually be concluded as a necessary consequence. [Note that this however was not I believe something that Lorentz sought to show in his early focus on seeking to account for Michelson's result (rather, even the contrary?), but, rather cleverly, Einstein, usefully for his own ends, found those results could nevertheless be so interpreted to support fully his own view regarding the velocity of light! [Yes, but it isn't just thevalidity of this law that one should be concerned with but with a way of accounting for or explaining that surprising validity!! How does one explain why light's speed appears the same whatever the velocity of its source or one's own velocity? Only later if and when anyone might accept that time and space are in fact relative does this odd outcome become marginally explicable. What did de Sitter say about it?] Other physicists at the time therefore tended initially to reject instead, said Einstein, the principle of relativity (presumably when seeking to resolve the above dilemma although I'm not aware of just who, if anyone, was so addressing this matter then (ca 1903-1904) nor recognised that it was a problem to be so addressed, other than Einstein himself, nor in fact whether there was yet a general acceptance of the certainty of the law about light (which, as noted, seems difficult to account for), nor the relevance or potential rejectability of the little mentioned principle of relativity in this regard).
474. It was, says Einstein, at this point "...that the theory of relativity (and thus his own thinking in this regard) entered the arena" (ie seemingly in late 1904/early 1905). He seems to be saying that because he had himself, if no one else, been troubled by the fact that the apparent constancy of the speed of light was not compatible with the usual conception of the principle of relativity (the essence of which had always applied without exception in the sphere of mechanics), some adjustment of it or of its bases was needed (somewhere) in order to allow the valid acceptance and interaction/compatibility of both these laws or principles (as argued above).
475. He then states: "As a result of an analysis of the physical conceptions of time and space, it became evident that there is no incompatability whatsoever between the two principles of present concern." The joint application of these two principles or laws is indeed valid in terms of his theory of relativity - by means of its necessary adjustments in the magnitudes of time and space. But this is really only concluded by virtue of the reasoning not given until Section 11. He doesn't say at this point why and when he considered such an analysis in the first place nor what the steps were, if any, in that analysis to reach this important conclusion.
476. By replacing the former ideas of the nature of time and space as used in classical mechanics with this new conception (whatever led him to it), he would show that both principles had in fact always functioned together compatibly - and thereby resolve the dilemma (of the apparently wrong answer arising by the application of the old Galilean transformations). The constancy of the speed of light and the properly applied principle of relativity had thus (on this basis) always been compatible and thus had always been the case, as had the relativity of time and space on which this newly recognized compatibility (and 'unification') had always been based; it was just that the manifestation of the latter was too slight in its effects at the slower speeds at which most bodies move to ever have been noticed, measured or considered. That class of 'mechanical' phenomena was the apparent reality that had always been ever so slightly in error. This new conception is elaborated as the special theory of relativity in the next Section (8) where he presents the fundamental ideas of the theory in which these new concepts are more fully explained. [This gets us back to the lightening striking the railway lines and the matter of the perception vs reality of time and space! We should try to discover just when and why Einstein first focused his attention of the concepts of time and space as being particularly germane to the problem before him; and also, just what that problem was seen to be prior to that new focus! It was essentially that of the constancy of the speed of light not proving consistent with the mechanics of the past, I believe.]
The blue summary of Section 7 (finally!):
477. The motion of all bodies was previously thought explicable in terms of classical mechanics alone, with its associated principle of relativity, upto and until the motion of the 'body' of light was found not to be so explained. This awkward fact "plunged the thoughtful physicist into considerable intellectual difficulty". This was the 'problem' or 'difficulty' so ofter referred to in his writings. It was exemplified in terms of the addition of velocities which, when the velocity of the body of concern should not be variable (as in the case of light), gives an invalid result when dealt with by means of the Galilean transformations (ie when one doesn't 'hold fast' to light's constant velocity). It doesn't meet the requirements of either itself or, therefore, the principle of relativity. It seemed that one of these laws or principles must be incorrect as they appeared to be mutually incompatible. But because the principle of relativity is so natural and simple and has such strong supporting evidence, at least in the mechanical domain, the law about light's velocity would seem more likely to be the one at fault. But, as this latter law arises within Lorentz's excellent electrodynamic studies as a necessary consequence (despite this not being a goal of that study apparently), prominent theoretical physicists were therefore more inclined to reject the principle of relativity - even though there was no empirical evidence (indeed, only the contrary) on which to justify such a decision.
478. It was at this juncture (ca 1905), says Einstein, "that the theory of relativity entered the arena", as we've mentioned above. This theory showed that both the two principles could be adhered to (ie they weren't really incompatible). Without explaining how, he simply states that his theory shows their actual compatibility became apparent as a result of "an analysis of the physical conceptions of time and space" (again, without stating why he decided to analyse these concepts in particular to this possible end) on which his theory is based. That is, what pointed him in this direction and when ? As a consequence, the addition of velocities could now give a correct result even though the velocity of light can be shown to remain the same (as it must according to its law and thus to the principle of relativity also) whatever the velocities of the two reference frames involved. The discrepancies from the previous analysis (as the man walking along the train) can be accounted for in terms of the new proportional variabilities of time and space - when perceiving their magnitudes on one frame from another moving at a different velocity. Holding fast to the constancy of light's velocity (in any mathematical analysis of same) forced this newly appreciated reality to be manifested. They (time and space) became the variable factors involved, as it were, rather than the velocity of light having to become so. And the informational lag implicit therein would account for the proportional velocities so arising for moving bodies on moving systems M - whatever the total combined moving speeds of such bodies and systems - when observed/measured from system S. When that total was at or near the speed of light (in theory) the proportion would be total while for any body moving in a stationary system, it would be zero.
479. By this means, his theory can be seen to be based ultimately on the universal principle contained in the single integrated postulate: 'the laws of physics are invariant with respect to the appropriate transformations between any two or more inertial systems'. That is, which display what was later termed 'universal Lorentz covariance' in recognition of Lorentz having derived such transformtions (based on v/c) before Einstein - although they were then based upon invalid premises and entailed quantitatively fitting results into such a model.
480. [We may usefully note here that in his 'Autobiographical Notes' (1949), Einstein, when reviewing his thinking back in 1903/4, mentions similarly that by about 1900 neither mechanics nor electrodynamics had achieved complete validity; there were certain anomalies and inconsistencies still present in research outcomes in both fields; the existing laws governing these domains did not fully account for certain findings. When seeking some way to resolve this, he decided that constructing new explanations from existing facts was inadequate and that only the discovery or application of some universal general principle that applied fittingly in both areas would lead to reliable results. To this end, he decided that if he was going to accept the validity of the constancy of the speed of light (and as the maximum possible speed), as concluded from his analysis of Maxwell's equations, he would have somehow to fit this and the motion of all other bodies into the dictates of such a single, broader principle. This turned out to be a principle of relativity that was more fully generalized and encompassing than that of Galileo or Newton. The motion of all bodies, in both mechanics and electrodynamics, would be better conceived as falling within the same broad spectrum, a single conception of all 'bodies' in motion - governed by a single general principle. [Later, in his general theory of relativity, he sought to broaden this principle even further. See Sections 15 and beyond.]
481. The light speed and relativity principles would thus be his only two postulates - ones he felt he could rely on. He asked himself where was it that light's velocity constancy had caused a problem when applying the old transformations of mechanics ? He eventually deduced that whenever these transformations were applied in the past, Newton must have tacitly assumed that time and space (always involved when considering velocity) were always constant. To allow the principle of light's constancy to be compatible with the relativity principle, new transformations were needed (based on the ratio of v/c which was concluded to be relevant (as discussed later) and this in turn resulted in time and space proving to be variable. [The sequence of the logic here may need further analysis.] In any case, this would become the crux of the special theory of relativity; for such 'new transformations', based on 'appropriate new relations', resolved the difficulties (when converting the measures of time and space of an event (whether concerning the motion of light or of any other body) in one system of reference to those when measured from another - moving at a different velocity). Such transformations, based on v/c, would necessarily imply variable perceived magnitudes for time and space (when so measuring velocities) and become fundamental and axiomatic to the theory. To derive these new relations and their transformation equations entailed determining the precise quantitative functions by which the time and space measures in the different systems so relate. This is elaborated below.]
482. In Sections 8, 9 and 10, Einstein then discusses the ideas of the relativity of time (in terms of simultaneity) - in Sections 8 & 9, and distance/space - on Section 10, and hopefully explains why and when he first decided to examine/analyse these particular concepts - as fundamental to his theory designed specifically to solve 'the problem' caused by light's constancy - with respect to the principle of relativity. [Note: this is relevant to what was alluded to in the preceding paragraph and may help clarify how the new relations and transformations were arrived at. Conveniently, he does so (re time) by means of the same thought experiment regarding simultaneity as described earlier - that is, of two bolts of lightening striking a railway line some distance apart at the same instant. [See also the summaries on Time by Kaku and Folsing - now below.]
[Note: The rationale of why and when he first analysed the roles of time and space in this regard might be partly answered in his remarks in Section 11 which begins 'Is there a thinkable answer to this question...etc'. I believe he was led into this analysis only after he had his 'eureka' moment about time with the Berne clock - even though he studiously avoids these proto-aspects in his rather 'clinical' presentation of the 1905 paper. But the reasoning behind the Berne situation and the lightening strikes is probably the same. They are important in that they provide Einstein with the means of accounting for the variabilty of time (and space) that he required without resorting to positing some influence by an ether. While he may have concluded the likelihood of such variability on the basis of his theoretical analysis - when holding fast to the unchanging value of c in the equations, he still needed a more empirical explanation as to how this variation actually comes about (ie by the Berne-inspired and lightening-inspired information lag); while these may well be related justifications/explanations, both appear useful.]
483. To verify that the above lightening strikes are indeed simultaneous, he indicates that it is necessary to place a stationary observer (or recording apparatus?) with an appropriate arrangement of angled mirrors at the measured mid-point between the locations of the two strikes. If he reports that he perceives (or it is recorded that) the two flashes arrive at the same instant, then, says Einstein effectively, they may well have arisen at their source (presumably moments before) simultaneously - just as the perception of them on his retina (or receoding receptor) did likewise moments later. However, he points out that it is necessary in addition to be confident that the velocity of the light emitted from those flashes over the two equal distances is indeed the same. But this would apparently require a means of measuring the time that such flashes of light take to travel those (equally confirmed) equal distances - that is, to measure light's velocity. As this is apparently not readily available for this purpose (why not if it is a thought experiment?), he says that it is nevertheless satisfactory to define the observed outcome as validly representing the simultaneity of those flashes (both as they struck and when perceived) providing one simply stipulates that light does indeed take the same time to travel those same distances - ie that its reliable constant velocity is taken unquestioningly to be the case (ie as a postulate?) for this present purpose - ie to so define simuultaneity. [Does such a 'stipulation' somehow differs, and significantly so, from such as an hypothesis, postulate, assumption or supposition?] In any case, this stipulation is a part of his definition seemingly. While it may imply total faith in his analysis of Maxwell's reasoning (and others' findings) about light, its acceptance here is primarily to arrive at a clear definition of simultaneity.
484. Next, he seem to imply that because this definition can apply (and give an exact meaning) equally to the timing of any number of events (along the same axis?) - at locations A, B and C, etc, not just to two (A and B), then it can be used also to establish a definition of 'time' in physics. [We must note that both for simultaneity and for time per se, Einstein is here restricting the meaning of these concepts (ie 'by definition or stipulation') which could well provide him with a 'built in' guarantee that those concepts he seeks to establish thereby must prove consistent with same. Can we be sure that such definitions (and the 'exact meanings' he is thus restricting one to) are necessarily the only or most valid ones - in the wider scheme of things ? Is it the case that his main thesis rests on his remark that we can't measure the time the information about the lightening (itself light) takes to travel from the point of the strike to the observer's retinas (or to any other recording instrument) ? That is, that such times will always be uncertain (and vary with the generally unknown distance) and therefore one is only left with the practical experience of one's perceptions ? If so, should this not be spelt out more clearly ??]
485. Thus, by this means, he states that we (?must) understand by the 'time' of an event (ie when it occurs not its duration seemingly) the position of the hands of a clock in its immediate location (say C) which has been synchronized (set to be, and continues to be, simultaneous) with the position of the hands of identical clocks in more distance locations (say A and B) along the same coordinate axis - such simultaneity being as here defined. By this means, one can obtain reliable times for events occurring at any of these separated locations on that straight axis. [Again, doesn't the time the light signal takes to synchronise the clocks enter into consideration? Yes, but only in a so far as defined here for this purpose.] So, he has thus defined Time - a rather important element in his thinking and theory - on the basis of 3 'events' along an axis on which he first defined simultaneity (as at two such 'events') based upon the stipulation that the velocity of light is a constant and is so at an agreed velocity. The outcome seems to be that "the perception of time is (or can be) relative (as here defined) - at least as further qualified - in terms of the extent of the different velocities of the reference systems concerned (but not otherwise?).
486. Einstein then asks if two events at points A and B (as two strokes of lightening) which are simultaneous (vs being reported to be perceived as such presumably?) with reference to a stationary railway embankment are (in this same sense) also simultaneous with reference to an observer on a train moving along that line from A towards B at velocity v. He then says he will show that the answer must be no (and seemingly will be so due to some effect of this arrangement on the magnitude of time thus clarified by means of analysing simultaneity.) Firstly, he defines more exactly what the simultaneity of the two strikes means or implies. That is, the light emitted from the two lightening bolts as they strikes the line at A and B arrive at location M on the embankment exactly mid-way between A and B. When the lightening strikes at A and B, positions A' , M' and B' on the moving train coincides with positions A, M and B on the embankment and hence position M' on the train is also mid-way between its positions A' and B' (and between A and B also) at that moment. (We should be clear just when that moment occurs; is it the time the lightening strikes the line or when perception of same occurs (in the retina and brain of the observer at M ?) In any case, all such events are being timed by identically synchronized clocks ; both the lightening strikes and the clocks' hands are simultaneous.
487. Observers at positions M and M' are thus exactly opposite each other at the instant the lightening strikes the line at A and B. As the light rays from A and B approach the two observers, the one at M' on the train is moving at velocity v towards the rays approaching from B and away from those from A while the observer at M on the embankment awaits the simultaneous arrival of the rays from both A and B without moving towards or away from either. They should reach him at the same moment and so both his perception of same and his assumption of their actual occurrence moments before would be reported by him to have occurred simultaneously - whereas the moving observer will report that the rays from point B reached (ie were perceived by) him before those from A did so and thus he would not report his perception of same as being simultaneous (although he might reasonably report that they may nevertheless have actually occurred simultaneously at A and B in so far as he would (we must assume) be aware that he was moving with the train toward one point at which the lightening struck (B) but away from the other point). However, if he was not privy to the arrangements of this thought experiment, such an observer may well be (I believe) 'in the dark' as far as being able to give an estimate of whether or not the two strikes were or were not in fact simultaneous. He would however presumably know he was moving towards the earlier perceived flash, although may not know the distances or speeds involved.
488. Possibly it is on this latter basis (of being 'in the dark') that Einstein then states that the observer on the moving train (at position M'), who takes the moving train as his reference body, must therefore come to the conclusion that the lightening flash at B (not just one's perception of same) took place before that at position A. [I still can't accept this: the observer at M' would indeed perceive the two flashes as not perceived simultaneously but, because he is moving independently of the reference body (the embankment and lines) at which the bolts struck, he need not conclude that the actual flashes 'must' therefore also not have occurred simultaneously. There is still room for them to have been simultaneous.] Nevertheless, Einstein concludes on the basis of this explanation that "...events that are simultaneous within one (eg 'stationary') reference frame (as observed at M) are not simultaneous with respect to a reference frame that is moving relative to that other one (as observed at M'). [One would feel more confident in this conclusion if the word 'necessarily' was placed between the words 'not' and 'simultaneous' and that he qualified 'events' with 'perception of events' or replaced the term 'are' with 'appear'.] In any case, he then claims that "..every reference frame has thus its own particular time".
489. That is, time is relative to (is dependent on) the relevant reference frame in the perception of motion/velocity of any event. (I believe Lorentz called this the local time.) This seems to indicate that if one was using clocks instead of lightening strikes, the time perceived in a moving reference frame would differ from that perceived elsewhere. One's clock on the train may show it to be exactly 10 am, say, but the clock at point A could show it then to be just before 10 and the one at point B just after - that is to the observer on the moving train taking that as his reference. Thus, says Einstein, 'events which are (taken to be?) simultaneous with reference to the embankment are not (taken to be) simultaneous with respect to the moving train, and vice versa.' Again, one might be more comfortable if the word 'are' was replaced with 'appear' but the die seems to have been cast now (ie by 1916). The 'malleability' of time has, ironically, now been set in concrete - as a 'fait accompli'.
490. The time at which an event is reported to have actually occurred must, it seems, be based solely on (and somehow equates to) the time at which it is perceived or 'judged' to have occurred - even if the (?local) time at which it may have actually occurred is possibly or likely to have been different from the former - since there may be no way to establish or verify the latter. One notes in this regard that Einstein significantly qualifies his statement that "...when the flashes of lightening occur..." with a footnote on the same page thus : ie ...occur... "As judged from the embankment". For 'judged', we may assume 'perceived' at the observer's retina and visual cortex. If a 'judgement' is then required (by the frontal cortex and memory banks) to 'estimate' the actual occurrence (as replacements for the earlier terms 'appear', appearance', etc), we would again seem to be on rather shaky ground. We may however be required to accept this restriction but it seems most odd that not withstanding this seeming imperfection, one can nevertheless apply the resultant theory with consistent success in its many predictions!!? [Yes, but might it be as validly accounted for by some other theory ?] Moreover, if the basis of this limitation on our definition of time is as described here ('time and distance can only ever be what we see the information about them to be...full stop!), why are they typically never simply explained or qualified in these terms (other than on one or two occasions typically - at the start only of any description of these phenemena ?] Without knowing to which reference body a statement of time (or length presumably) refers, there is, says Einstein, no (valid) meaning in such statements. Well, no pre-defined, exact physical meaning in them certainly - when compared to those based on clearer definitions and stipulations, it appears.
491. Prior to the theory of relativity, from the days of Galileo and Newton, it had, says Einstein, always been assumed in physics that statements concerning time did not have to be qualified in regard to whether one was measuring it within the same frame of reference as the event and observer moving together uniformly were situated - or as it appeared to be in a frame moving differently to the observer - where the (apparent?) time and distance moved of the event concerned would somehow depend upon or be relative to the difference in their respective uniform velocities. For if time was thus, for some reason, considered now actually to be relative to such a difference in motion (or simply 'to be relative'), then the previous conception of time in such situations could be considered to have been 'absolute' or invariable in magnitude - uninfluenced by anything else - including in particular motion of the relevant reference systems, although 'motion' is of course what Einstein is primatily concerned with here and thus why he stresses that this particular factor was overlooked previously (as if, of all possible factors that might conceivably be over-looked and affect such results, this one should have been the first to be considered). Thus its value was assumed to never depend on (wasn't relative to) differences in the velocity of the observer and the event measured. It was assumed to be quite independent and everywhere the same, whatever different speeds might be so involved in such motions of event and observer. That former conception was thus incompatible with the conclusions Einstein arrived about time - at least as based on what he describes as "the most natural definition of simultaneity" (presumably meaning that concluded on the basis of our 'natural', day-to-day experience when we typically don't (even ?can't) have all the relevant information). That is, that it wasn't independent but rather 'depended on' the motion of the reference system from where it was measured. [Note: It doesn't appear to follow that (as mentioned above) time and space would show this characteristic within one and the same (and only) reference system however.]
492. Again, we may ask when and why Einstein (or Poincare for that matter) first examined the concept of simultaneity in this regard (through which he concluded that time was relative and thus that it was in conflict with the classical conception of absolute time). We must assume that he was confronted with the apparent incompatibility of the two laws described above (when considering matters in this general sphere - for reasons which we should set out more clearly) and, amongst various approaches to the problem, eventually decided that the principle of relativity could and should take on a more general significance than it had theretofore such that even the law of light's constancy should hold for all reference systems whatever their motion (as was the case in its more restricted sense for all mechanical laws). He eventually focused on the possible role of the values of time and space in improving that principle's appropriateness in this wider regard. This may have followed a kind of trial and error methodology until he 'happened' upon it - it being the only route left. In particular, we may imagine that his thoughts about the Berne clock and tram may well have set his thinking along the lines of the simultaneity implied whenever one notes the position of a clock's hands in relation to local events and to distant ones. His subsequent ideas were likely influenced also by Lorentz's adjustments of time and space - as his way out of the Michelson result! But Einstein doesn't seem to point too directly to that source of his new suspicions. Nor to Poincare who was apparently also concerned (for other reasons) about simultaneity.
493. [To me, this definition of time (even that by Einstein) seems to lack the usual rigour of physics and is more a compromise that places ?fallible 'human experience' rather to the fore; but what else is there - to interpret nature?] By discarding the assumption that time is not absolute but relative (but in only certain circumstances), one can, says Einstein, show that the constancy of the speed of light and the principle of relativity are not incompatible (as they previously appeared to be). 'Showing' this must entail more than this considerable jump in the line of reasoning, however, with several steps apparently missing, but for now we might suggest that such 'malleable' values for time (and for length) are the means by which this can be so shownand so allow an explanation of how and why the speed of light is in fact not 'boosted' (as would otherwise be expected) but remains as the constant value that Maxwell's law indicates (and thus as the principle of relativity requires) - when its speed is measured in relation to a fast-moving source from the standpoint of a relatively stationary frame of reference (and equally why the speed of anything travelling much slower than light is also, pro-rata, not boosted as much as formerly believed in such situations).
494. [A fuller explanation of how acceptance of relative time (and length presumably) allows resolution of the incompatibility described above will, I believe, be provided in Einstein's next two Sections. For this apparent incompatibility is described in that statement in terms of rather general concepts that would benefit in being broken down into their more specific and pertinent elements, if one is seeking a clear and full explanation. For example, should one not re-emphasise the fact that the perceptions of time and space from either differently-moving reference systems have equal relevance and primacy and that therefore (somehow) they, respectively(!), are the only 'reality' ? Also, I would like to see more reference to the fact that light begins its existence - to travel at its one and only speed - only at the moment of its release from its source; what speed and direction that source may be is of no consequence to it. And how an observer (or recording/measuring apparatus) - whatever their own speed - can only perceive its speed to be c, also needs explanation.]
495. Before we consider this, however, it may be useful to consider a comparable situation to that of the two simultaneous (or apparently not) lightening strikes. As pointed out by Einstein, it is possible to synchronize clocks along such as the railway line at points A, B and C (with C as the mid-point between A and B). We may imagine the observer M perceiving precisely when the minute and second hands of two synchronized clocks at A and B 'struck' exactly their positions of 12 o'clock noon, say, and was thus able to calculate, knowing the speed that the information was conveyed to him (ie at the speed of light) was identical for the respective legs of those two journeys, that both clocks had arrived at 12 noon simultaneously a few precisely calculated moments before so perceiving that (delayed) information, and confirmed by his local clock at C. In the case of observer M', moving on the train towards clock B at velocity v, he will perceive that clock showing him it is 12 noon slightly earlier than would observer M - as the light conveying that information would have a slightly shorter journey to travel to him - even if also receiving it a little after clock B 'struck' noon; but it would be not quite as long after M would perceive this same information. On the other hand, M' would not receive the information from clock A as quickly as would M nor as quickly as he would himself receive that from B, as he would be moving away from those 'messenger' light waves. Again, the clock at C (or C') could provide confirmation. The two clocks at A and B would thus not be perceived as being synchronized or simultaneous. Again, however, if he knew the velocity of the train (v), and that of light, he would be able to calculate that his reported non-simultaneity of the noon-striking clocks (as perceived) would however not imply that the two clocks did not actually strike noon simultaneously. His calculations would indicate that they did indeed do so simultaneously.
496. If we combine the two imagined situations such that observers M and M' can fully perceive the incoming information-conveying light equally about the clock times and the lightening bolts (magically 'set' to strike simultaneously at 12 noon) at A and B , it would appear to follow that both observers would be able to report that the two lightening bolts did strike at A and B simultaneously even if , for M', perceptions of same did not. For he would now have not only the values for v and c but the times at which the bolts were perceived to strike and when they actually struck, as well as the times at C (or C').
497. The foregoing would seem to indicate that our perceptions are not our only accessible reality in this particular sphere although the exposition of this - to be understandable by children or the layman - must, it seems, be typically confined to not fully explained assertions and to careful definitions and stipulations ex cathedra, albeit backed up ultimately by seemingly inexplicably valid predictions. So, what to think ? I would imagine that there must be some flaw in my analysis even if, at the moment, I can't see it. It may be that in reality, one typically doesn't know all the variables on which to anaylse the slight differences entailed and so it becomes neecessary to accept that the apparent non-simultaneity is as near to being truly the case as it is possible to get. We may thus continue with Einstein's exposition in which he seeks to explain that the apparent conflict between the two pillars of his theory - the constancy of the speed of light and the principle of relativity (the meaning and significance of that latter principle possibly needing further explanation with respect to the present problem) - can be fully resolved if we simply discard the earlier tacit assumption that time (in particular seemingly) is not absolute and constant in magnitude whatever the motion of the relevant frames of reference (ie where it occurs and where it is perceived and/or measured) but is somehoe truly dependent on those differences.
498. Thus, one can no longer accept that as measured from a slower-moving platform, the velocity of any body with a given velocity of motion relative to its own immediate reference frame will show an increase in its velocity fully equal to the difference between the velocity of those two frames (ie the latter being faster than the former), but only some portion of that increase in velocity. This, says Einstein, is because the time as perceived from the two frames is not the same; each time (eg durations of the seconds or minutes concerned) is relative to or dependent upon the relative velocities of their respective frames (as concludes from his earlier discussion about time). The greater the difference in velocities, the slower the time is perceived to pass pro-rata. If the difference were of the order of the speed of light, it would (in theory) stop altogether. But, as velocity is a compound concept or measure - of both time taken and distance covered - Einstein points out that the latter (distance or length) was in the past also tacitly assumed to remain unchanged in magnitude when measured as described above. He addresses this matter, albeit rather briefly, in the next Chapter (10) but again, still more in terms of time than space (distance) seemingly.
499. [Note: It is in this regard that my idea concerning how, if Lorentz was right - the motion of any body through the ether causing its length to contact (with a seeming comparable, ie equal, effect on time (a slowing) that was however not as adequately explained) - fits rather well with the idea that Einstein might be wrong - in trying to account for various awkward results in terms primarily(?) of a variability in time (that may be only in the perception of time) with that of length somewhat subsumed (and only explained) within this possibly suspect time idea. Could one not explain such results totally in terms initially of a contraction of length - with no effect on time (almost the mirror image of Einstein's view) ? That is, can velocity differences be accounted for in any asymmetrical way (eg as mostly due to an effect on length or even on time) or do time and length have always to share ?'equally' (?comparably) in this ? Must the transformation equations necessarily give them equal influence? How did Lorentz (and then Einstein) come up with this symmetry of influences or effects when deriving their functions (the 'relations')? Or is velocity - as 50 mph vs 100 mph - always just a function of one or other of these two variables - ie distance per set time or of time per set distance ?? Probably! [God! As if I didn't have enough to be concerned with !]
500. In Section 10 (to return to the matter of the brief account of the role of the relativity of the (perception or conception of) the distance (spatial) element in measures of velocity taken from different frames of reference), Einstein contrasts the length measured by spatial measuring rods between two separated points A' and B' of a long train moving at velocity v, when the frame of reference is taken as the train itself, with that same distance as judged from the more stationary embankment. To do this, he suggests that one must first determine the position of the points A and B along the embankment that exactly coincide with points A' and B' on the passing train at an exact time t - as judged from the embankment (presumably using a clock local to the embankment). This determination of the positions of points A and B (and their distance apart) - ie all spatial measures) is effected says Einstein not by measuring rods but by the "applying the definition of time given in Section 8" (and hence the apparent need to have the time when this is so determined). This definition was the empirical one based on a definition of simultaneity which was, in turn, based on a 'stipulation' that light takes the same time to traverse the two paths from point A to the mid-point between it and point B and from B back to that same mid-point. This definition of simultaneity, thus 'restricted', can be used to give an exact meaning not only to two events but to three or more and this in turn, says Einstein, leads us to the definition of time in physics - the application of which also allows us to determine the spatial position of points A and B that we are seeking. That is, space as determined indirectly by means of time and velocity, not directly by spatial measuring rods.
501. This seems an oddly indirect way to determine (the distance between) two points. After obtaining the distance between A' and B' on the train (whether moving or still) with measuring rods, one wonders why he didn't simple mark our that same distance along the embankment such that when the train comes along the time (if needed ) can be noted at the instant the two sets of points coincide. However, Einstein explains that we cannot, a priori, be certainty that the distance A - B on the embankment as determined by his method will equal that as measured on the train between A' and B'. This implies that the length of the entire train as measured from the bank may be (not 'appear') different from that measured on the train itself (as the length A' to B' was a major part of that length). This, says Einstein, provides us with a second objection to the assumption that we can add the full value of velocities when measured from a different reference point than where the velocity of the moving body concerned is boosted by the speed of its own reference frame. In Einstein's words, "...If the man in the carriage (as described earlier) covers the distance w in a unit of time - as measured from the train - then this distance - as measured from the embankment - is not necessarily also equal to w."
502. That is, both time and distance, say Einstein, are altered in such a situation and presumably contribute equally to the non-linear addition of the velocities involved. [Or, is it only the perception (or even the conception) of these elements of velocity that are so altered?] One wonders if he is relying too much on his conception of time to account for the 'malleability' of both that element and that of distance (rather than having a more specific and independent justification for the latter)? We are reminded of the remarks made above regarding the kind of converse image that Einstein's conception presents when compared to that of Lorentz - with his more spatially dominant explanation, with time rather more neglected. Apparently in Einstein's manuscript sent to the journal which published his 1905 paper, that portion dealing with the spatial element was allegedly somehow misplaced (by Einstein). His transformation equations incorporated this aspect of course (seemingly dealt with in terms of the length and velocity elements, but that doesn't clarify the logic of its equally significant premises or evolution to the layman. Could time have been handled in the same way - as a consequence of more thoroughly explained distance effects - and still convince the reader ? Similarly, could Lorentz have made his case in terms primarily of the effect of motion through the ether affecting time and then only partially explain a comparable effect on distance - but essentially in those time-dominated terms ? Well we ever know ?]
[Blue summary of Sections 8, 9 and 10 here:
503. The foregoing is so comprehensive (and wordy), it is unlikely that any new or more pointed slant can be provided in this briefer summary. It appears that we must just accept that the appearance of the variability of time and space is itself sufficient evidence for the validity of this conclusion. While we can appreciate that information about the actual magnitudes these fundamental factors of motion cannot be perceived instantaneously, but only after the lag necessarily arising by the albeit exceedingly fast and reliable light signals conveying that infomation about the 'actualities' concerned, we are nevertheless not able to take account of such resultant inherent discrepancies but must instead accept that the altered values so perceived (symmetrically by those on different reference frames) are our only available 'realities' and that all predictions based on such accepted 'perceptual realities' somehow always prove correct in predicting various actual(?) outsomes in physics. Thus, when measuring time and space (as involved with any moving body) on one reference frame from another moving at a different velocity, their magnitudes not only appear to be dilated and contracted, respectively, for all intents and purposes they must be accepted as actually being so altered - in 'reality' - at least from what they would have (again) ?appeared to have been otherwise. Again, we may have to keep in mind how the fact that all such perceptions have equal 'merit' whether perceived on system A from, say, system B or vice versa; this equality seems to force acceptance of a relative, perceptual answer rather than any single, absolute, truly 'real, valid' answer - unavailably 'out there' somewhere.
We may now continue (finally) with Section 11:
Section 11 - The Lorentz Transformations.
504a. Einstein begins by stating that the apparent incompatibility of the law of light's constancy and the principle of relativity (as understood by Galileo and Newton) was, as explained in the previous Sections, eventually concluded to be due to the tacit but erroneous acceptance that time and space were independent of (not affected by) the relative motions of the bodies of reference concerned in their measurement**. One wonders if Einstein honestly believed that there was ever any rational reason why anyone previously would ever have considered examining or questioning those tacit assumptions - on that particular basis (or indeed on any other basis)? Why would they ? Was the term 'arbitrary' not a little harsh here ? And, more importantly, why did he do so?
[** NB. One was very curious to know (before later readings) just how the different relative motions of the reference systems concerned so affect (make dependent) the actual (or only available?) values of time and space. And why this was ever considered that this was where the problem may reside. How do they so affect such measurements (or even their actual quantities, if they so do ? Simply by 'delayed perceptions' ? Yes, apparently so (as understood from those 'later readings'! And therefore they don't themselves (it would seem) directly affect their 'real' magnitudes; rather, it is our perception of them that is so affected. The words 'the actual values' should probably read 'our actual (estimated/judged/perceived) values ...'. Unless someone knows otherwise ? And are they so affected within system K itself, however it may be uniformly moving - by those also therein ?]
504b. I can only assume the reason Einstein initially questioned the tacit assumptions about time and space was that when he analysed some body's imagined motion but 'held fast' to the value of c (which his analysis of Maxwell's equations suggested was a reasonable thing to at least consider), he saw that this would be possible only by virtue of the time and space values somehow adjusting' (by whatever 'mechanism')!] As mentioned above, it may be helpful if we analysed just what is implied more specifically in these two laws or principles, stated in such general terms, that were both considered ro be fundamental and inevitable and yet - as incompatible. Just how do time and space enter into consideration with respect to the seeming incompatibility of our two crucial principles or laws? How did Einstein initially come to focus on these particular factors or components (of velocity) - as the possible source of the difficulty ? [Possibly only after his 'eureka' moment ? and 'Is there a thinkable answer...?' and by seeing what must come out when he holds fast to the constancy of c (as shown in his derivation of the equation) ie only Time and Space were left as the objects of adjustment !! Also, Newton had taken the other position - at least tacitly - and possibly in error - while Leoentz's solution had aslo entailed altering conceptions of time and particularly space.]
505. p;> 506. The principle of relativity stated originally that all laws of mechanics should operate exactly the same in all uniformly moving frames of reference. This meant that even if judged from a frame moving more slowly than the one on which some body had been forced into some degree of greater motion, the extent of that boost in motion in relation to its own frame was unaffected by the relative speed of that local frame - or by that of whatever reference frame it may be measured from. If the former's greater speed (relative to that from which it was being measured) was subtracted from the combined speed of the body and that of its frame, the body's speed would be found to be exactly the same as it would have been on the slower-moving frame from where it was being measured. This expectation was always found to be the case - at least within normal errors of measurement at those speeds - simply by subtracting the extent of its increased speed due to that of the faster moving frame. However, when the body concerned was a pulse of light, very precise measurements should, if it was truly constant in its speed (in contrast to all other known bodies whose speeds can and are varied according to the magnitude of forces applied to themand to the relative velocity of their reference frames), find that any increase in its reference frame should however not increase light's total speed - as it is believed (?stipulated) to be constant (based on 2 or 3 facts quoted by Einstein, as de Sitter, etc) no matter what the speed of the external observer and his frame.
507. The two 'laws' are thus, on this basis, incompatible. If light is truly constant in its speed, then the original principle of relativity must somehow be wrong. There was at least one law (if not 'of mechanics', then, more generally, 'of nature') that does not accord with that principle - at least as it was usually applied. The increased speed of the frame of reference from which the light was released was not added to that of the speed of light, so subtracting it from the speed of light as measured would give a net result that was exactly that much too slow to validly represent the dictates of this particular law (re the inevitable constancy of the speed of light) and so did not meet the normal definition of the traditional principle of relativity as expected. ,u>This was 'the problem' - first appreciated in this form at least by Einstein.
508. The principle of relativity (when restricted to the sphere of mechanics) states than when we measure the effects of applying any law of that sphere to a body on a frame of reference moving itself at any uniform velocity (relative to any criterion), the results as determined by that law, when measured on that same frame, will be found to be the very same if it was applied to such a body on any other frame of reference moving at a different (say faster) velocity to the first one, again as measured on that differently moving frame from a slower moving one, say. If the outcome on that first frame is measured from the second (say faster-moving) frame, any apparent difference in the outcome would thus be due to the difference in the speeds of the two frames (and not to any difference in the effects of the law concerned per se (on velocity) operating on the body in frames moving at different speeds; their different uniform, straight speeds per se should have no effect whatever on the operation of the law concerned). Any seeming difference in outcome can be fully accounted for by applying the relevant transformation equation - which, in this case, would normally entail subtracting the full difference in speeds of the two frames. As mentioned , this is the case where the mechanical law concerned produces a change in the body's velocity - albeit of equal value in the differently-moving frames (and so accord with the principle of relativity) - but this identical outcome in a velocity change due to the law concerned is, as shown, only revealed by subtracting from the compound totality of its now apparently greater velocity in its overall velocity as measured from the slower moving frame that portion due only to the greater speed of its frame relative to the other frame. The outcome of the action of the law concerned per se thus fulfils its requirement - as required by the principle of relativity. [See Section on the Addition of Velocities about here.]
509. If the mechanical law concerned was however one that produces an effect that doesn't directly effect (a variation in) the velocity of the body concerned - ie within its own frame (if such a law existed), this same (non-velocity variation) effect should again be found, according to the restricted principle of relativity, if measured within another frame moving at (say) a faster (or slower) velocity and, moreover, it should also be found to be of that same (non-velocity variation) magnitude and form if measured from the other, differently moving frame - since the difference in their respective velocities does not enter into the measurement of the (non-velocity variation) effect concerned. But when we apply the principle of relativity to another sphere of physics or nature (not restricted to that of mechanics) - ie to that of electrodynamics (as light's velocity) what do we find ? In this case, the effect concerned is a velocity effect (like our first example). We would therefore expect the effect of its law, again, to be manifested identically when light's velocity is measured within different frames moving at different speeds. Their respective different velocities have no effect on the outcomes of that law within either of those respective frames. However, (as stated above) when its velocity in one frame is measured from another having a different velocity, the law of the constancy of light's velocity requires it to be measured to have the same velocity as it shows within either of these two frames - without having to consider (eg subtract) the difference in the velocities of the two frames. For that is what this law requires (as justified by an analysis of Maxwell's equations, amongst other rationales (and de Sitter's findings), although this needs greater clarification, I believe).
510. The situation is thus more comparable to the 2nd example in that there is no real velocity effect; that is what the law of light requires and hence that is what the principle of relativity should also require. The latter is therefore should not be incompatible with it. It is simply that the wrong transformation equation was applied. It worked when there was a complete addition of the velocities involved but would not be appropriate when no addition of velocity of the relevant moving body was the case - ie in the unique case of light when newly applied 'forces' don't in fact boost its speed as they do for all other bodies whose speed is not restricted by any constant (although they may be so restricted by the maximum possible speed). This fact will pertain to the need to conceive of the movement of all bodies as being placed along a single continuum of possible motion (as discussed further below) rather than dealing with the constancy of light as a one-off exception - with its own tailor-made transformation equation (which would subtract or add 0% of any reference frame velocity differences) - and with all other bodies which move much slower (and non-constantly) and so continue to be handled validly by means of the equivalent of the usual Galilean equations - which subtract or add the total difference (ie 100%) between reference frame velocities. A single continuum could possibly entail a sliding (?linear) scale between the two extremes of this spectrum. (Indeed; see below).
511. Although this law (of light's speed) does concern an outcome (when activating and releasing a source of light) that entails a velocity effect, the difference in the velocities of different frames when the velocity of light released in one of them is measured from the other, effectively does not have to be taken into consideration - when seeking to determine the actual direct effect (of releasing the light). If one did, by subtracting the greater speed of the faster frame (as occurs when applying the usual transformation equation as above), it would (as already mentioned) result in a conclusion that light's speed would now be less than its generally accepted one and only constant speed - of c. And that would conflict with both its own law and with the principle of relativity which requires that the law of light's speed, as all laws of nature, be realised - so that it is not only the same whatever may be the respective different velocities of the frames in which it is released and measured but also (and almost magically or inexplicably) whatever is the difference may be in the velocities of different frames if its velocity in one is to be measured from the other. Clearly, the speed of light cannot be boosted by being released from a fast-moving frames of reference as its velocity must always be the same (being a constant); the reason being that that constancy happens also to be the maximum speed at which anything can travel. Thus, it can't be boosted for two reasons.
512. Moreover, if measured from a faster moving frame, its speed from the slower frame will still be at its one and only constant and maximum speed; adding the speed of the faster frame by way of a Galilean transformation would now produce a speed that was too fast by exactly that amount. Why doesn't it appear to go slower (or even actually do so) when so measured ? Again, it is because its speed is not only the maximum at which anything can move but a constant and thus, in effect, that speed is also the minimum at which it can travel. That some body can not move faster than some maximum might be readily accepted as being due to some limitation in the mechanism of propagation through space but for it to never be able to move at any slower pace than that same maximum constancy - from wherever measured - is less easily accounted for in such mechanical terms. One would assume that whatever is the correct explanation, it must apply equally to the explanation of the upper limit of its speed as well - rather than the seemingly reasonable one given above. Presumably, it has to do with something inherent in Maxwell's equations. And thus, no matter how fast one may seek to travel past a source of light, once the light there is released, it will still move away from/past one at its one and only speed. Thus, if one moved past the Sun at near the speed of light towards the Earth and a beam of light was released from the Sun at that very moment, it would take that beam 8 minutes to reach the Earth and while it may be expected that the fast-moving observer would take, say, 9 minutes (moving about 10% slower than the light), this would not be the case. Rather, whatever speed one travelled at, one would not 'catch-up' with the light in the slightest. Somehow, the light would still reach the Earth at 300,000 kms faster than one's own speed!
513. One might handle this situation by utilizing one set of transformation equations for all mechanical problems and, as touched on above, a different (unique) type for that pertaining just to light's speed. However, Einstein makes a case that all such problems in nature (from both spheres) should be handled by one internally consistent method; with one set of transformation equations, based on just one principle of relativity, for all laws of nature involving velocities of (all) moving bodies. In the case of the body which moves in response to some mechanical law, that outcome can and does show variability - eg of velocity - depending on the force applied and the (relative) velocity of its frame which effectively serves as another 'force' (when measured from beyond that frame). In that case, it is consistent to remove that portion of its total velocity due to the frame's added relative velocity to ascertain the effect of the force alone, as the added motion does not affect the value of the former - whatever the frame's velocity - this being exactly what the principle of relativity and the law of inertia effect. [This was utilised by Galileo to prove his ideas on the motion and position of the Earth and Sun, etc. in terms of that latter law. He had no need to pursue it further seemingly. Actually, there are minute differences in the 'added ' effects - as explained elsewhere.] But when the 'body' concerned moves in response to an electromagnetic law, the outcome does not show (this same) variability - being by definition a constant which is therefore not varied whatever may be the 'force' or increased speed of its source that might otherwise influence it. Its consequent velocity is due entirely to its own inherent electromagnetic character at the instant of its 'release'. Whatever may be the difference in the velocity of the different frames concerned, it is not now appropriate to subtract the extent of that difference from the observed velocity of the light - as it remains the same regardless (as per its own law) - which, again, accords with a principle of relativity which accepts that it must (and does) accord with all laws of nature.
514. It turns out that, in effect, only zero percent of the extra speed of the faster frame may be subtracted from the observed speed of light (due to its constancy and 'maximumness') while that to be subtracted from the speed of any other (mechanically) moving body will vary from 100% (at zero speed difference, or on one and the same frame) down to approaching 0 % (the same as for light) if and when it could ever attain such speed. To place these two extremes into a single framework (ie that is neither mechanic nor electromagnetic but something that encompasses both spheres in one conception of nature - as 'electrodynamic mechanics?), a new transformation equation is needed which can recognise and adapt to the total variation found within this wide but single range - of any and all bodies that can move in nature . Something had to 'give' which allows just that flexibility to this desired generality. [The rationale of the appropriateness/justification of that desire might be usefully explored about here.] Because we're dealing here with velocities - which are comprised of measures of distance per time, it may be appreciated that these latter would be reasonable factors for any physicist, including Einstein, to address when seeking to resolve this particular quandary. But did he come to this realization by this route or ? We may never know.
515. Einstein points out that this incompatibility of the two laws is thus only apparent, not real, and that this 'mis-perception' was eventually appreciated to be due to the former belief that time and space were not dependent upon (did not vary according to) the difference in the velocities of the frames of reference concerned - as he eventually hypothesised they were. [However, why would anyone ever consider such a possibility ? Indeed, how did Einstein come to so consider it?] Presumably therefore he will show us how such dependence (relativity) translates into altering the principle of relativity such that it can and does prove to be more obviously compatible with light's constant velocity than was formerly apparent. This must require acceptance of the logic of his reasoning that this is not due to just the perception of time and space being relative but to their being truly so in reality (as treated within the analysis of electrodynamics in contrast to the use of the absolute equivalents within mechanics). [No. I believe now that the two must be accepted as the same!] Seemingly, the extent of the dilation of time and the contraction of length - according to the magnitude of the difference in the velocities of the frames concerned (whereby the information at least - of time and length - is thereby altered) - will account for an addition of velocities as described earlier being less than calculated where times and lengths were (wrongly) assumed not to so vary. The predicted total lack of any boost (addition) given to the velocity of light by a frame moving faster than the one from where it is measured - as implied by the law of the constant velocity of light - would thus be accounted for similarly. In all such cases, this up-dated version of the principle of relativity relies upon the previously argued case that time and space per se (and not just perceptions of same) are (somehow) relative and the arithmetic for the transformation equations must alter accordingly. [Or, is it still just the perceptions??]
516. As Einstein states, if we drop the assumptions that time and space are not affected by the motion of the reference frames (as detailed elsewhere) (and/or of the bodies concerned themselves??) then the theorem of the linear addition of velocities would no longer hold. There wouldn't be a full one to one addition but rather, some proportional addition only - becoming increasingly less as the difference in the speeds concerned becomes greater. At the speed of light, the addition of velocity of any body (including light) due to any increase in the velocity of its reference system would thus be perceived as zero - with time appearing infinitely long and length infinitely contracted (at least in theory). At half the speed of light (still extremely fast), the proportion of any such extra velocity would still be rather minute as such a body would (be perceived to) move across a very contracted distance over a very prolonged time. The exact magnitudes of such perceived dilations and contractions for all velocities between zero and the speed of light were effectively calculated by Lorentz when he derived his transformation equations based on the hypotheses of an actual (nor just perceived) contraction of any body's length as it moves through a still ether (as on the Earth at its known velocity) and an associated dilation of time which was less mechanically accounted for. [It is sometimes inconvenient to always refer to these new transformations - as derived by Einstein on his own basis and for his own purposes - as 'Lorentz's since that useage can lend itself to confusion as to which person's theory and pertinent transformations are being considered. I prefer (for now) to call them 'new' transformations.]
517. The magnitudes of these effects for all velocities were based initially on the quantitative results calculated from the measurements made by Michelson and Morley in their light experiments of the 1880s - seemingly then generalised from that of the Earth for all possible speeds from zero up to that of light. From these 3 values he was able to conclude that the effects of increasing velocity of a body through the ether was determined by some function of the ratio of that velocity to that of the speed of light (ie f(v/c). Einstein was thus aided when he derived the same equations even if based on different premises as to the cause of such variations in space and time; but, happily, they would be quantitatively identical. The extent of the alleged contraction of Michelson's apparatus arm at the speed of the Earth through the supposed ether (as well as the known speed of light?) provided the vital measurements by which the general equations could be calculated (by Lorentz) for all velocities up to that of light - ie in terms of the ratio v/c. The effects were essentially proportional according to a function of that ratio and we may assume that Einstein concluded the same equations by means of his own independent derivation (or was he essentially guided by that of Lorentz ??). It appears (from Appendix A), that he used only the 2 reference points - of zero speed and that of light (c) - assuming linearity between.
518. Einstein also describes in this Section the derivation of (his version of) the Lorentz transformations verbally - in terms of his usual example of the moving train and the stationary embankment. How may one, he asks, find the place and time of an event in relation to the moving train, when we know only the place and time of that event with respect to the embankment? Can we answer this (ie find their valid 'relations') such that the velocity of light remains at its accepted constant value in relation to both the train and the embankment and thus does not conflict with the principle of relativity? [This restriction on the calculation would be the one way to 'nail' the idea that time and space 'must' prove to be the only factors (now variables) left to so vary.] Well, this conflict was apparently not resolved when the usual ('absolute') values of time and distance (of Newton) were assumed to underlay that latter principle (as suggested above in terms of the addition of velocities). Einstein says that the answer to the question is a positive one and we may reasonably conclude (from the foregoing constrained calculation) that it must entail a conception of time and space that is not absolute and unvarying - but relative and thus variable (as defined above). For there is nothing else available to manipulate within the stipulated confines of the phenomena concerned. And this analysis and answer (given that one believes that all motion including light should be conceived as operating within one single (limited, not 'open ended') continuum, as discussed above) will also lead one to a set of more appropriate general transformation equations that would allow one to determine exactly the extent by which the increase in the speed of any body due to a faster reference frame must be constrained, according to the (?combined) speed of that body and latter frame, such that a body moving at or near the speed of light before that increase would gain no increase at all whatever the speed of that frame in relation to that of a slower moving frame from where it may be measured, while bodies moving at all slower speeds would gain some proportional increase only. Such variation in velocities would necessarily reflect the variation in their constituent elements - of time and space - and do so to an extent determined by the quantitative perceptual functions that correctly apply.
[Note: I have moved 3 paragraphs that were formerly placed somewhere above to here - to be integrated or removed as required. Viz:
519. The transformations so based (as considered in Section 11) were required to account for theoretical expectations under certain conditions, and the actual empirical findings predicted thereby - and later found to be correct (ie despite being based seemingly on perceptual not actual alterations in the magnitudes of time and space - under particular but significant circumstances - of practical relevance in understanding actual nature. Such transformations serve to remove the additional velocity due to a difference between the velocities of a reference system on which the event occurred and that from which it was measured (and so reveal the actual velocity due just to some law of nature acting on a body in that first system). For the same one transformation equation to provide the valid result, it must cope with bodies moving at very slow velocities on a system moving very slightly differently (if at all) from the one from which it is measured (or, more often, from the same system) - to one (ie light) which moves at the fastest possible speed of any 'body' - whatever the difference in speeds of any two systems concerned, and for all velocities of bodies and systems between. And in so doing, the amount of the extra velocity thus accounted for may turn out to be more or less than the amount of the original difference. For velocity to be so variable, the magnitudes of its basic constituents, distance and time, must also (and with precedence) display this same variability or malleability. Holding fast to the constancy of the speed of light 'forces' this newly appreciated malleability of time and space and hence of velocity to reveal themselves. If the former is accorded primacy and inviolability, it must follow that our conception of time and space must be accepted as variable. (Did he only then look for any plausible explanation of this surprising phenomenon ??)
520. Except for light, this domain of moving bodies was considered initially to fall within the category of mechanics and its laws and the results of measures of such motions to be thus 'handled' (analysed) accurately and so revealed by its own (Galiliean) transformation equations, while light fell within the sphere of electrodynamics in which its measured velocity required no subtraction (or addition) to reveal its true velocity; it being totally revealed by virtue of its constancy in any case. However, a more accurate analysis required a single new transformation equation which treated all such moving bodies as falling within a single domain of what might be termed 'electrodynamic mechanics' - from a virtually stationary body to one (light) moving at the maximum possible velocity - which latter speed was a constant so that it couldn't travel at any speed less than that either - and to every speed between, with all those velocities below that of light having to adjust to the latter's velocity ceiling. To do this, the equation had to find a way to vary the effect of added speed of one of the systems concerned such that it was graduated (eg reduced) proportionally as the speed approached that of light (where there could be no such (ie zero) addition) rather than be ungraduated right up until the speed of light was virtually attained and then having an adjustment that would have to become total extremely suddenly - as though one was suddenly coming under the influence of a different physical principle just at that point. The basis of such graduation was inherent within the equally varying (ie 'graduating') magnitudes of (the perception of?) the relevant measures of time and space (and hence of velocity) - now conceived as relative, according to the extent of those velocity differences between differently moving reference systems (when measuring activities on one from the other).
521. In other words, if the principle of relativity requires all laws of nature to perform as their respective laws dictate whatever the differing uniform speeds of different reference frames on which or from which they are operating or being measured, this can be exactly accounted for both light and for any other (mechanical) bodies moving at any other (slower) speeds - by applying the appropriate transformation equations based on values of time and space that are relative (to those velocity differences - as a proportion of the maximum possible speed) and not absolute. In Einstein's terminology, this is described as..."when 'changing over' from one body of reference to another." If absolute values were used, the results would suggest that the value of light's speed when measured from a slower frame should be that of its usual speed plus or minus that of that faster or slower frame and this would conflict with his conclusion (and hypothesis/postulate) regarding light having only one possible speed - from wherever measured. On that basis, light can not exceed (or be less than) its one constant and maximum/minimum speed. The values found by this approach for all slower moving bodies would however be the same as found before applying these new equations but these would now be incorrect as they would not be consistent with there being no increase shown or possible for light's speed if measured from some fast-moving reference point. It was to derive equations which could accommodate the constancy of light's speed with the variability of the speed of all other, slower, moving bodies whatever their speed, that was required. [Did Lorentz reason that the extent of the contraction of space (ie how much it varied from its prior absolute values) depended upon the velocity (v) of the body concerned - as a proportion of the maximum possible speed (c), which provided that upper limit below which all speeds had now to 'fit'? But, did he not believe still in Newton that there should be no such upper limit ? And what was his reasoning with respect to the dilation of time ? How did v/c relate to that element of velocity ? And on what reasoning ? ]
522a. [NB: There seems on the face of it to be effectively two bases for the 'necessity' of time and space (and thus velocity) to be variable (ie as gleaned from the 'later readings' referred to above): 1. the existence of a ceiling or limit on how much two or more velocities can added up to (thus providing a kind of 'compression' effect - to squeeze all such combined motions into a smaller, proportional 'fit' than thought to be the case previously) and 2. the signal lag inevitably occuring when perceiving information regarding a body's velocity (via time and space measures) from a differently moving reference system. Is it both or are these simply related aspects of basically one 'cause' ?] We may point our here that the idea of the need to 'fit' all velocities somewhere within the restricted velocity range zero to c (rather than zero and infinity) - the extent depending on what proportion v is of c - was subsequently noted in a dialogue on one of those physics chatrooms in which the term 'compression' was also used. We might reasonably say that time and space must adapt to such compressive/proportionalising influences accordingly. However, this could well be a rather seductive graphic aid rather than reflect the actual the actual mechanism concerned. On the other hand, while the theory of relativity is noted primarily as the source of the new idea that time and space are actually variable (rather than invariable), it is also the source of the idea that the total energy held by any body is a function not only of its velocity but also of its mass (even if not moving).
522b. In this regard, it imay equally be the case that the increase in speed of a body is ultimately limited (constrained) by virtue of its mounting mass and that further increase in its speed (and thus mass) by the application of more energy (force) becomes impossible at or near the speed of light. Thus, as its speed increases, the information of its velocity (with its associated times and distances) takes proportionally longer to reach any observer and that velocity (based on time and space measures) is proportionally squeexed as its upper limit is approached. Thus, as any body that is already moving with its reference system at a relatively high speed, say, can have that speed further boosted by a force applied to it within that system but with increasing difficulty as the combined (total) speed appraoches that upper limit, as its mass necessarily increases. There would thus be a corresponding 'information lag' regarding time and space measures (so that one was seemingly dilated and the other contracted) and a 'velocity (time per space) limitation' with comparable ceilings/limitations and consequent compressions'. Are they somehow both the same thing ??]
523. In our present example, the question to be answered was posed in terms of the time and place of an event on the embankment being used to find those same values for that event on the train. The train tracks or line can be taken as essentially the same one coordinate axis (eg called x). Einstein wishes to generalize this conception by imagining a 3 dimensional coordinate system of x, y and z axes, the latter two being those at right angles to the horizontal x axis - ie for the vertical and depth dimensions. This may be described as the Stationary coordinate system K (for the embankment) and that for these same axes relative to the train as Moving system K'. The values of y and z do not enter into consideration for the problem discussed here but, along with the dimensions for time (t) and for the speed of the train (v), they do prove necessary for a proper derivation of the equations required generally. These same values for system K' are shown as x', y', z' and t'. On this basis, the question to be answered becomes 'What are the values x', y', z' and t' (ie the places and times) of an event with respect to K' when only the magnitudes x, y, z and t for the same event with respect to K are given ?' That is, in Einstein's terminology, 'what are the 'relations' between these two sets of values'? Any equation which calculates x' , say, (or y' and z') in terms of some function that it is (or they are) of the values for x, v , t and c - that is 'the exact quantitative relations' between them as shown by such an equation, 'must'*, says Einstein, be 'chosen' (?somehow calculated) such that the law of the constancy of the speed of light (c) is satisfied with respect to both system K and K' (ie their speed differences can have no effect on light's constancy). Such calculations would seem thereby to also calculate the resultant velocities implied. (Seemingly, by having to meet this criterion, the values for all other (slower-moving) bodies are necessarily also found - ie by means of the new transformation equations in which such 'relations' are revealed and provided. [* 'Must' in the sense of the logic provided earlier regarding the primacy (as it were) of the generalized principle of relativity insisting on the recognition of the law of the constancy of light's velocity.]
524. [This law regarding light 'must' meet the criterion described for reason(s) not mentioned here by Einstein - but given elsewhere. It is, he claims, obligatory in physics that all motionally relevant laws in nature (including that for light) to meet the requirements of the proper principle of relativity; that is, must be found to apply as its law denotes in all uniformly moving frames of reference whatever their relative speeds and from wherever they are measured - ie after applying the appropriate transformations which take account of the differently perceived speeds of those mutually relative if differently moving systems of reference. For light, this means that if measured on a system moving at some considerable speed - of say 10,000 mph - from one that was relatively stationary, the subtraction of that extra speed from that found for light, wouldn't provide the tequired speed for light that would be found when it was measured within that slower or stationary system - for its law requires it to move at its one and only speed (c) in relation to any and all such systems and their observers or recorders - not one that is 10,000 mph slower than that - which result is what such a subtraction would obtain. If the speed so measured was that for some other body (with no law of constancy pertaining to it), such a subtraction would appear to be quite acceptable; its actual speed devoid of the boost given it by the faster moving frame, should (as formerly understood) be found to be the same as it would be when measured on the slower-moving frame by those on that same frame (given that it was propelled to that same speed by the same degree of force). As such, it would be seen as meeting the requirements of the original principle of relativity. But the restriction forced on all motion by having a limited upper velocity mean that all other bodies' motion had to be adjusted accordingly - to 'fit in', as it were.
525. That is, the different speeds of the two frames was formerly expected to have no effect on the resultant outcome - once the usual transformation was applied - ie of subtracting all the extra speed of the faster-moving frame from the total speed noted for that body in relation to the slower-moving frame. Once that speed difference was so accounted for, the result was always thought to show that the particular law of mechanics governing that body's motion (in response to that force) did indeed apply equally in the differently moving frames (whatever their speed differences and the speed of the body concerned) as the original principle of relativity required. But such a subtraction would not result in the speed of light at least to be found to still meet its law's requirement - of always remaining the same - for that calculation would give a result that was 10,000 mph too slow for light. However, it was later concluded by Einstein that the amount of speed of the faster frame to be subtracted in the case of the non-light body was in fact not 100% of that speed but, rather, some smaller proportion of it - depending on what proportion it (v) was of the ultimate speed (c) - of light (being thus some function of the ratio v/c) - which would be increasingly less the greater the speed difference approaches that of light. ]
526. One might of course treat light as some kind of exception and just accept that its speed will always be the same no matter what the speeds may be of the relevant frames of reference. But following the explanation given to the Michelson-Morley experiment by Lorentz involving light and the seeming contraction of length and a less convincing alteration of time he suggested to account for the different velocities found, Einstein (and others) seem to have sought a more coherent explanation - as witnessed in the question posed above by Einstein. Treating light as an exception was not further considered.
527. Einstein then gives us - ready made - the equations which satisfies the demand described above (ie finds that required function) - rather than showing us, step by step, the logic behind this derivation (at least at this point). Thus the values of x' and t' are simply 'given us' as:
This system of equations is known as the Lorentz Transformation - which Lorentz derived when seeking to account for the unexpected result of the Michelson-Morley experiment referred to earlier. In it, is implied the variations in the values of space and time which Lorentz related to the contraction of a body's length as it travels through the ether and to an associated dilation of time which apparently occurred due to some other less well explained phenomena concerning local vs 'non-local' time. The associated and resultant velocity would theerefore be likewise affected. Einstein concluded that variations in space and time of identical quantitative values actually arose instead when moving bodies on a given reference frame are observed and measured from a differently moving frame - without any actual structural influences so arising due to passage of bodies through a still ether. Rather, he accounted for such identical quantitative effects in terms of an alteration in the necessarily delayed perceptual information regarding these elements of velocity when so observed. Thus, he was able to apply transformation equations of identical form and effect when, as he phrased it, one is "changing over from one body of reference to another" - even if derived on the basis of entirely different premises. We may add that other physical parameters pertaining to bodies moving at velocity v, such as their kinetic energy and mass will also require analysis in terms of relevant transformation equations. [See also Einstein's own derivation of same - in Appendix A.
528. Thus Lorentz apparently got the right answers for the wrong reasons and thus interpreted their meaning incorrectly. He felt that space contraction (and to some extent time dilation) occurred due to motion through an ether which effectively masked the variation in light's speed otherwise expected by most (including himself initially) when it was produced and measured on the same reference system. Einstein, on the other hand, concluded that (?perception of) space did contract and (of) time did dilate - when one measures a body's motion on one reference system from a different one (moving at a different speed) except for light which, because it was already moving at the maximum speed possible, was always found to move at that one constant speed from wherever measured (ie c/c = 1); at that speed, any extra speed that might otherwise be reasonably expected due to its release from a faster-moving reference frame(when measured from a slower moving one) was not at all manifested. Otherwise the extent of the contraction and dilation of relevant lengths and times did depend on the extent of the difference in the two reference systems' velocities - but with the proportion of such 'boosts' becoming increasingly less the nearer that difference approached the speed of light (v/c). This proportionality provided by the new transformation equations was what differed from the previous (classical) method of analysis; it allowed a new interpretation to encompass the motions of all bodies from the slowest to the fastest possible - that of light - within the one consistent set of fully generalised equations.
529. The extent of any body's boosted speed (say) due to the speed of its reference frame (relative to that of a slower frame from where it was perceived and measured) was thus effectively reduced by an extent which depended upon the combined speed of that body and its reference system. This contrasts with an unreduced boost that was previously assumed to have occurred under a Galilean adjustment when the full (vs proportional) amount of that assumed boost would be subtracted to provide the actual net speed expected for that body (due to some added force previously applied to it). As its speed approaches that of light, so the additional speed due to that of its immediate reference frame would approach zero since (?perception of) the magnitudes of the components of that extra velocity (space and time) would accordingly contract and dilate towards zero and infinity, respectively. That is, no greater velocity can be perceived and measured for a body already moving that quickly - as by any fast-moving reference frame. He gives a more general derivation of his equivalent transformations (replacing trains and embankments with theoretical coordinate axes and systems K and K', etc) in Appendix A - based again on his own premises rather than those of Lorentz.
530. [Note: I must better differentiate what difference there is in the contributions of the fast moving body per se and that due to the speed difference of the reference frames (M vs S) to the apparent contractions and dilations concerned as they approach the speed of light. Thus if a train moved at many thousands of miles per hour (or even per minute) and a bullet fired within such a carriage was also (itself) exceedingly fast (for sake of argument in theory at an impossible many thousands of miles per second, say) then, when measured from a slower train, are the transformations applied with respect to both the train and the bullet or to their joint combined speed ? And does one consider the contractions of both the train and the bullet.....or what?? The answer to this probably available in Section 13 of Einstein's book - in that he re-states the theorem of the addition of velocities there, but now in terms of the theory of relativity - with 'Lorentzian' (ie ?Einstein's 'new') transformations rather than the Galilean ones used in his 'pre-relativity' example (of the man walking on the train). By this means, one can see that it is the combined speed of the frame and the body (ie W = v + w) which is reduced - by virtue of its division by an amount that is slightly more than 1 (which itself would leave it unchanged) by adding to that value of 1 in the denominator the value of the ratio of the multiple of these two velocity values over the multiple of the velocity of light multiplied by itself (ie squared). (But the reduction appears nevertheless to be applied only to the velocity element (v) of the reference system - not to that (w) of the body concerned. [Is 'squaring' and then taking sq roots only done to eliminate negative values or...? One asks this because it is noticed that in his famous equation relating energy (E) and mass (m), the immense value of c squared is a major factor; how did this meaningful value come out of the calculations IF the squaring of c (in the transformations at least) was normally always associated with the similar squaring of v (followed by their square rooting)and, effectively, only a mathemetical device ? However, we note that in the foregoing example, the square root of c^{2} is not then taken and thus we have an example here of c squared arising - unrelated to that device !?]
531. Where the velocities concerned are relatively slow compared to that of light, their multiple becomes miniscule compared to the square of the speed of light (an enormous number that can hardly be imagined) so that their combined speed W would be divided by an amount that is barely greater than 1; the amount of the extra speed (boost) added to that of the body concerned by the faster moving frame would thus be only very slightly less than the full 100% of that extra speed. If, however, the combined speed of system M and of the body moving within it begins to approach that of the speed of light (which would probably be most unlikely), their multiple would much more closely equate to that of the multiple of the two equal values of the speed of light (ie c squared) and so approach unity itself which, when added to the 1 already there, would approach a value approaching 2. This would halve the resultant value of the combined speed W and so indicate that virtually no extra speed would be provided by the speed of the frame, in such an unusual case.]
532. The first equation above shows how the position of the event of concern on the moving train, say (which may be a perceived contraction), can be obtained in terms of the known (uncontracted) position of that event in relation to the embankment. It will be seen that position x' is a particular function of the position of x from which the value of the train's velocity v times the time t taken - divided by the square root of a value slightly less than 1 - which will itself be slightly less again. Such a division would thus increase the value of the numerator term depending on the value of vt. The greater that value, the greater would be the extent of the contraction of the distance represented by x'. The dilation, if any, of the time concerned is then shown by t' and is a function of the value t - reduced to the extent represented by the term v (the train's velocity) over c squared times x - all divided by a value slightly less than 1 - which thus increases the value of the numerator term t accordingly. Thus, for high values of v, the value of t' will be increased (dilated) - by an amount determined by the v to c ratio for the initial distance x. Time will thus have slowed just as space contracts (in both cases as perceived and measured on M from S) - with increasing velocity of the train.
533. [It is at about this point that one might consider just what the basis is of the values by which the equivalent quantitative values of Lorentz's and Einstein's transformations, respectively, are - given that they have different qualitative bases. That is, by what means was it that both Lorentz and Einstein came up with the same formulae by which the value of W (w+v) is modified according to the extent by which it is divided by a varying amount over 1 that is determined by the ratio of the velocities' multiple divided by the square of c ? What was Lorentz's logic as the basis for arriving at this (his) particular formula which so proportionalizes the effect - that is, that followed from his ideas on the qualitative basis for such transformations (eg contraction due to the ether) ? How did he come to utilze c in this derivation ? And did Einstein just accept that same logic - even if his qualitative basis was that of the altered (?delayed) perceptions of length and time, etc - or, did he have a different basis for arriving at this same method of obtaining these varying proportions ? [See Einstein's derivation of the Transformations in Appendix 1.] See also my idea about how Lorentz used 3 points of velocity to obtain his equations and Einstein seemingly only 2. Also, fit in the idea about Galileo's transformations being derivable from Lorentz's - if c is allowed to become infinite (or at least the possible velocity of anything) and thus not constraining (theoretically if not practically) - so that the proportionality factor is (erroneously) not there as it should have been to so vary (?perceptions of) time and space. Also, see on what Voigt and/or Abraham based their derivations of v/c.]
534. It is not clear why Einstein made a more explicit case for the apparent relativity of time - in that he argues persuasively the case for its dependence on the motion of the reference systems concerned - while the basis of his conclusions regarding spatial distance in respect of such variability is less apparent. But the analagous parallels for the motion/speed of any body seem evident; velocity being a function 'equally' (?always) of distance and time (or, at least, of some ratio between them). Moreover, while he used as examples to analyse these seeming variables quite independent situations, it would seem more efficent to do so with respect to some particular motion (event) to which both pertained equally and in association (as they always are with respect to any velocity). In any case, he then re-iterates the idea that it was the false assumption that both time and space were only ever rigidly 'absolute' (and our definitions of same determined thereby) so that their roles in questions concerning velocity of motion led to the seeming incompatibility as between the constancy of the speed of light and the principle of relativity. By replacing these false a priori assumptions and definitions with his latest ideas that these fundamental factors were in fact both relative (ie not absolute or constant but varying and the definitions based upon empirical realities), the actual compatibility between the two 'laws' (and thus the wider relevance of this) could be revealed and accepted. In effect, it was the 'rigidity' of light's constancy that required the 'malleability' of time and space and not the other way around. [Or, were they both determined by some greater common factor??]
535. By dropping these two false assumptions, the dilemma of the incompatibility disappears since the wrongly accepted theorem of the simple addition of the unyielding velocities of classical mechanics becomes invalid (as Einstein anticipated). [But how and when did he first recognize that they were false and unwarranted !?] The basis for this invalidation of the wrongly assumed incompatability by this means is the crux of the theory of relativity (as he eventually 'saw' - even if probably arrived at initially by unknown means. In the discussion concerning the classical addition of velocities, we are concerned with places and times relative both to the train and to the embankment. If we know the place and time of an event with respect to the embankment, how can we find the place and time of that same event with respect to the moving train (ie as perceived by those moving with it)? Is there a thinkable answer to this question, asks Einstein, which allows the constancy of the speed of light to not contradict the principle of relativity? This would clearly entail analysing just what motion 'variables' are involved in any relevant calculations to which these 'laws' of motion apply. [Note: Any such 'thinkable answer' to this question was seemingly unknown by Einstein before his imagined tram journey of May 1905 with its revelations regarding time. It may have been - by Einstein, or several others - had the need to drop these two unspoken 'hypotheses' - in association with considerations about the apparent incompatibility of the two 'laws' of concern - been appreciated earlier. Apparently they weren't, at least as far my reading of same is concerned.]
536. Clearly, such an analysis must have focused primarily on the two components of motion - time and space. Its conclusion eventually amounted to seeking to find new (non-absolute) definitions (values) of time and space (empirically based) by means of which the velocity of light does not increase or decrease according to the speed of its source - as viewed from differently moving frames of reference as required by its law (actually 'by its inherent nature'). But we may ask just when this conclusion was reached? Before the constancy of the speed of light was a consideration, the means by which such positions and times were determined was through the approriate (Galilean) transformations which were very straight forward - as the time and distance values used were understood to be identical in the differently moving frames of reference and there was no known need to adapt to laws of motion in nature which didn't vary according to the velocity of its source. In his 1916 book, Einstein puts the matter where such a law (ie regarding light)is now known to apply - with the question: "Can we conceive of a 'relation' between time and place of the individual events (involving some motion of a body) relevant to both reference frames, such that every ray of light possesses the constant velocity c relative to both the embankment and to the moving train?" (and thereby satisfy the requirements of the principle of relativity)? This is the same as the 'general rule' that we had concluded (about 10 or more paragraphs above) was needed to accommodate all motions - both mechanical and electromagnetic.
537. Before explaining just what that relation must be, Einstein informs us that there is indeed a positive answer to this question and that it will be realized in his derivation of the more generalized transformation equations which can accommodate the law of light's constancy within the confines of the (adjusted) principle of relativity. By this means, the correct but unknown magnitudes of space and time can be determined for events occurring in one reference frame (for which these values are known) as seen (measured) from another moving relative to it, and vice versa. How the unknown values are calculable (and by how much they may differ for all bodies in motion according to the differences in the speeds involved) from the known ones is revealed in the particlular way the former prove to be a function of the latter. That is, x' = f (x, v, t, c) and t' = f(t, v, x, c); these particular functions will be revealed (by means of differential equations apparently) in the new transformation equations. Moreover, either set of values can be regarded symmetrically as the unknowns - as viewed from the other reference body. And just as Lorentz had simply to calculate these so that they exactly neutralised the extent of the unexpected result of Michelson's experiment, so too did Einstein have simply to derive their values so that the speed of light remained = c despite a difference in the speeds of the reference systems compared. He may have had Lorentz's equations as an heuristic model to guide him but needed to derive the values concerned in terms of his own conceptions of the effects of motion on such variables (as shown in his examples of the lightening strikes or Berne tram, etc).
538. [We might recall however that this approach (allowing time and space to vary) seems to have been what Einstein only later realized was what it 'all came down to in the end' but his earlier struggles may not have proceeded in quite this more generalized, logical and succinct fashion. Once he was convinced that a way had to be found to maintain the primacy of the principle of relativity for all laws of nature (that is, that his search was (in future) now to be essentially 'directed' by this requirement), the necessary mathematical derivation to answer the above question would prove to be no problem; its solution being at about the level of 6th Form algebra. But the greater problem was to discover a logical justification for varying such values of time and space - ie over and above seeking to remove the apparent incompatability (which, however, may have been justification enough). He had as a kind of model Lorentz's equations* for that researcher's particular needs (which weren't Einstein's) which may well have suggested the direction he should be considering; but he still needed, I feel, an additional rational basis for altering the values of time and space (where no ether proved relevant) to those which his algebra would then easily derive, if it hadn't done so already. This additional basis would seem to have been his realization, after his tram journey, that the hypotheses regarding the absolute nature of time and space was not justified. For he 'saw' how all non-local time in particular depended on the distance from and/or the difference in speeds of the reference systems to which it applied.
[*Lorentz had calcualted these to exactly neutralize the basis, as he saw it, by which the speed of light failed to vary by a predicted amount to verify the existence and influence of the still ether. With Einstein, it was rather the opposite! Ironically, Lorentz's 'local time' provided his answer. He had focused more on the Space(length) dimension while Einstein focused more on Time.]
539. To this point, Einstein has dealt with the two reference frames of the embankment and the moving train as being immediately beside one another and essentially in the one horizontal plane. To generalize his conclusions, he would derive his equations in a more theoretical arrangement in which all three planes of both coordinate systems are imagined as mutually perpendicular - horizontal, vertical and 'depth' - which systems occupy the same space yet can theoretically move 'through' one another (which of course wouldn't be possible if they were composed of the solid frameworks that such planes are assumed to represent). In addition, the time of the events so considered will be effectively a 4th perpendicular (orthogonal) dimension or plane in each system. Thus, the embankment is represented by a coordinate system K of these planes and the train by system K', with its planes, moving at velocity v relative to K along the same horizontal 'line'. The location of an event with respect to system K will then be represented by the position of the three perpendiculars x, y, z of its planes (as measured by the necessary measuring rods) and the time of its occurence (or duration) by the time t using the clocks of that system. That same event, but with respect to system K', would be represented by the four different coordinates x', y', z', t', again measured similarly. [Note: in the present examples concerning the train and embankment, the y and z dimensions are not relevant (the event's motion or otherwise being restricted to the x axes) but they would be if the events were such that they occured, as they might, anywhere in 3-dimensional space. Moreover, the transformations hold equally for any other positioning of the reference systems, with light moving in any direction. These more generalized derivations are beyond the scope of the present account. (Where all 3 spatial dimensions are not relevant in any discussion, one of them may be used as the time dimension to illustrate the latter's independence (ie being at right angles).]
540. The question to be answered in the more restricted present case can (again!; see parag .... above) be formulated somewhat more generally but precisely as: 'what are the values of x', y', z', t' of an event with respect to K', when the magnitudes of x, y, z, t of the same event with respect to K are known (those for y and z in both systems being in the present case identical)? The required 'relations' between the two sets must be so chosen that the value of c remains constant for the same ray(s) of light with respect to both systems K and K' - the speed of either system relative to the other having no effect (by definition) upon the value c. [We may recall here Einstein's thought experment at the age of 17; the answer could now be seen to be that the light wave wouldn't be dark and 'frozen' beside him. Rather, Maxwell's theory was the more correct view (devoid of its ether) for whatever his speed, light would continue to move away from him at its usual value of c as, effectively, his own speed had no absolute meaning, only relative; he may think he would be moving at near the speed of light (which does have absolute meaning) but may actually have been moving at virtually any (relative) speed less than that of light, or even none at all - relative to any appropriate frame of reference (as one moving quickly away from him).]
541. The problem of obtaining the correct 'relations' (once it was appreciated that the former ones were incorrect) to satisfy the latter requirement (that the calculations must maintain the value of c - rather than maintain the converse influence of the ether) is by means of deriving the appropriate transformation equations (which he likely already had a good 'model' of - from Lorentz). [The term 'relations' here might be usefully replaced by something that communicates better.] While Einstein simply states these equations at this point, we should, for a complete understanding, attempt to discover the basis of his derivation of them (which is, I had assumed, as already implied above). [How did Lorentz derive them?] We can possibly approach this (again) by first applying the normal Galilean transformation equations to such as the foregoing abstract problem on the basis of the long-held view that all laws of nature fall within mechanics and that times and distances are only ever absolute (ie always the same in all circumstances). For application of these equations (ie where x'= x-vt; y'=y; z'=z and t'=t) this would result in an answer in which the speed of light would appear to vary according to the speed of its source, which is no longer acceptable by Einstein - the speed of light being now accepted by him as a constant. But as we have seen, holding fast to the one constant value c for light appeared to result in answers which do not accord with the dictates of the principle of relativity (which requires all laws to perform identically in all inertial systems) - at least when the values of time and space applying therein remain constant/absolute. And, as with the constancy of the speed of light, experience appears to indicate that it too is a valid principle or 'law'; so both must somehow 'hold' true. Rather than having the speed of light appear to vary by being a function of any such variations in the speed of its source, it is now required to find the extent of the variations which must result in the values of time and position (distance/space) that result as a function of holding the speed of light to its one and only value in all situations - there, after all, being no other 'variables' available in matters concerning motion which could take up the unavoidable consequences of that insistent constancy.
542. The new transformation equations (in which the required functions and relations are inherent) are thus as follows (at least for x' and t'; those for y' and z' being, as mentioned, not relevant in the present context; the symbol \/ here represents sq rt):
[Note: These equations have already been given above. We may again note that these effects on distance (x') and time (t') of a body's motion observed on systen M (from S) as compared to those (x and t) observed on system S itself) pertain ultimately to the (comparative, perceived) velocity of the body concerned. There appears to be no single (combined) transformation which removes the velocity values per se, only the latters' two constituents - by means of their own pertinent transformations; I should check this out further however. Thus, what about the equations for V (and W) ? That is, where V = c + w / 1 + w/c = c shows why two velocities can't sum to more that the speed of light (c) and where v' = v + w / 1 = vw/c provides a combination of the two transformations shown earlier for distance (x') and time (t') individually ? But, I need to think these through more.]
543. Following the failure of Michelson's experiment, Lorentz decided that Michelson's expected result could still be the case if he 'adjusted' the relevant values of time and length (for the latter of which at least he eventually found a plausible electrodynamic basis which entailed an interaction between the atomic constituents of the moving body (measuring arm) and the hypothesised still ether, although not for the former). In his later developments of this 'theory', this effectively entailed the derivation of the appropriate transformation equations by which the speed of light could be accepted as being what they understood it should be in that particular experimental arrangement - ie as influenced by such a still ether - a kind of equal and opposite factor to offset light's now apparent constancy. This was premised on an acceptance of that ether's presence. But this derivation, while producing equations which were quantitatively correct for this specific purpose (and in fact were identical to those yet to be derived by Einstein), they were not derived on the same qualitative basis as would be the latter's - seemingly by "dropping the two unjustifiable hypotheses of classical mechanics regarding the absoluteness of the magnitudes of time and space (ie as though they were quite independent of the motion of the frame of reference)".
544. [It was of course his eventual insight into why these hypotheses were not justified and what would effectively replace them that provided the key to understanding the origin of his theory.] This would resolve the incompatability dilemma (of which Lorentz was seemingly unaware or unconcerned) - by virtue of invalidating the awkward theorem of the addition of absolute velocities assumed for Galilean mechanics. But happily, the speed at which Michelson's apparatus moved (30 kps) would (it seems to me) provide Einstein with one reference point (with the speed of light (c) the other?) with which he could presumably check his own still developing transformation equations - derived otherwise. This more relevant derivation would appear to have followed on from Einstein's insight into the appropriate measures of time and space (and velocity) after his imagined tram journey in Bern in May 1905 at an enormous velocity. Once he appreciated that our conception of (?others') time would vary according to the motion of their reference frames, he may well have put this idea to the test by adapting the Galiliean transformations accordingly to find that he thereby derived the same equations as advanced by Lorentz - of which he was no doubt aware (although possibly (as he later implied) not of their Michelsonian origins?). While these particular adjustments in space and time appeared necessary when dealing with the awkwardness of light in whatever sphere, the true source of such adjustments would be discovered in terms of the orientation taken by Einstein and not that by Michelson and Lorentz; they were oriented within different 'spheres of influence'.
545. As a proof that, on the basis of his derived transformations, the speed of light remains the same whether it is measured in terms of either the stationary or the moving frame of reference, Einstein offers the following analysis: A light signal is sent at its velocity c along the x axis for a time t and so covers a distance of x = ct. From the transformation equations we see that the relation between the distance x and the time t for the system K may be considered in terms of the similar relation between x' and t' of system K'. If the value of x is substituted by that of ct in the transformation equations, we get values for x' and t' in these terms. That is:
and from this, by division, the expression x' = ct' immediately follows. When referred to the system K', the propagation of light proceeds in distance accordingly. From this it may be seen that the speed of light relative to system K' is also equal to c (and not to such as c+v or whatever). If calculated for any other direction, it will be found that the light's speed remains at c. Einstein suggestd that we shouldn't be surprised at this since the relevant equations were derived to conform to this point of view. That is, by appropriately varying the magnitudes of x' and t' (ie space and time) c will be shown to have remained at its one constant value, regardless of the velocity of any frame of reference relative to any other. While Einstein refers consistently to his new equations as the 'Lorentz transformations', it is not clear whether he felt that a different analysis of their derivation could be interpreted as showing that his own view as to the basis of their derivation could also be discerned within Lorentz's own thinking (if the role of the ether was excluded). This seems unlikely in that Einstein's premises not only excluded the ether but dropped the assumptions of constant values for time and space and held to a truly constant value for c, while Lorentz's incorporated the ether, retained absolute values for time and space and appeared (?initially) to tolerate variation in the speed of light.
546. As introduced earlier, Einstein imagined his trip on a tram from the Berne Square at the speed of light (in about May 1905). He then appreciated for the first time that the time he would view on the Berne clock would remain at the time it showed when he began that journey - at say 11.00 pm. Moreover, he soon realized that anyone observing Einstein's fast-receding watch on his tram (if able to see it) would equally see it continue to show the time as 11.00pm. But to Einstein, his own (local) watch would, at say 5 minutes past 11.00, show the time then to be just that - 11.05, while in Berne, the observer there would see his local time - on the Berne Square clock - also show him it was then 11.05. It was this experience which apparently allowed Einstein finally to resolve the difficulties facing him over the previous many months or years. All the foregoing analysis (of seeking a means whereby the incompatability problem could be resolved) must therefore have only come to fruition after this experience. And while in his imagined circumstances, he would of course realize what the actual time should have been (or even really was) 'back there', the crucial point was that in all normal situations, this certainty is not the case. Without such prior knowledge, one simply wouldn't know what, with confidence, the time may be in others' distant localities. [I feel that this point is rather fundamental and wonder why it isn't underlined more; maybe I'm wrong here? Also, the matter of the essential symmetry in regard to current induction being a much more general principal in nature whereby the crucial relevance of the symmetry between the motion or non-motion of all reference systems is manifested is not mentioned or integrated into Einstein's 1916 elaboration - despite it having been introduced at the very start of his 1905 paper. Its 'relative' motion that is relevant and happily so as, apparently, no absolute rest place or motion exists.]
547. Blue summary for (the extremely lengthy) Section 11 analysis here?:
And then (at last!):
Section 12 - The Behaviour of Measuring Rods and Clocks in Motion.
548. Einstein next describes the behaviour of (rigid) measuring rods and clocks in motion' - presumably in terms of the new transformation conceptions introduced above. One might have preferred if it was called 'the perceived behaviour'...of these (rod and clock) means of assessing space and time. In any case, he discusses these in terms of his now more formalised method of analysis - with the coinciding systems K and K' and their relevant spatial and temporal dimensions of x and t. While he begins with the spatial (distance or length) dimension first, in his original insight following his tram journey, it was his sudden awareness of the influence of motion on time's perception that provided the breakthrough, so we shall describe this dimension first. Seemingly, he soon realized that the effects of delayed light signals on his perception of time would be of similar effect on that of distance - which we will thus describe subsequently. In regard to time, he places an initially 'stationary' clock at the origin of system K' (the soon-to-be moving system which is akin to his tram) when the time t' = 0 seconds. A second later, t' = 1 second (of elapsed time) on that clock in K', as observed thereon. In terms of an observer at the origin in system K however (eg the observer in the Square at Bern), the time at the start of K''s intended motion (to move at velocity = v) is t = 0 (ie on the clock on K'), while 1 second later it has, as per the appropriate transformations, become t = 1 / sq rt 1-v2/c2 - ie in terms of system K. This would be a slightly longer period of time than the 1 second it would remain as shown on his own clock in system K (or on system K''s clock), the extent of time dilation perceived by the observer in K depending on the magnitude of v.
549. In Einstein's original imagined tram journey, v = c and the imagined effects were therefore such that each succeeding second of the journey would see the Bern clock remain at its initial time, rather than simply go more slowly than usual. Hence, 5 minutes later, it would still showed the time in Bern as at the start (eg 11.00). Equally, the K' clock would show the K observer it was still 11.00, and vice versa, That is, the effects were equal and symmetrical; either system can be viewed ar the one moving or 'stationary'. {Einstein later realized that the velocity v could in fact never, in theory or practice, attain or exceed that of c so time could never 'stop' completely but v could in some circumstances be some large proportion of c - when perceived time would be significantly slowed.]
550. In regard to the effect of motion on perceived distance or length, Einstein exemplifies such in terms of a required 'length-measuring' rod (just as a clock is a necessry 'time-measuring' mechanism) which he places along the x' axis of the moving system K' so that its one end is, initially, at the origin where both x and x' = 0 and the other end is at x' = 1 (meter). He then asks, "what is the length of the rod relative to (ie as seen from) system K" ? The answer requires us to consider at what time t (of system K) this question is asked (and the rod measured) and at what velocity, if any, it is or will soon be moving (away). That is, where are the two ends of the rod positioned in respect of system K at a given time t for any magnitude of v. At t = 0, the transformation equations will show that these two positions will be x = 0 times sq rt 1-v2/c2 (for the end of the rod nearest the origin) and x = 1 times sq rt 1-v2/c2 (for the leading end of the rod) - so that its length would be sq rt 1-v2/c2 (seemingly just before it begins to move). When it is moving at velocity v relative to K, its length will = sq rt 1-v2/c2 of a meter. That is, it would be seen as shorter when in motion than when at rest, and the greater its velocity, the shorter would the rod appear (and for all intents and purposes would actually be shorter). [This latter qualification requires a more detailed explanation; see later.] For the velocity v = c, we would have a length = sq rt 1-v2/c2 which would = 0.
551. But for this result and any that might be based on a value of v greater than c, the sq rt becomes imaginary such that we may conclude that with respect to the principle of relativityty the value of c is a limiting velocity which can not be exceeded, or even reached by any moving body - even if an attempr were to be made to 'force such a body's speed to be 'boosted' from its basic speed by virtue of a fast reference system. Their combined speed could never exceed the speed of light - even if arithmetically their respective speeds may sum to a greater value than c. This is confirmed by the transformation equations which would become meaningless for values of v + w greater than c. [Note: light must traverse distance (space) to convey information, not only utilise time to do so. With a very large (long) frame of reference, such factors could prove relevant regarding distant events within same where the local time and distance in one area had not been stipulated/synchronised for it all - even if events in no other moving frames of reference were involved.]
552. If we consider the meter rod at rest on the x axis in system K, its length as viewed from moving system K' would also = sq rt 1-v2/c2. This too accords with the principle of relativity in the present context. The new transformation equations allow us to learn the altered physical magnitudes of bodies'lengths and of clocks' times expected in one reference system from another moving (very fast) relative to it, or vice versa, in terms of the altered values of the relevant dimensions of x and t - which those transformations imply. If Galilean transformations were applied instead, the moving body (the measuring rod) and clock would not, of course, have been expected to show any shortening or slowing, respectively, as a consequence of its motion.
553. We re-consisder Section 12 further here: Einstein shows how the (?apparent) length of a measuring rod and the time shown by a clock are affected when perceived from a differently moving frame of reference - with respect to which the measuring rod and the clock would of course be moving slower or faster relatively. This difference in their relative movement is apparently sufficient to account for the perceived alterations in the length and time concerned. These results follow from the application of the new transformation equations. Einstein has still to account for these by his own, non-Lorentzian derivation, I believe, based upon his conception of the effects of perceiving and/or measuring these values from a differently moving reference system (or does the 'glass ceiling' of maximum possible velocity enter into consideration here?) . Otherwise, the alterations in length and time, at least as described in this section, are simply assertions that rely ultimately on Lorentz's falsely based premises. [Does his Appendix derivation touch on this? And, as touched on above, does he explain how he (or Lorentz on different premises) arrived at the particular ratio formula by which the exact extents of these altered perceptions are calculated as a proportion of the speed of light squared ? And how do these altered perceptions of time and length contribute equally to the effects obtained (50:50?) ? Are they effectively determined by their necessary 'equal' confounding within any measure of velocity ?]
554. As mentioned, Section 13 (later) might provide further insight into these matters. It seems too easy for Einstein's logic to always begin by simply accepting the Lorentzian transformations (as though the logic and arithmetic thereof, even if based on the wrong qualtitative reasoning, was somehow self-apparent and needn't be further elaborated). Or, is he using the term 'Lorentzian transformations' (in deference to Lorentz's precedence in deriving them ca 1895-1900), if on faulty premises, even though he actually means his own, later derived, transformations (of equivalent values) - if differently based ? If so, then he still appears to rely in his reasoning (in his exposition to us) too much on the primacy of such 'Einsteinian' transformations as the basic premise or foundation from which the subsequent logic must necessarily follow. One would prefer to have a better breakdown or analysis of that particular basis, something more fundamental as to how the effects are a function of some particular proportion of the speed of light say (by which the denominator of W is, accordingly, varying degrees greater that 1. How was the basis of the particular rate or pattern of that varying quantity (according to the extent that the difference in the velocities of the two frames is a proportion of the velocity of light) arrived at? Was it based on Lorentz's faulty analysis of Einstein's quite un-related 'altered perception' theory ? Possibly this was already provided by Einstein in an earlier Section - which should therefore be reviewed ? Or, is this implied within his method of describing the dimensions of a one meter measuring rod in this present section ?
555. For he asks in this Section what are its dimensions relative to a system K if it was first determined to be of unit length (eg one metre) along the x' axis of a system K' moving at a different velocity to K ? To answer this, we only require, says Einstein, where the two ends of the rod lie with respect to K at a particular time t - of system K. At the time t = 0 (in K), the value of these two points x (beginning of rod) and x (end of rod) would be 0 times the sq rt of 1 - v2/c2 and 1 times that same sq rt - so that the distance between these two points would thus be = sq rt of 1 - v2/c2. But the rod is moving at v relative to system K. It therefore follows (ie if we accept the basis of his thinking (the validity of the transformations used) that the length of the (albeit rigid) rod moving in the direction of its length at velocity v would be = sq rt of 1 - v2/c2 of a meter. As the square of 1 is still 1, it appears that taking the sq rt of both it and of the subtracted v to c ratio, simply removes the quantitative effect of the squaring - (which may have entered into the formula in order to neutralise the effect of negative numbers or...??).
556. In any case, one can appreciate that the length of the rod is reduced from its magnitude of 1 metre when measured/perceived within its own reference frame, according to the extent of that latter ratio; the greater the velocity of one frame compared to the other (K vs K'), the greater would be the reduction in the perceived/measured unit size of the rod. As the velocity v approaches that of light ( c ) so the reduction in length would become almost total. The speed of light is the maximum velocity possible before length of any body would in theory disappear; as it couldn't become negative. This limitation is also apparent from the Lorentz transformations themselves, says Einstein, as they become meaningless if v is chosen to be greater than c. ('chosen', as this analysis is theoretical not empirical presumably). But if the body to be measured was of some unknown initial length, the use of the measuring rod would be of no value since it would 'shrink' comparably and so would show the body to have exactly the same dimension as before!? Proof of the effect must therefore rely on some other method. [What is it?] We may note that the use of the original Galilean transformations would have resulted in a conclusion that the measuring rod was not so shortened but then neither would be the body to be so measured. This seeming equivalence suggests that there is no practical difference even though we are assured that there is, really.
557. By like reasoning, it turns out that the length of time between two consecutive second-ticks of a clock (like points at the two ends of the rod) that is situated at the origin of an axis x' in K' (ie as the frame whose velocity relative to any other frame is the same as that clock) may be describes as having values (t) of 0 and 1, respectively , but when measured from a frame K moving at a different velocity, the length of that second between 0 and 1 seconds of time becomes = 1 divided by a value that is slightly smaller than 1; namely by sq rt of 1 minus v2/c2. The length (duration) of that second would therefore become greater than 1 and time would have slowed (been dilated). The greater the value of v, the longer would become the time (duration) of the second being so measured. If v could equal c, the length of the second (that is of time per se) would become infinitely long - ie the next tick would never occur. Nothing could 'happen'; it (and time) would take 'forever'.
558. While the physical dimensions of the (rigid) moving ruler are apparently not truly shortened in reality by virtue of being perceived and measured from such a differently moving viewpoint (that is, there is no 'compaction' of the molecular structure of the material concerned), nor seemingly does the clock's previously engineered mechanism 'go' more slowly so that there was less than 24 hours in a day, one's 'perception' and measurements of these dimensions are so altered - but presumably are so in terms of a particular conception of 'perception' that must differs from the more usual ideas about that mental process. This particular conception is related to the idea that there is no 'real' or absolute magnitude of any quantitative measure in the universe. There is no absolute 'standard' against which any quantity can be assessed, measured or compared. They are all completely 'relative'. This, indeed, is why the theory is called as it is presumably. If everything in our world or universe suddenly doubled, or halved, in size or length, whether concerning space or time, in an instant (relative to present magnitudes), there would be no differences perceived whatsoever - if we knew nothing of our prior sizes (which in any case had no absolute magnitudes themselves). A meter ruler would still have 1000 mms along its length (chosen by reference to some internationally agreed reference) and they would have exactly the same utility and meaning as they have now in measuring anything.
559. The same would apply to any such change in universally agreed timing systems as well, based for example on the period of the Earth's revolution around the Sun. Thus, the apparently altered perception of length and time when viewed from the differently moving viewpoint must have exactly this same kind of 'altered reality'. It apparently comes about because of the limitations inherent in how we can get information about same (by light) between such differently-moving frames of reference. As such, these perceptions of time and space are always only relative - to the reality of that limitation (as a consequence of constancy of the speed of light). It is as though there was such an instantaneous change in size or duration of time - but of just those dimensions subject to that informational limitation due to differing velocities. This explanation by Einstein thus differs from that advanced by Lorentz - albeit for alterations in length and time of these very same magnitudes for the same values of velocity v - eg by an action of the assumed ether in the case of length and by.......(?) for time.
Section 13 - The Theorem of the Addition of Velocities - 2: Fizeau's Experiment
560. In this Section, there is an elaboration of the matters described initially in Chapter 6 concerning the man walking along in a train when his velocity was measured relative to the train and to the embankment. On that occasion, his velocity was measured by means of Galilean transformations with its assumptions that there was no alteration in lengths or in times when different reference frames were thus involved. A simple theorem of the addition of such velocities was thereby derived. Such alterations were considered in the previous section individually but these would enter into consideration jointly and reciprocally whenever velocity is of concern. Most bodies, including clocks and measuring rods, can only be moved at velocities that are very slow compared to that of light so that we can not in practice test the implications of the suggested alterations in length and time theorised in the previous section nor seemingly the (classical) conclusions regarding the velocities concerning the man walking along the train. His velocity in relation to the embankment was concluded then to be that of his velocity relative to the train plus that of the train relative to the embankment. But we were forewarned by Einstein that this seemingly correct result, based as it was on the time-honoured Galilean or classical analysis, was in fact not the case.
561. In this present Section, he reproduces an equivalent situation, similarly analysed, but in which the walking man along the train is replaced by the idea of a point in space (as if representing the core 'essence' of a moving body) moving relative to a coordinate system K' - along its horizontal axis x'. Thus its position on x' after any given period of time (t') will equal its velocity (w) relative to K' times the time (t') so moving. That is x' = wt' . To assess this movement relative to the horizontal axis x (of a slower moving frame K), we can again try using the Galilean transformation equations to give us the position on x = (v+w)t - where v = the velocity of system K' relative to K. The motion of the point relative to the slower moving system K is thus equivalent of the motion of the man relative to the ('relatively still') embankment. In both cases, such 'motion' can be usefully quantified in terms of their respective velocities, here symbolized as W, which would = v+w. However, we may recall that Einstein informed us previously that using the Galilean analysis for such problems of motion did not really give us the correct answers (whether in terms of the distances or times involved nor therefore of the velocity so calculated). [He did not give the reason, at this point, why that method of calculation was not now considered valid.]
562. He simply points out that we can also (?instead) analyse this calculation in terms of the theory of relativity - with its 'new' transformation equations. That is, we transform thereby the statement that x' = wt' (ie with respect to system K') into its equivalent x = (v+w)t in relation to system K (ie for the motion or velocity of the point being considered) so that W now = v + w (as before) but now with that latter sum divided - by a value of 1+ vw/c2. That is, by an amount that is greater than 1 and hence the sum of the velocities contributing to the overall velocity of the point - relative to the slower system K - must be less (ie slower) than this total as calculated by means of the Galilean transformations. And the greater the multiple of v times w - in ratio with the value of c squared, the more this slowing would become apparent. This corresponds with the similar 'theorem of the addition of velocities' calculated in section 6 but is now calculated instead by means of the transformations underlying the theory of relativity - which provides an answer at variance with that based on the classical method. Clearly, he has suggested this other analysis as a superior one - putting it forward on the basis of his theoretical analysis of such problems of motion. As a theory, it can only be verified or supported by proving to be consistent with any evidence when the alternative isn't.
563. Hence, Einstein then asks 'which of these two forms of this theorem of velocity additions (and hence by implication views on the relativity vs absolutivity of the components of velocity - space and time) better accords with any empirical evidence' ? Such evidence is fortunately available, says Einstein, in an experiment performed some 50 years earlier by Fizeau (possibly to a different but related purpose) and repeated since by other well qualified experimental physicists. In this, the moving body or point is represented by a pulse of light considered initially as travelling through a motionless liquid (as water held in a long glass tube) which it is known to do at a given constant velocity w. It would travel at that same velocity relative to the tube as well. At what velocity will the light move when the liquid flows in that same direction with a velocity v - again relative to both the water and to the tube ? Einstein points out that in accordance with the principle of relativity, light should move, at least in relation to the now moving water, with the same velocity w as it did with respect to the still water and tube - no matter whether the water is moving relative to the tube.
564. This follows in the same way as we accept that the motion of any body relative to a given local reference frame isn't affected by the (uniform) motion of that frame; a ball thrown with a given force in a stationary train carriage travels the same distance in a given time relative to that carriage as it would if the carriage was moving steadily at any given speed. If this velocity is to be confirmed from a differently moving frame, the velocity of the local frame must be subtracted from the total velocity (ball + train) measured from the other frame. Under the former Galilean hypotheses (of constant or absolute magnitudes of space and time) this subtraction would (slightly erroneously) be of the full value of the velocity of the train; under the relativity hypothesis, however, that value would be a function of the proportion that that velocity was of the velocity of the light and as such, would be a slightly lesser but more valid value. This applies equally in the present case of light moving in water which, unlike light in a vacuum, there isn't the same constancy constraint on its velocity and thus it is appropriately treated as just another moving body. [Presumably, there are now other examples of empirical evidence of both this matter (the addition of velocities) and other aspects of special relativity. Yes; these are discussed in Section 16 which we might usefully consider next after this present Section (13) and before those of Sections 14 and 15.]
565. In any case, as we now know that velocity of light relative to the water, flowing or not, as w (and not c, as it is not a vacuum) and also the velocity of the flowing water relative to the tube as v, we may reasonably ask what is the velocity of the light in the water relative to the tube ? The use of the symbols w and v, we might reasonably assume, equates to their equivalents in our two previous examples - of the walking man on the train moving relative to the train and/or embankment and of the point moving relative to systems K' and/or K. The tube plays the part of the embankment or system K, the water the part of the train carriage or system K', while the light represents the man walking along the carriage or of the point relative to system K'. We seek to find the velocity of light relative to the tube which, as before, we represent by W. In the previous cases, this was found either by using the equations based on the Galilean or the Lorentzian (relativity) transformations, with the latter taken by Einstein as being the valid one - at least on theoretical grounds at this point.
566. What does the empirical evidence of Fizeau's experiment better support ? It agrees very precisely, says Einstein, with the latter, 'relativity-based' analysis, utilizing its associated transformations (and presumably therefore not with the classical version). According to the most recent measurements by Zeeman (ie when Einstein wrote his book), the influence of the water flowing at velocity v on the velocity of light relative to the tube (ie on W) is most validly represented by the equation W = v+w/1+vw/c^{2}, and not by the Galilean-based equation W = v + w. That is, it does so by reducing the sum of the velocities v + w by an amount that depends on the 'effect' of the extent of the ratio of these velocities (v and w) to that of the velocity of light in a vacuum (c ). The greater they are as a proportion of that latter speed, the greater is the reduction in the value of W, that is the velocity of light relative to the tube - when the light is moving within the flowing water. The reduction is not by subtracting v but by subtracting the complex (based on the ratio) that so characterizes the new transformations. [Are perceptual anomalies involved here?]
567. Finally, Einstein mentions that the Fizeau experiment was also used earlier by Lorentz to support his own theory regarding the velocity of light relative to a water filled tube. But he did so in terms of his theory regarding the electromagnetic structure of matter based on Maxwell's equations. Einstein points out that the latter theory in no way opposes the theory of relativity as an explanation of the Fizeau result however since relativity was based upon a new combination and generalization of the same hypotheses, formerly independent of each other, on which the electrodynamics of Maxwell was built. The implication seems to be that Einstein's theory is simpler but of wider scope and hence better approximates reality. He discusses this further in Chapter 16.
568. Section 13 has again presented the new transformations as givens or at least as the more valid suggested answers (hypotheses) to the problems of light that were of increasing concern around 1900. One would still like to see a fuller analysis of the rationale for presenting these particular answers even if they do appear to account for certain phenomena 'better' than other theories and to lead to verifiable predictions. Lorentz had his premises and postulates and Einstein had his others - the latter based on his two united principles and resulting interpretation of the perceptual realities arising from measuring activities arising on one reference system from another moving at a different velocity and which (informational lag) manifests itself quantitatively in terms of the proportion that certain velocities therein are of the velocity of light. This may be touched on later.
569. In the present Section 13 (summarized again here), Einstein addresses the more realistic situation in which bodies and clocks can actually move only at velocities that are typically very small compared with that of light. (The foregoing examples were theoretical ones in which the velocities were such that the effects produced were in theory more obvious.) Are there any examples where these surprising theorized effects might actually be measurably manifested and so confirmable in reality? He approaches this by deriving other predictions from the theory - ones which should be, and indeed have been, confirmed by experiment.
570. The prediction so derived pertains to the theorem of the addition of velocities, the negation of the two underlying hypotheses of which proved to be crucial in deriving the new transformations and thus the theory of present concern. That theorem can be derived for motion (in just one direction) on the basis of the Galilean transformations (for which the unspoken hypotheses of classical mechanics on time and space are assumed; see Chapters VI and XI). This was done earlier in terms of the man walking along the train carriage but may be repeated in more theoretical terms by replacing him with a point (as a small 'body') moving along the x' axis of moving system K' such that x' = wt'. If we assume that classical mechanics can handle this analysis, we can express x' and t' in terms of x and t - so that x = (v+w)t. This expresses the extent of the motion of the point moving in moving system K' but with respect to 'stationary' system K. We may call this velocity W which, as before, = v+w and so represents an addition of the two velocities concerned as per a simple Galilean transformation based on the original principle of relativity which of course assumed unchanging (absolute) values for time and space (the necessary components of all velocities concerned).
571. But in view of the later ideas about the apparent limitations of treating such additions in this classical way, we may now determine the result on the basis of the theory of relativity in which the principle of relativity has been adjusted (perfected) by a recognition that time and space are not absolute and independent so that the appropriate transformations are those in which this new reality is incorporated. Thus, in the equation x' = wt', these values of distance and time in system K' will be expressed in terms of x and t of system K by making use instead of the more generalized transformations in which these components are allowed to vary accordingly. In this case, W becomes a slightly smaller value (as distance shortens and/or time lengthens) to an extent depending on how much the addition v+w is reduced by the magnitude of its denominator - as per the relevant transformation equation: W = v+w / 1+vw/c2. This is the theorem for the addition of velocities in one direction that arises when calculated in terms of the theory of relativity (ie where the actual variabilities of time and space are recognized). (The variables of the denominator term are not first squared, to bring them to a positive sign, and then the square root taken - to remove negative values (as in the typical equations shown above) - possibly as this calculation is for one direction only. Nevertheless, one notes that the value of c is squared.)
572. Is there a way, he asks, by which we can decide which of these two theorems better accords with experience ? Einstein refers us to an experiment conducted much earlier (ca 1865) by the French physicist Fizeau, and later confirmed by Zeeman (ca 1912). In this, a beam of light is sent through a clear liquid contained in a long glass tube when the liquid is either still and light's velocity through it = w (which compares with its speed in a still vacuum = c), or when the liquid is flowing at a velocity v, in the same direction as the light. The question arises - what is light's speed in the latter situation? Is it faster than w in the flowing liquid or is it just the same as when the liquid was still? Now, the principle of relativity would require the speed of light to remain constant (for a given medium) relative to that medium - whatever velocity the latter may be, relative to its own surroundings. That is, that the two velocities should be the same. (This recalls Einstein's own thought experiment when he imagined he was travelling at near the speed of light (relative to the ground, say) and wondered how light travelling beside him, in that same direction, would appear to him. His later conviction that Maxwell's equations were correct answered that question: it would, nevertheless, still move forward from him at its one constant speed - his own relative speed having no absolute meaning or bearing; all one could say was that his speed was always 186,000 mps slower than that of light but that it could have been anything (below the speed of light)...or nothing - relative to other reference bodies moving relative to himself.)
573. To establish this theoretical expectation, Fizeau's experiment may be consulted, says Einstein. In it, the tube can be equated with the railway embankment or coordinate system K while the flowing liquid becomes the moving carriage K' and the light becomes the man walking along the carriage (or the moving point similarly). Is the velocity of the light W better represented by the Galilean result = v+w or that based on Einstein's more general transformations of his theory where W = v+w / 1+vw/c2 ? Fizeau's and Zeeman's experiments both indicate that the latter, not the former, provides the correct result to a high degree of accuracy. Einstein points out also that results such as those of Fizeau and Zeeman (and possibly that of Michelson?) were explained previously by an electrodynamic theory advanced by Lorentz long before the theory of relativity was produced but that the basis of that former theory "...does not in the least diminish the conclusiveness of such experiments as crucial tests in favour (also) of the theory of relativity [possibly differently interpreted?] since that basis (the electrodynamics of Maxwell-Lorentz) in no way opposes the theory of relativity". Rather, says Einstein, his theory was developed by combining the two formerly independent hypotheses on which electrodynamics itself was built and generalizing this new conception. His analysis focused initially upon the kinematic logic of this new conception, argued mostly by theoretical means, in which considerations entailing the actual mass, forces and energy involved in the motion of bodies, and hence the wider electrodynamic implications as based on these conclusions. were not addressed. These were presented in Part II of his 1905 paper but aren't addressed as specifically here (in our resume of the less technical 1920 account) as have the foregoing kinematic aspects of Part I of that earlier paper. The electrodynamic considerations are described more thoroughly below in a later resume of Part II. (see paragraphs ... to ...). Meanwhile, the 1920 version continues next from Section 14 and beyond, although we cover firstly Section 16 (with some electrodynamic aspects) as it seemed (on first consideration) to fit better here than in the original sequence (although this could be altered in time).
Chapter 16 - Experience and The Special Theory of Relativity.
574a. Consideration is given here to some of the electrodynamic evidence in support (or arising out) of the theory of relativity, before returning to similar matters discussed in Sections 14 and 15. As pointed out in regard to Fizeau's experiment, such evidence is however difficult to obtain since the velocities required to allow a ratio with that of light which is practically measurable are rather rare. Seemingly that of light through water did however provide one such means. But because the theory arose in effect from the prior electromagnetic theory of Maxwell, as interpreted by Lorentz, all the evidence supporting that theory can also be cited as support for the relativity theory. The latter theory can thus be cited as correctly predicting certain effects on the light reaching the Earth from the fixed stars due to the relative motion between the two. The yearly motion of the apparent position of the stars results from the relative motion of the Earth around the Sun, referred to as 'aberration'. The colour of the light from the stars is also affected by this relative motion, called the Doppler effect, compared with same here on Earth. There are other experimental results involving light which the Maxwell-Lorentz theory can, admittedly, account for better than other earlier theories but not, says Einstein, as well as does the theory of relativity.
574b. [We place here a simple definition of the Doppler effect (as noted in Wikepedia) in the event that we may wish to elaborate on this topic as one of the bases of Einstein's later derivation of his famous equation E = mc^{2}: The Doppler effect (or Doppler shift), named after the Austrian physicist Christian Doppler who proposed it in 1842, is the change in perceivd frequency of a wave for an observer moving relative to the source of the waves. It is commonly heard when a vehicle sounding a siren approaches, passes and recedes from an observer. The received frequency is higher (compared to the emitted frequency) during the approach, identical at the instant of passing by, and gradually lower as it recedes. For waves that propagate in a medium, such as sound waves, the velocity of the observer and of the source are relative to the medium in which the waves are transmitted. The total Doppler effect may therefore result from motion of the source, motion of the observer, or motion of the medium. Each of these effects is analyzed separately. For waves which do not require a known medium, such as light or gravity, only the relative difference in velocity between the observer and the source need be considered. The relativistic Doppler effect is the change in frequency (and wavelength of light caused by the relative motion of the source and the observer (as in the classical Doppler effect), when taking into account effects of the special theory of relativity. The relativistic Doppler effect is different from the non-relativistic Doppler effect as the equations include the time dilation effect of special relativity (arising out of the implications of using v/c rsther than just v presumably - with its time (especially) and space components) and do not involve any medium of propagation as a reference point. They describe the total difference in observed frequencies and possess the required Lorentz symmetry.
574c. Understanding relativistic Doppler effect requires understanding Doppler effect per se, time dilation and the Aberration of light (which we shall elaborate on elsewhere). As a simple analogy, consider two people playing catch. Imagine that a stationary pitcher tosses one ball each second (1 Hz) at one meter per second to a catcher who is standing one meter away. The stationary catcher will receive one ball per second (1 Hz). Then the catcher walks away from the pitcher at 0.5 meters per second and so catches a ball every 2 seconds (0.5 Hz). Finally, the catcher walks towards the pitcher at 0.5 meters per second and now catches three balls every two seconds (1.5 Hz). The same would be true if the pitcher moved toward or away from the catcher. By analogy, the relativistic doppler effect shifts the frequency of light as the emitter or observer moves toward or away from the other. We may imagine either an emitter or an obsever traveling to the right. While the color shift appears similar, the aberration of light is opposite. To understand this effect, again imagine two people playing catch. If the pitcher is moves to the right and the catcher stands still, then the pitcher must aim behind the catcher. Otherwise the ball will pass the catcher on the right. Also, the catcher must turn in front of the pitcher, or the ball will hit on the catcher's left. Conversely, if the pitcher is still and the catcher moves to the right, then the pitcher must aim in front of the catcher. Otherwise, the ball will pass the catcher on the left. Also, the catcher must turn to the back of the pitcher, or the ball will hit on the catcher's right. The degree to which the pitcher and catcher must turn to the right or left depends on two things: 1) the instanteous angle between the pitcher-catcher line and the runner's velocity vector, and 2) the pitcher-catcher velocity relative to the speed of the ball. By analogy, the aberration of light depends on: 1) the instanteous angle between the emitter-observer line and the relative velocity vector, and 2) the emitter-observer velocity relative to the speed of light. In many of the equations applicable in understanding and analysing the above effects more precisely, the usual ratio of v/c plays its typical special relativity part as compare to the case under the classical Doppler situation (where any observer typically moves with the source of the light concerned).]
575a. In addition to the above two effects (of Aberration and Doppler), there are two other classes of experimental results which Maxwll-Lorentz seek to explain similarly but which require the introduction of auxiliary hypotheses - which the theory of relativity can apparently again better explain without such extraneous aids. The first group concerns the behaviour of electrons whose negative electric charge tends to force them to disperse from one another when they are produced as a stream of such discharges. Their motion can be studied very exactly under the deflecting influence of electric and magnetic fields. It was eventually discovered that the dispersion of such electrons is to some extent opposed by the effects of an attractive force between them due to their mass; this was considered to be a form of gravity. But Einstein appears to focus more on the fact that the consequent relative distances between electrons which are assumed to remain unchanged during their motion leads to a law of their motion so based which, however, does not agree with the evidence.
575b. To overcome this, Lorentz, in his electron theory of 1904, advanced an hypothesis similar to that first suggested by Fitzgerald in response to the apparent failure of the Michelson experiments of the 1880s, and independently promoted by Lorentz himself to this same end in the 1890s, in which the dimensions of the electrons (vs that of the measuring arm of Michelson) contracted in the direction of their motion (in consequence of an hypotheised electrodynamic effect of their motion through the ether). The magnitude of this contraction was calculated (on the basis of what would be required to exactly eliminate the troublesome discrepancy) to be = sq rt 1 - v^{2}/c^{2} is, an amount that depended on what proportion their velocity (v) was of that of light ( c ). There was , says Einstein, no basis for this hypothesis within the then known facts of electrodynamics although a contraction of just this quantity did exactly overcome the problem in the law of electron's motion which, by this theoretical adjustment, was however now in agreement with the factual evidence. This suggested that such an apparent contraction or something like it may well have been the case (and one for which the method of calculating that particular magnitude may not have been inappropriate) but which nevertheless likely required a different (?more acceptable) explanation, noted Einstein, than that advanced by Lorentz - being due to the assumed electrodynamic effects of the motion of any body (with its constituent molecules and electrons) through the assumed still ether. Seemingly, Einstein didn't accept that such an explanation could be justified in terms of the laws of electrodynamics as they stood in 1904, as well as the uncertainty regarding the actual existence of the ether required.
576. The theory of relativity, on the other hand, leads to the same accurate law of the motion of the electron - but without requiring any special post hoc hypothesis regarding the structure and behaviour of the electron - as its physical contraction or that of any body in motion (nor indeed the existence of a causative ether), as will be discussed later. This proves comparable to the situation regarding Fizeau's experiment where no special hypothesis was required with respect to the physical structure or nature of the moving liquid. In like fashion, Einstein describes the second class of experimental facts which the Maxwell-Lorentz theory does not account for without applying a similar ad hoc hypothesis. This concerns the attempt to measure the motion of the Earth through space by experiments performed on the Earth itself. This seems to refer (if obliquely) to the experiments by Michelson in the 1880s as alluded to above. However, such experiments were motivated by an interest by Maxwell in confirming the role of the ether in his latest theory of the propagation of light. Presumably, they may also be described, not inaccurately, in the terms used here by Einstein. In any case, he has previously referred in Chapter 5 to all such attempts proving negative.
577. The major attempt in this regard was of course that performed by Michelson in 1887. Before the theory of relativity was put forward, it was, says Einstein, difficult to become reconciled to these negative results. But Fitzgerald and Lorentz sought to rescue this situation by independently hypothesising that the arm of the apparatus used by Michelson contracted due to the effect on its atomic/molecular structure of its motion through the assumed ether of an amount that exactly compensated for the very slight difference in the time concerned from that expected if the ether had indeed been present and still. Such a contraction (or apparentt contraction) of a body because of its motion would indeed be the correct one, says Einstein, as would be predicted also by the theory of relativity (see also Chapter 12 on this) providing that such an apparent contraction did not involve any physical change in the structure of the body concerned (as the material of the apparatus arm) whether due to the effects of an ether (as a unique coordinate system hypothesised by Lorentz) or by any other physical means but was due instead, says Einstein (if in rather brief, abstract, general terminology), to the effects of his unification of the two fundamental (and not incompatible) principles of the theory of relativity - which required no such extraneous hypothesis.
578. By this, he means that there is an apparent contraction (of sorts) but the prime factor in its manifestation is not the motion of the body concerned relative to an ether (which in any case he says doesn't exist) but rather its motion relative to a differently moving reference system as chosen for the case in point. Thus, says Einstein, for a coordinate system moving with the Earth, the apparatus used by Michelson is not shortened in any physical sense (nor would it be in the sense meant by Einstein) but it would ('appear' to) be so 'shortened' in that latter (relativity-based) sense for a system which is at rest relative to the Sun (that is, from the perspective of the Sun or anything of comparable 'stillness' relative to the moving Earth).
579. Thus, to Lorentz, any body in motion is moving through and relative to an ubiquitous still ether - which 'ether-passing' motion has a physical effect on that body's molecular structure effecting an actual contraction - to become truly shorter than it was before - that is, in some absolute, physical sense. To Einstein however, such a body only 'appeared' to have become shorter - not shorter than it actually was before, but shorter than it had always 'appeared' to be before - since any body has no absolute, objective size or length but only the relative size we have previously accorded it - relative to the relevant reference systems chosen when examining and perceiving its size or length. Additionally, this apparent effect has no need for an ether to so account for it. Rather, it is accounted for by....(to be completed). Lorentz made the oversight in not appreciating that time and space (and thus velocity) were not absolute but relative (as did most physicists before 1900). But just what they are relative to (dependent upon) should always be made clear: namely,....
580. The foregoing statement of the theory implies that such had always been the case but that it hadn't been previously recognised - especially as such effects are not manifested within measurement capabilities for most normal velocities of moving bodies. Also, the effect on a body's velocity, as a proportion of that of light, must entail time as well as distance and in fact, Einstein's breakthrough came, he later claimed, when he first realised that it was indeed the relativity of time especially that had been so overlooked. The perception of both time and space require the information of their status to be conveyed by means of necessarily non-instantaneous (eg light) signals - as the quickest and most reliable means. This necessitates that their perception (the only reality of their relative natures that we can know) is a function of the difference in velocity between the reference systems involved - as a relative proportion of the velocity of light - being the upper restraint of all velocity. It's as though all possible velocities are part of an enormous cake of finite size and none can take up any (eg high) values they wish and still somehow squeeze (only near the limit) under that upper limit; rather, all velocities, no matter how small, have to be some proportion only of that ultimate upper limitation - even when very much less (slower). And the mechanism by which this proportionality is manifested (arises) is that of the information lag which is necessarily always an integral part of such perception and measurement.
581. In Section 16 (summarized again here), Einstein discusses the extent to which the special theory of relativity can predict or explain natural phenomena not accounted for otherwise. [See also earlier discussion of this Section above.] This begins with the question: 'To what extent does 'experience' (empirical evidence) support the theory of relativity ? To Einstein, 'experience' seems to represent all non-theoretical (empirical) data that arise 'out there' in the actual world - from observation and experiment. And where it proves to be well founded and consistent, it serves as a necessary final arbiter in establishing the validity of the associated theories. He mentions that this question is not easily answered for reasons he has already touched on in his discussion of the experiment by Fizeau. That is, that......] In this regard (seemingly), he points out that the relativity theory 'crystallized' out of the Maxwell-Lorentz (M-L) theory concerning electromagnetic phenomena (which should probably be more explicitly described here) and as such, may be presented as accounting for much the same empirical evidence seemingly, if not better. Whatever supports the former also supports the latter theory. Thus certain effects on the light reaching the Earth from the fixed stars if explicable or predictable by the M-L theory are also explained by relativity theory - which equally predicts such effects as noted by experience precisely. These effects are complex to describe verbally but are well known by the terms 'Aberration' and 'Doppler effect' which involve the motion of the observer relative to the source of the wave phenomena (as light) concerned, or vice versa. In any such case, the actual direction (position) and/or the frequency (radial components) of the light at its source (as the fixed stars) may be perceive differently by the observer due to that motion (see brief discussion earlier). The arguments based on the electromagnetic M-R theory to account for these phenomena are numerous but apply equally to relativity theory - ie where the observer is not associated with the moving sources concerned. They again entail the appropriate application of the ratio of v/c in place of v alone by means of the new (non-Galiliean) transformation equations. It appears that in the case of these two phenomena at least, the M-L theory and that of relativity provide the same results - with no apparent superiority associated with the latter interpretation since the phenomea are relatively straight forward with no additional ad hoc hypotheses required by the former.
582. But, says Einstein, there are two other related classes of empirical facts which while seemingly also explicable/predictable/accountable by M-R theory, are so only after the introduction of auxilliary hypotheses which can be seen to be 'extraneous' (and thus imply that the reasoning behind that theory may well be faulty and incomplete) once the more efficient self-contained relativity theory is applied to them. Amazingly, the description of the second of these returns us to the original Michelson-Fitzgerald-Lorentz interpretations concerning the measurement of light as propagated through an assumed ether. But the first of these two questionably explained classes concerns the behaviour of certain 'rays' now known to be streams of negatively charged electron particles which travel extremely fast. The law of motion as it affects these may be conveniently studied when they are subjected to electomagnetic fields. From a theoretical point of view, the negative charges of their constituent parts should cause them to separate due to the mutual repulsions expected of like charges but some other 'attractive' force such as gravity appears to account for the fact that they in fact remain cohesive and stable.
583. In terms of classical mechanics, their size should thus remains constant during their motion due to such fields. But certain experiments have apparently concluded that they must have contracted in size and Lorentz hypothesized that it was their motion (seemingly through a still ether) which accounted for same and that this occurred in proportion to their speed of motion according to the now well known expression: sq rt 1 - v2/c2. This 'law of motion of the electron' was subsequently confirmed with great precision even though it could not be justified in terms of any known electrodynamic facts. However, again, the theory of relativity leads to this same law but without the need for auxilliary electrodynamic hypotheses concerning the structure or behaviour of the electron itself. This is quite analogous to the predicted explanation of Fizeau's experiment where no hypothesis concerning the physical nature of the moving liquid medium was required. In both cases, the experimental results could be thus more elegantly accounted for in terms of a different theory - devoid of any such extraneous hypotheses about physical matter, even at the atomic level.
584. The other (related) class of facts which Einstein's theory can account for without use of such hypotheses pertains to the question of whether or not the motion of the Earth in space can be made perceptible in terrestrial experiments. In Section 5, it was pointed out that all attempts of this type (seemingly including those of Michelson) led to negative results. That is, that it was not found that light's speed when sent through an assumed ether from its source on the moving Earth varied (which would indicate that such an ether did exist and so confirm (also) that the Earth moved in some absolute sense relative to such an absolutely still ether (thus assumed) - rather than only relatively with respect to the Sun, say) but had in fact remained stubbornly constant. But then Fitzgerald and Lorentz advanced similar hypotheses designed to explain such negative results such that the original expectation (that.....) had in fact been confirmed. Otherwise, it was difficult to accept this negative result, so firmly had the concept of the ether and its role in light's transmission become entrenched. Moreover, Michelson's technique was so precise, it seemed to leave no room for doubt.
585. Einstein then describes what appears to be the period of experimentation just before the Michelson experiments. In these, he notes that the assessment of motions in one coordinate system from another (ie as analysed by kinematics) traditionally proceeded in terms of the usual Galilean transformations (with its unspoken assumptions about unvarying time and space, and thus of the unlimited extent that the velocity of any body could attain). But where the M-L equations for electrodynamics hold for the (usually) stationary system K in such early studies, it appears that, unexpectedly, they did not hold with respect to a system K' moving uniformly with respect to that stationary one as they should have seemingly - at least where the usual Galilean equations were applied. One interpretation was that the allegedly stationary reference system K in such studies, of all Galilean coordinate systems, was somehow unique. It was thus interpreted that K was 'at rest' with respect to an assumed ether - itself stationary in space - ie as though K was (effectively) that still ether (I think). All coordinate systems K', K'', etc moving uniformly with respect to this K were thus seen as moving with respect to that still ether but that this unique system K did not display any motion relative to any other coordinate system(s). Certain 'complicated' laws were then suggested (eg by Lorentz) that affected K' due to its motion relative to that system K ether - which helped expain the unexpected results in all such cases. If K' was considered to be the Earth - moving relative to that ether, the existence of the assumed effects of this motion (called 'ether-drift') were then sought in a number of such experiments to so account for the anomalies typically found.
586. In one of the most notable of these attempts*, Michelson devised a method which being so precise was thought to be decisive. But the experiment gave a seemingly anomalous negative result which, as implied above, was most perplexing to physicists. Fitzgerald and then Lorentz sought to rescue the theory by suggesting that the body of the measuring apparatus had contracted in the direction of its motion relative to the ether just enough to compensate for the unexpected difference in time found in the experiment. Now, interestingly, Einstein states that this solution to the problem was (as seen from the point of view of the theory of relativity) actually the right one - by which I assume he means was 'quantitatively' the right one. [See Section 12 for the basis of this conclusion.] But that latter theory provides a means of interpretation of its qualitative basis that is much more acceptable and satisfactory than that of Lorentz and Fitzgerald. For it doesn't entail acceptance of any 'specially favoured' coordinate system that requires belief in an unproved stationary ether system or a need to somehow verify its existence. The relevant laws of electrodynamics can (and must) remain the same for all such applications.
587. According to the theory of relativity, an pereceived 'contraction' (of sorts) of the bodies concerned does indeed occur but this follows from the operation of the two fundamental principles of the theory, without the need for any extraneous hypotheses. [Ie differently-moving reference frames can have no effect on the operation of natural laws.] And the prime factor in such contraction is not the motion of the body itelf ('rubbing up' against the still ether) - for which there is no proof and hence no 'meaning' - but rather is the motion with respect to the body of reference chosen for the particular case in point. Thus, for a coordinate system moving with the Earth, the Michelson apparatus is not in fact shortened - although it is shortened (in the sense that Einstein accepts such contraction) for a coordinate system which is at rest relative to the Sun (ie as theoretically observed/perceived from that point of view). No reference is made by Einstein in this discussion to the comparable dilation of time which, in Lorentz's theory, occurs without any explanation - again simply to make the calculations come out right. As mentioned earlier, Einstein maintained that he was unaware of Michelson's experiment before 1905 and these latter accounts of it were apparently only considered by him latterly, in retrospect.
[*Note: Michelson's experiment was in response to Maxwell's suggestion and interest in such terrestrial methods of confirming the speed of light as an electromagnetic wave of ether to this end (and who, according to Einstein, must therefore have been one of those early workers so interested.)
Section 14 - The Heuristic Value of the Theory of (Special) Relativity.
588. Einstein sums up his special theory of relativity and its implications at the most general level here and points out that when so conceived the theory should have the added advantage of helping science in its search for more general laws of nature. He proceeds to this conclusion by first reviewing how 'experience' has led us to accept that both the special principle of relativity and the constancy of the speed of light are in effect facts of nature which, while appearing independent, may be united (by reasoning which effectively constitutes his theory) to (necessarily) obtain a new set of transformation equations when measuring the motion of a body in one system of reference from another moving at a different velocity (where significantly either system may assume either relative status). The two principles were united (despite having been formerly assumed not only to be independent but mutually incompatible) by means of treating the motion of all 'bodies' within a single continuum of velocity in which light was simply at one very fast extreme end - being at that point the maximum possible velocity of anything and a constant at that. All velocities of moving bodies were thus effectively constrained to 'fit into' this limited totality (velocity spectrum) as it were rather than have the freedon to be virtually unlimited; and the information of same must necessarily reflect a proportionality of the velocities concerned and not the unlimited values and instantaneous transmission of information (by light waves) as formerly assumed. As such, one must focus on and measure the exact magnitudes of the constituents of such motion , that is, of its velocity - namely those of distance or space (along x, y, and z) and over time t (as in theoretical coordinating systems K, K', etc). For one doesn't normally have velocity magnitudes per se available directly, they being a compound of these two sets of their fundamental elements. He then describes the events of such motion (ie of moving bodies) as constituting the processes of nature - as governed by their laws.
589. He then provides us with some insight into the order in which his thinking evolved prior to formulating that final conception of his theory. For he says that while acceptance of the law of the constancy of the velocity of light is justified by our (prior) empirical knowledge and plays an important part in this 'process of thought' (ie in its unfolding we might assume), it (only later) became possible to unite this principle or law with that of the principle of relativity "..once we were in possession of the 'Lorentz' transformations" (with their implications that time (and space) were relative - to the difference in the velocities of different reference frames as a proportion of the velocity of light. [Review here how Lorentz came up with these.] Now, Einstein himself apparently called these transformations after Lorentz because he had indeed derived them (albeit on a different basis) before Einstein. However, this considerate gesture does have the disadvantage of sometimes confounding the interpretation of such a statement as the foregoing. Were the transformations he so needed to 'possess', while called by Lorentz's name, not those Einstein had already derived (?semi-independently) for his own particular requirements ?
590. While he certainly derived these equations on the basis of his own (valid) premises concerning the relativity of time and space (unlike Lorentz), he may (it seems) sometimes refer to them with the latter's name when in fact he is referring essentially to those of his own derivation and basis. However, one may equally interpret the foregoing statement to mean that it was indeed in Lorentz's own initial transformations that Einstein finally 'saw' the means by which his two undeniable principles could properly manifest their compatability. (But they may or may not have been the means by which the effects of relative time and space (as defined elsewhere) were revealed and were first manifested). He was then able to obtain/derive the same quantitative transformations but now through that 'unity' and not by the same qualitative basis (utilizing an ether concept) as had Lorentz. It was as though the model or structure (and the equations?) of Lorentz's transformations (despite their basis using the velocity of the Earth through an assumed still ether and the assumed contraction of the measuring arm from that self-same reference frame) were useful to guide their subsequent derivation - from his own point of view. The rest of the theory then fell inevitably into place.
591. Thus, that final theory can then be summed up, says Einstein, thus: "Every general law of nature must be so constituted that it is transformed into a law of exactly the same form when, instead of the space-time variables x, y, z, t of the original coordinate system K, we introduce new space-time variables x', y', z', t' of a differently moving coordinate system K' the relation between the magnitudes of which is provided by the 'Lorentz' transformation equations. In brief, this states that 'General laws of nature are, according to the theory of relativity, co-variant with respect not to the Galilean transformations but to those named after 'Lorentz' - at least where they pertain in one way or another to motion and velocity, one presumes. In effect, these take account of the difference in the velocities of the two systems concerned such that the basic physical processes (general laws of nature) involved proceed identically in the different systems. As such, says Einstein, the theory becomes a valuable aid in the search or confirmation of other general laws of nature, particularly within the scope of electrodynamics. [Clearly, there is some foreshadowing here; he has obviously already derived some such general laws of nature in this field by this means.] This may initially involve adapting existing laws or interpretations to now incorporate the implications of the transformations of special relativity - utilizing the sq rt of 1 - v^{2} / c^{2} in place of a conception of unlimited magnitudes of velocity (v) in all those equations regarding the motion, momentum, mass and energy, etc of moving bodies. And if any such law is found, or adapted, whether through this guidance or not, and it turns out that it did not satisfy this assertion then at least one of the two fundamental assumptions of the theory (and their compatability), namely those of the constancy of the speed of light and of the validity of the principle of relativity in all circumstances, would have been disproved. He then suggests that one next examines whatever more general results the theory has thus far evinced or pointed to. Have they in fact been 'co-variant with the new transformations', as suggested they should, thus widening the scope of the theory and thereby further consolidating its validity and rigour ?
592. In Chapter 14 (summarized again here), Einstein thus sought to express the significance of his new theory in its most general form. Hence, he states that given that experience (empirical evidence) indicates that both the principle of relativity and the constancy of the speed of light are both valid and true, their necessarily combined application/function leads to a set of transformation equations for the spatial and temporal coordinates of motion/events (importantly when compared across differently moving (and symmetric) reference frames) which constitute the actual processes of nature. This statement does not appear to speak directly of the situation in which such coordinates are not sought for one coordinate system from another moving relative to it - ie by means of such transformations, with their relative values for time and space when so compared now fully appreciated. It is as though that 'cross-over situation' is a special case - by which such universal truths concerning their mutual equivalence were discovered - but that the underlying principles thus revealed somehow imply that this is a general reality for any and all, even single coordinate systems - where there is no apparent involvement with any differently-moving comparative systems. It is simply how it is - everywhere and at all times - in which all of the events of mechanics, now fully encompassed within electrodynamics, are only fully analysed in terms of such potential perspectives and their required transformations. If so, I don't myself quite 'see' this yet. That is, are time and space always relative even in one's own 'local' environment where no transforming is involved, or only when 'transforming' with respect to another system - moving relative to oneself? Is that latter situation so universal, that we can (must) conclude that such relativity is of virtual universal significance - whatever our 'object' frame of reference (as our local one only)? What is the answer here ?
593. Possibly the answer to my confusion is provided in the next section: He says there that having (as it were) first accepted that the velocity of light was indeed a constant, it was only ?after then deriving the generalized transformations (or possibly becoming aware of them as formulated earlier by Lorentz (?) - to account for a different phenomenon), that this reality could be combined with the (?later realized) principle of relativity such that the theory of relativity could (?only) then be (?developed or, at least, its essence) 'summed up' as follows: 'Every general law of nature (entailing motion?) must be so constituted that it is transformed into a law of exactly the same form [which is hardly a transformation at all!?] when, instead of the space-time variables x, y, z, t of the coordinate system K, we introduce new space-time variables x', y', z', t' of a coordinate system K'. The relationships between the two sets of variables is given by what he later called the 'Lorentz' transformations' (in recognition that Lorentz had published them first - even if so derived on unsustainable premises). Or, more briefly: 'General laws of nature are co-variant with respect to Lorentz transformations' (but possibly only 'when' they are so required - for differently moving systems). This is a definite mathematical condition that the theory of relativity demands of any natural law and can thus have heuristic value in the search and confirmation of such laws. If one was ever found that did not satisfy that condition, then at least one of it two fundamental assumptions would cease having support (and so be effectively disproved). None have been found.
594. Fine. But - just WHEN is it that we in fact do this? How common or general is this situation? If it isn't very common, how is it that nevertheless general laws of nature acting in accord with the theory of relativity can predict and account for various new realities in the universe - seemingly of a universal and general kind - when the theory is premised on these rather special kinds of 'cross reference system comparisons'?? Maybe Section 15 will provide the answer. For in that, he notes that to come into line with the new theory, classical mechanics had to be 'modified' although for motions at typical speeds this was bearly evident (but were nevertheless still there!) - but again, presumably only when one was 'crossing over' (to abbreviate this condition possibly too succinctly). But at very rapid speeds, as apply to electrons and ions, where the value of v in relation to that of c is not small - the laws concerned would be modified to greater effect. Now, are these modified laws concerned only with cross-system comparisons (eg concerning electron motion) or are there more general, universal implications? Are there no situations where Galilean transformations alone still correctly apply? For example, while considering activities occurring in the moving train from the embankment now requires use of the new transformations, what is required if one was restricting attention to actions within the embankment environment or the train alone? If these do not entail more than 'local' times and distances, is there any need for these to be variable depending on the ratio of v to c ? Presumably not, but then no transformations would be required at all and any calculations would be simply and direct.
595. If not, then one would wonder just how extensive would be the applications for which Einstein's great breakthrough are required. Even Galileo's example of activities on the moving ship concerned comparisons with those on the dockside, or vice versa. [Yes, but his results were minutely wrong as they still had to accommodate restrictions imposed by the constancy of the speed of light having to meet the 'commanding demands' of the principle of relativity (based now on the variabilty of time and space).] His transfomations should really have been the later derived Lorentzian/Einsteinian ones for absolute precison. But what about all the activities in life that don't entail 'cross-coordinate system' comparisons? Surely Einstein's great theory has more generality than just the latter? But quite possibly it was never touted to have more generality than that? I must keep an eye out for any reference to just how general are the applications of relativity theory for events within just one local coordinate system.
596. But that degree of generality, whatever it is, was nevertheless apparently sufficient to have several significant implications, Thus, some modification of the relevant laws were required, for example, with respect to the mass of such fast-moving small bodies as electrons and ions. And, in Section 16, we considered the prediction concerning the path of light passing a large body as a means of verifying the theory. It is quite possible that all such predictions and implications are in fact restricted to those in which relative motion (between coordinate systems) is involved. But then, is not all motion ultimately relative and therefore this may be the answer I'm seeking as to the basis of the apparent generality of the theory of relativity! Science is interested in 'happenings' and all happenings entail motion - which is always relative. But where does 'local' time and distance (events) in one coordinate system fit in? Possibly the motion of all local events and their times and distances (ie within their own system K alone) are always 'relative' - but on a smaller scale where other mini-systems therein may be assumed? Possibly as general laws operate identically in all relatively moving coordinate systems (and there are no such systems that aren't moving relatively to some other system(s)), they may well operate validly with respect to activites in any one system but the values of time and space involved in such are those which could only have been arrived at in terms of what it proved they had to be when measurements pertaining to such laws were made from the perspective of one moving system in respect of activities in another. Otherwise, I'm foxed! But can they never occur just in their own system ??
Section 15 - General Results of the Special Theory of Relativity. (Includes version #3)
597. Einstein first points out here that his theory of (special) relativity arose out of the fields of electrodynamics and optics (even though he felt required to explain it only after setting out the essential kinematics). While predictions based on earlier theories of those topics may continue to prove valid, their theoretical structures were complicated by a number of independent hypotheses. These were, with advantage, considerably reduced in his new theory with its simplified structure. The validity of the Maxwell-Lorentz theory of electrodynamics in particular was made much more plausible and trustworthy by the implications of relativity theory. In the case of classical mechanics, the alterations in measured results (when these are as perceived from a frame of reference moving differently from that of the body concerned) by the replacement of its 'laws' with those of special relativity, with the new conceptions of time and space which arise as a result, are so minute at typical velocity values of moving bodies, relative to that of light, that they are, admittedly, of no real practical significance - if even measurable.
598. However, these do become significant (and measurable) in the case of the 'mechanics' (ie electrodynamics) not only of light but of other very fast moving phenomena -such as electrons and ions, whose velocities are a much larger proportion of that of light. In their case, the new 'laws' demanded by relativity (in those same circumstances) are more appropriate and productive in revealing further aspects of nature and more accurate measures of same. Thus, the kinetic Energy (E_{k}) possessed by a body of Mass m moving at velocity v when treated by classical mechanics has typically been given by:
[We might usefully review here when and by whom this was first derived. It was apparently so concluded empirically after a number of experiments in the 18th century involving dropping different weights (balls) into mud from different heights. One recalls that Force has a similar formula - viz: F = ma]
But if v is not a small proportion of c (as it is not with electrons and ions), the electron's energy is more accurately given on the basis of the theory of relativity by the equation:
The derivation of these relativistic versions by Einstein will hopefully be shown below. However, no particle can achieve a velocity to equal that of light (c) no matter how great the energy applied to same since the kinetic energy so possessed by such a particle (where v/c would equal 1) would (by this relativistic equation) have to approach an impossible infinite magnitude.
599. Einstein then states that "if (without explaining why) we develop this relativistic expression for kinetic Energy in the form of a (?Taylor) series, we obtain:
and then points out that when v/c is small compared with unity, the 3rd of the terms so revealed is always small compared with the 2nd and the later ones even more so. For most situations, the kinetic energy for reasonably fast moving bodies could be accepted as that given by the 2nd term alone (even if based on this more accurate relativistic formiula) with its other terms. This in fact comprised the total equation (without such other attendant terms) for such energy when calculated by the classical formula shown above. With the fuller equation, the 1st term - m.c^{2} - would normally not be considered further - at least when concerned essentially with the kinetic energy of a moving body of mass m - for no mass can move at the velocity c (even if not squared as here) whereas they can and do move typically at some lesser velocity v (although the squaring of same in either term needn't necessarily be achievable by the body concerned; these values simply pertain to the calculation of the energy thus acquired). However, that 1st term will prove most significant in a later context below - and this may account for why he bothered to present this particular (expanded) formula for kinetic energy at this point. But he still denies us an insight into the sequence of his reasoning to this end.)
600. Without further development, he then follows the foregoing by stating that "the most important result of a general chatacter to which the special theory of relativity has led is concerned with the conception of mass". In the classical conception, physics recognised mass as the proper and only subject of the law of the conservation of mass, just as it saw energy as the proper and only subject of its similar but independent law of the conservation of energy. But the theory of relativity has shown (although we haven't been exposed to this as yet) that these conceptions are not independent ones but are in fact the subjects of just one law - that of the conservation of 'mass-energy' - a new, single conception. He then considers (albeit 'briefly') how this unification (by himself!) came about, and its meaning.
For this, he reverts back to his very fundamental 'principle of relativity'. This requires that, as any law of nature, the law of the conservation of energy too should hold not only with reference to an inertial coordinate system K, but also with respect to every other such coordinate system K', K'', etc which may be in a state of uniform motion relative to K. And the proof that such is indeed the case is only provided by calculating these identical net values by means of the appropriate relativistic (Lorentzian) transformations. [We may usefully emphasise here that what is termed as system K should not be considered as some kind of 'first among equals' of such frames of reference - from which situation those on all others should be judged; rather, every such system is, in effect, a 'system K' to itself - with exactly the same justification of assuming effective precedence regarding its own perspective as any other. This seems absolutely basic in comprehending the theory of relativity.]
Einstein then continues by noting that 'By means of comparatively simple considerations (which, possibly typically, are not expanded upon), we are, says Einstein, led to draw certain conclusions from the above premises (in conjunction with Maxwells equations): that is, that 'a body moving with velocity v which absorbs an amount of energy E_{0} in the form of radiation has, as a consequence, its 'energy' [note: this is not qualified here as being necessarily kinetic or of any other form] increased by an amount equal not to that absorbed amount of radiation E_{0} per se - but, typically, to a larger amount of energy indicated by:
[Note that if the values for v and c were always taken to be positive, there would be no need to complicate the formula by squaring and square rooting these values; the denominator is essential a subtraction of v/c from the unity that would otherwise be assumed 'under' any value in such equations.]
If we recall the expression given above for the kinetic energy at least (that accords with the theory of relativity) - ie:
we will see that, following algebraic manipulations, the 'required' energy of the body (*+*)becomes:
which, following further algebra, shows that the body then has the same (kinetic or total?) energy as a body with mass equal to (m = E_{0}/c^{2}) when moving with velocity v. From this it follows, says Einstein, that 'If a body (or system of bodies) absorbs an amount of (?radiation) energy E_{0}, then its inertial mass increases by an amount (E_{0}/c^{2})c^{2}'. Thus, the inertial mass of same is not a constant but varies according to any such changes in its (?overall) energy content - ie through gaining (or, equally, losing) such (but what kind of?) energy. Because of this apparent reality, 'the mass of any body or system of same may be regarded validly as as a measure of its energy'. [Again, no qualification is given as regards any particular form of such absorbed, lost or final total energy.] From this, Einstein asserts that "the law of the conservation of the mass of a body (or system of same) becomes identical with the law of the conservation of energy" (of same) - when the body/system concerned does not vary its status in these regards.
The total energy for any body or system may thus be expressed (relativistically) by the equation:
in which we may note that the term m.c^{2} (which we first met above in the expanded equation for kinetic energy) must be nothing less than 'the energy possessed by the body before it absorbed the energy E_{0}'. [This conclusion will require that we can appreciate how this term represents the same thing in the two formulae; ie by virtue of saying the same thing, essentially, in these different ways. Can we appreciate that ? This matter relates to the need to show the derivation of the equation of E = m.c^{2} more clearly - something that at this juncture at least Einstein doesn't appear to have attempted. {Where is it shown ??] However, he does say that the absorption of E_{0} by a body of mass m (seemingly whether moving ot not) increases that mass by a very small amount of E_{0} divided by an enormous number (c^{2} and, by virtue of the revised form of the energy equation (m + E_{0}/c^{2})c^{2} / sq rt 1 - v^{2}/c^{2}, he states that the term m.c^{2} represents the energy possessed by the body before absorbing that small amount of extra energy and so would represent an enormous amount of initial energy (seemingly non-kinetic). In this sense, he has stated both the qualitative and the quantitative relationships between mass and energy although we may have to analyse further just how the latter term emerged out of the foregoing ('required energy')when written in the form shown above at (*+*).
When Einstein came to his (theoretical) conclusion regarding mass and energy (before the 1920s), he points out that it was not then possible to verify same through experiment, in that the magnitude of the changes (increase or decrease) to which a body's or system's energy E_{0} may be subjected were not large enough to be perceptible in terms of a consequent change in its inertial mass. The value of E_{0}/c^{2} is too small in comparison with the mass m to which such energy was applied or removed. It was, says Einstein, because of this (non-testability) that classical mechanics could and did hold that a law of the conservation of mass was (apparently) independent and valid (and likewise wouldn't have suspected the relationshuip between energy and mass as eventually derived by Einstein). The experimental proof was eventually forthcoming - in 19.... by.................. .
[The following was an earlier version of Section 15 (ie version #3) and may now have to be altered (and numbering revised)]:
The relativistic versions of the energy, mass (and momentum) equations seem to follow from the fact that just as the velocity of a moving body must be perceived in terms of its upper limitations being that of the velocity of light (c) - and thus when measured from a neutral viewpoint the appropriate transformation equation is Lorentzian not Galileian (with their inevitable infomational lag consequences) - so too must these latter measures of a moving body - as they too are a function of that velocity limitation. So their magnitudes when so perceived must also be limited by virtue of the reality of that same v to c ratio (rather than v alone) - and the associated Lorentz transformations and explanatory 'mechanism' based on same.] From this, it may be seen that as v approaches the value of c, the value of mass and energy could in theory approach infinity. To accelerate a body of initial mass m to such a velocity would however require more energy (ie Force) than is possible (indeed, an infinite amount) - so large (?dense) would the mass (and its associated kinetic energy) become. From where was the energy derived by which a given electron's immense velocity (per unit of mass) was first provided ? Was it a function of its velocity and mass (however small the latter) - possibly first 'instilled' (?conserved) at the time of the big bang? Ditto with that of the photons (and/or electromagnetic waves) of light ? ...Or... ?
600. We may note that in this Section (concerning the major results and implications of his special theory), Einstein has thus broadened his perspective from those kinematic aspects pertaining specifically to the uniform velocity of moving bodies per se, including light, (through time and space) to now incorporate topics concerning in particular the mass and energy also associated with all such moving bodies, which have thus far been unconsidered (in the present 1920 version). However, he had addressed some of these issues in Sections 6 to 10 of his main 1905 paper and did so more specifically in his later paper of that same year when considering the Mass- Energy relationship in greater detail. We may ask why this particular path of enquiry eventually came about ? Was it a part of the original problem for which the theory was advanced and developed or did it reveal itself only subsequently - as any more general results or implications of the theory were gradually considered, deduced or suggested themselves? It was seemingly the latter but, as touched on earlier, quite likely in conjunction with the implications of generalizing Maxwell's equations of electrodynamics - as they were a crucial starting point; hence the title of his paper. But Lorentz's interest in the motion of electrons must also have suggested to Einstein that there may well be other such related applications and analyses also suited to his new theory that had not been previously considered in this light. It seems to have 'come to him' more clearly during August 1905.
601. [Note: the content of the present Section (15) - as written in 1920) appears to derive in part from both those latter Sections (8 to 10) of his main 1905 paper (as submitted on 30 June that year) and from that shorter paper (more specifically on Inertia (Mass) and Energy submitted on 27 Sept, also 1905.) This would seem to point to July/August 1905 as the period in a which he likely developed these latter ideas further. In any case, we may anticipate that it was this line of analysis (seeking to validate the theory by noting its relevance and validity with various more general electrodynamic applications (re-interpretations) where the unqualified parameter of velocity (v) is relevant, that eventually led to the formulation of his famous equation further linking these two concepts - namely E = mc^{2} - an outome which, in addition to the unexpected conclusions about time and space, is what most people seem to think of when the theory of (special) relativity is mentioned. But that equation and the relations between mass and energy (and any applications of same) wasn't particularly obvious in the previous classical formulae for these parameters. It would seem to have come about as a deduction within his analyses of the implications of the new relativistic forms of these same measures - involving appreciation of a role for the ratio v/c in further spheres of physics.
602. [Note: In this regard, we may note that as any moving body's mass, energy and momentum are all functions of the velocity of that motion, the classical formula for each of these contributory parameters for slow velocity values would need to be modified (if full accuracy is desired) - particularly where that velocity becomes a much larger proportion of the velocity of light (c) - ie by incorporating the crucial relativistic ratio of v/c (as touched on above). In doing so, it would become apparent that each of these 5 parameters (ie m, E, p, v and c) are often a complex function of one or more of the others and so relate to one another in ways which should now be derivable and revealed from within these same new 'relativistic formulae' - in forms that accord with the principle of relativity. Such derivations apparently reveal relationships which, amongst others, eventually identified the parameter of light's constant speed (c) in a squared form rather than simply as c itself (as immense as that was even on its own) as a crucial component in this newly appreciated relationship (between E and m). And thereby, through deduction within this 'parametric complex' - when Einstein concluded that "Mass and Energy (and their conservation laws) are therfore essentially alike - being but different expressions of the same thing" - that the equation of present interest was so formulated. And as such, we may appreciate (as later stated by Einstein in 'The Meaning of Relativity' (1922) that when a body increases or decreases in Energy, its Mass varies accordingly, and vice versa.]
While c^{2} does appear frequently within the important ratio of v^{2}/c^{2} - as the means by which such accurate transformations may be calculated and seemingly nowhere else (and thus why, for a time, one reasonably assumed that in the many algebraic manipulation involved in the analyses pertaining to this derivation, this ratio must be where that immense value arose), it turns out that there is in fact another source altogether for this massive quantity - one in which presumably it better reflects some real physical entity - in contrast to that in the briefly squared ratio (including c^{2}) so often mentioned which seems to be but a mathematical device to remove negative values - with no physical reality beyond that. It might have been less confusing therefore to have used |v/c| instead - which symbolism signifies the relevance of the numerical values alone, thus ignoring any negative values.]
603. Such energy (of E = mc^{2}) was thus eventually concluded to be the inherent (latent) energy of any such mass, not its more immediate, kinetic energy provided by means of its velocity as a moving body (nor even its 'potential' kinetic energy due to, say, some teetering 'pre-kinetic' position (eg ready to fall due to gravity). As such, it is not easy to appreciate how this outcome was arrived at in that we have thus far considered only moving masses whose inertia and energy appear to reflect primarily 'that mass's overall movement' - ie as kinetic energy or momentum only and its reluctance to change its velocity in the fundamental concept of inertia. Moreover (as mentioned above), the enormous quantity c^{2} appeared (to this point in our analysis) to have entered our equations (and/or those of Lorentz) only as a component of that mathematical device used to neutralize any negative velocity values (in ratio with c) - ie by squaring the values of both v and c in their ratio - and then (in virtually the same instant) taking their square root - eg after subtracting from 1. How does it 'suddenly' now represent a real physical value (of enormous magnitude) in its own right ? In the kernal of the answer to that question may not lie the answer to 'the secret of the universe' but it must certainly reveal the answer to why any bit of 'stuff' in that universe contains such an immensity of latent energy (power/force). After all, c^{2} in kilometers per hour equals a value of ninety billion (90,000,000,000) as the velocity-based 'exchange rate' between mass and energy - ie linear distance covered per time. We might note that when squared, this value reflects not just a very large number (based on a linear velocity) but possibly one that represents the coverage of a squared area at that velocity (whatever that might mean in terms of a still mass - covering no distance, or area, of motion but an intact entity in its own right - seemingly devoid of velocity). Or is that speed somehow inherent in the individual masses within its own borders ?? If so, I suppose that one could visualise the totality of all the atomic/nuclear binding forces (and hence energy) within each and every atom if spread out one quark thick, to entail an area characterised by c^{2} by which that (thin) mass must be multiplied to represent the total latent energy involved in its binding (and/or the velocity of associated electrons).
604. [Note: If the foregoing analysis could have as usefully been approached initially from the point of view of mass (rather than from that of energy), we might have considered that any mass (body) has first to be accelerated from a relatively resting state and as such it requires a force to do so and, having done so, that mass would then possesses an equivalence of energy and the greater the mass, the greater the force and/or energy required to so accelerate it - and effectively transfer that amount of (?latent) energy to it. Does its resting mass therefore reflect the extent of its reluctance to move due to some aspect of its existing 'compacted energy' associated with its internal micro-motions (and binding) - instilled originally...when, where ?? (That is, something like the resistance a spinning wheel manifests when one tries to alter its plane of spin.) Once set in motion (if it is), it of course gains kinetic energy as well. As there is non-linear motion involved in such initial acceleration of the resting mass (and its constituents), any comprehensive interpretation of same would, presumably, have to entail a complex calculus-based analysis.]
605. We describe here (for a 2nd time) Einstein's later (1920) exposition of the foregoing matters regarding Inertia (or Mass) and Energy as touched on more generally in #2 above) ie as provided in the Section 15 of that book. As noted, he begins: 'The formula for kinetic Energy in classical mechanics for small velocities (as shown above) - ie E = 1/2 m.v^{2} - 'turns out', notes Einstein, to be only the second term of an additive (binomial) series (which seemingly represent the total energy available (kinetic, latent and potential?) from an initially moving body when so interpreted). This fuller equation for the total energy of a moving body (E_{total}) appears thus:
606. When v^{2}/c^{2} is small compared to unity (as in most 'classical' considerations, involving both systems K and K'), then the third and later term(s) are always very small compared to the second, and so on - which second term alone was thus usually considered in classical mechanics to represent the resultant kinetic energy (vs 'total' energy) for bodies moving relatively slowly (which it did very closely). (One would assume that the summation of all those later terms (3/8, 3/16, etc) is what provides the accepted overall fraction ( 1/2) by which this equation for kinetic energy was necessarily divided in the calssical version. When considering such energy associated with a body moving at velocity v, the first term above, having no velocity element (for a tangible body at least) was, noted Einstein, generally not considered further (despite the odd immensity of its size??). However, it does have significant energy not associated with the body's overall velocity (but rather with just its own internal cohesive forces seemingly although this is not amplified here and in a sense, we are not yet privy to any explanation of such unexpected revelations). Any unmoving Mass simply 'has' (apparently) enormous latent Energy within itself (as will be concluded, we trust, from one of our several accounts (# 1,2,3, etc)) and (as we shall see) either property (m and E) 'can be transformed into' or 'is somehow the equivalent of' the other - provided the term 'c squared' is somehow recruited into quantitative version of the derivation !
607. Hopefully, it will be addressed further below when the source and derivation of this particular bipartite component (m.c^{2} by way of the above mentioned Taylor/binomial expansion series of an otherwise moving body) will be better revealed. It would seem to be quite fundamental and is hardly fully explained by the rather casual phrase: it turns out that...' as applied at least to its accompanying 2nd term. How and Why did such terms 'so turn out! ? And why, we may ask, was an expanded Taylor's series considered for this particular equation in the first place ? We may recall that in his 1905 Inertia-Energy article (considered above in #2), there was no reference to the enormous (nor indeed any) amount of Energy 'locked up' in any unmoving body's mass - at least as revealed in the equation referred to now. Rather, there was merely the conclusion that with an emission of light or radiation from a body, there was (likely or a ?proven) reduction in its inertia (or mass) and vice versa - due to some apparent equivalence or inter-convertabilty between them - but the total magnitude of same nor any explanation about it was apparently not addressed at that time. And, again, does the c^{2} element arise quite unrelated to the only place we have repeatedly noted it thus far - ie in its inclusion when squaring the crucial ratio v/c (and heretofore always immediately 'square rooting' same) - seemingly as a mathematical device only ? [The answer to this is apparently -'Yes', 'it does so arise elsewhere'. But, if not by way of some inclusion of or reference to a photon of light moving at this crucial velocity c (so providing a source for this factor in the derived equation, albeot then for some reason squared) then where? The relativistic formulae for the dynamic parameters of moving bodies (with their inclusion of v/c) seems to be the only possible source. Is it possible that because the formula for kinetic energy for a moving body entails (for whatever reason) the value v^{2}, that the need to replace v per se of the classical formulae with (v/c)^{2}, that v^{2} somehow becomes replaced by c^{2} ? While the energy in a given amount of mass is clearly much greater than the mass itself (as quantified in typical units), one would like to see just how that particular multiple (of c squared) emerges out of the derivation. What is the logic of it - in particular ?
608. Where else might c^{2} have entered our considerations thus far ? Possibly only when (as just mentioned) we noted that in classical terms, it was in a sense 'represented' by v^{2} itself which, in relativity terms, necessarily become represented by c^{2} instead. We may note that even for bodies moving at slow velocities compared to c, the correct formula should still be the one which utilises c (within the ratio of v/c) rather than v alone, even if there is no practical utility for doing so. But if c squared enters the final equation other than as part of that mathematical device (ie in ratio with v squared), what could possibly be its source in the first term of the binomial expansion of an equation for kinetic energy for a tangible moving body of given mass ? No light photons are included in the discussion about that body moving at velocity v, nor in association with it when its inherent latent energy is considered as a feature of its mass - when it is still and its velocity zero. And yet, the equation contains the value c^{2}. Indeed, why need we address this matter in terms of a moving body (with its kinetic energy) if we are seeking primarily to show that any body possesses much more energy in its unmoving mass - albeit somehow as a function of the square of the velocity of light (c) ? Is a moving mass (with its known formulae and equations - of a relativistic form) the only way to approach this conclusion - by which the moving mass, once slowed to a still state, now somehow possesses a previously unsuspected enormity of latent energy - simply 'left over' at the end - and this conclusion is not nearly so apparent if that moving condition is not first considered ? [Note: Adam's derivation may place less (but still some?) emphasis on this more typical approach.]
609. In this regard, we may note that Einstein does state at the start of this 1920 interpretation that this most important result 'of a general character' arising from the special theory of relativity was indeed concerned with how it may have affected our conception of the Mass of a body (and of the latent Energy inherent therein seemingly). These aspects of the theory had not entered our considerations to this point but of course every moving body (our basic concern) must possess a given mass and, as it is typically 'moving' at some velocity, we have the two necessary ingredients for calculating its kinetic energy, at least. This, in turn, leads one directly, says Einstein, to a prime example of the unifying nature of (so generalizing) relativity theory - namely, the fact that the two most important conservation laws - those of mass and of energy (?kinetic, latent, or both?) - previously considered as mutually independent - can 'now' (or will ?shortly) be appreciated as effectively united into one conservation law. [How this conclusion came about will, says Einstein, be explained shortly.] And, being a (single!) law of nature, the conservation of such a new 'mass-energy' conception should, according to the theory of relativity, hold not only for any given coordinate system K but for all systems K', K'', etc - moving uniformly with respect to K, since their differing smooth velocities should (by earlier definition of that principle) have no effect on the operation of all such general laws of motion therein/on- including this 'new' one (ie after applying the approriate transformation equations to confirm that the net values were indeed unchanged). These would typically entail velocity (and thus time and space), mass, energy and momentum and now do so in terms of the ratio v/c and thereby necessitate consideration of how that latter source of c affects the resultant equations. But surely we don't account for the enormity of latent energy in an unmoving mass by virtue of our information lag hypothesis alone ?? Would it magically disappear if the relevant information was somehow conveyable instantaneously ?? [It couldn't as c is the limit of same.]
610. This absence of any such (net) effects is, as mentioned, revealed through the application of the now required transformations with their proportionalizing effects on velocity (vis a vis the upper limit of velocity - c) and its two (time and space) components (again arising by virtue of our old friend the information lag seemingly - via its limitations due to the reality of c). Any moving body's mass, energy or momentum cannot escape this same re-assessment. But is there the same 'information lag' underlying the (?ultimate) explanation of any such new revelations/intepretations arising in this regard (concerning these three paramenters) and the major conclusions deduced from same ? That is, must (or should) the terms 'appears, seems and apparent', etc always qualify the relativistic statements concerning them as they should have (seemingly) about the time and space variabilities (in a body's motion/velocity per se) - whenever perceived/measured from a differently moving frame of reference? Apparently 'Yes', they should. But the nuclear bomb doesn't just 'appear' powerful but acually is so, from wherever observed !?
611. For the application of that theory, any translation from one such system to another of different uniform motion is, as we now know, when viewed from the one moving differently, proceeds by means of the Lorentzian transformations with their inherent v/c ratios - in contrast to the Galileian ones of classical mechanics (utilizing only v on its own - as if it could somehow be infinite if required). This means effectively that one must again take into consideration the effects not just of the velocity v of any such moving (and/or ?'non-moving') body (with now its mass and energy implications) but the ratio of that velocity to the ultimate possible velocity - as represented by that of light (c). Such effects become much more significant where the velocity of the body of concern (as with electrons, ions and even light itself - which may or may not have mass (apparently) but does have energy and velocity (so that, confusingly, v = c when considering light as the moving body of concern) is a large (or even total) proportion of that latter ultimate speed. As this concerns the rather abstract concept of 'energy' which appears throughout physics in various guises (forms) in addition simply to obvious masses moving with given velocities (and with which I have some trouble), I look forward to learning more about its interpretation within relativity theory. Energy seems so often to be treated rather casually as 'an unquestioned entity' of nature rather than as an albeit convenient 'abstracted communality' so derived which, in differing ways, facilitates the work (causing motion of masses) performed during various subsequent events) - which it seems actually to be). We may note that as of ca 1900, there was no knowledge of such as quarks, gluons etc and barely of electrons. How was latent energy accounted for then - or even by 1910 ? ?
612. [In this regard (of a too easily assumed 'entity' of energy), it may be germane to include here a recently discovered quote from Richard Feynman, which I rather appreciate:
"There is a fact, or, if you wish, a law which governs all natural phenomena known to date. There is no known exception to this law and it is exact so far as we know. It is called 'the conservation of energy'. It states that there is a certain quantity, which we call 'energy', that does not change [in effective quantity] in the manifold changes that nature undergoes [in which it may be affected/involved]. It is a most abstract idea, because it is a mathematical principle; it says there is a numerical quantity [but of what?] which does not change when something [involving it] happens. It is not a description of a mechanism, or anything conctete [although it does accelerate electrons, say]; it is a strange fact that when we calculate some number [pertaining to it] and then watch nature go through her tricks, and then calculate the number again, it is still the same. It is important to realise that in physics today, we have no knowledge 'of what 'energy' [represented by that number] actually 'is'. We do not have a picture that energy comes in little blobs of a definite amount [ie as if of pure, distilled 'energy-stuff' from which all its other forms derive]. It is not that way. It is an abstract [derivative] thing in that it does not tell us the mechanism or the reason for the various formulas." [Additions in square brackets are of course mine.] We may know what it 'equals' (m.c^{2}) but not what it 'is'. It is not, seemingly, some mass moving about extremely quickly (at m.c^{2}) although its effects may well be represented by that quantity. However, might we not say that what it is, is the equivalent of mass per se ? Or is it, rather, that which holds the constituents of mass together ? And what is that ? Part of the Forces of nature ? God ? Moreover, it seems that it is those constituents themselves that can become energy and not just the forces binding them; they are energy...and mass !
613. [In any case, it (Energy) thus seems something like the converse of Inertia - and conceivably might have been called 'Activia', say. Are they reciprocal 'properties' of Mass - as though one was 'latent energy' and the other 'kinetic energy', with the former requiring the latter in order to move and so realise that 'latent potential' ? No. We seem to treat both in terms of the derived quantities (of some other variables) that they (?qualitities) account for. One wonders if all the energy in the universe became available when, as hypothesised, the single body of immense mass (and heat) blew up at the time of the 'big bang' - out of which smaller bits of always reluctantly moving mass have since coalesced but with the two forms being interchangeable? As such, would not the energy of any moving body always require replenishment whenever its velocity (or any 'force') seeks to move other bodies ? That is, do 'work'. Hopefully, we may soon see.]
614. While Feynman notes that there is no known exception to the law of the conservation of energy (which is undoubtedly the case), Einstein has shown how it can be (or is) effectively unified with the comparable law of the conservation of mass. While that isn't exactly an exception, it surely is a relevant qualification and interpretation of its 'mechanism of action'. How can we account, asks Einstein, for the effective equivalence of the laws of the conservation of energy and that of mass so that their unification [which he has suggested is the case prior to actually establishing same here, I believe] effectively into one conservation law - of 'mass-energy' (concerning this unity of nature regarding same) may be better appreciated ? And we may enquire what significance or value would have been expected follows from being able to confirm this 'suggestion' or 'guess' (at this point) ? We may consider firstly that we know that the theory of relativity requires that as a general law of motion, that of energy conservation must, like all other such laws, hold equally in relation to whatever uniformly moving coordinate system (whether K, K', K'', etc) that may be considered (ie to which the energy (and mass?) status considered may be referred) so that when translating from one such system to another (ie perceiving and measuring such status from a differently moving system or viewpoint), the restrictions inherent in the new translation equations (which allows us to see that the effects are indeed unchanged (as the underlying inertia-based principle requires) in the differently moving reference system) must apply. That is, we must recognise that it is the ratio of v to c (and not just some unlimited magnitude of v per se) that correctly allows this equal and symmetric outcome to be appreciated. This reality when considered in conjunction with Maxwell's 6 fundamental electrodynamic equations, leads to the conclusion, notes Einstein, that:
"....if any body moves with a velocity v and, as it does so, absorbs radiation (?light/?heat) energy E(a) (from no explained source and for no explained reason) of an amount as judged from that same reference frame, its total energy (E(t) (as measured from...what reference?) is thereby increased by that absorbed amount (E(a) divided by a value which varies from almost 1 (which leaves the increase near that added amount only) to almost infinity (which equates to an infinite increase in the total energy involved) according to what proportion the body's velocity v is of the velocity of light c. That is, by:
615. For slow moving bodies, this increase in energy would thus be very little more than that of the absorbed energy E_{a} per se, as its value would be divided by virtually 1, whereas for very fast moving bodies (as an electron), it could increase greatly according to just how much the denominator of 1 is reduced by that essential v/c fraction - the greater the value of v and thus the nearer its value is to c (as in the case of an electron), the much greater would be the increase in its (kinetic) energy - due to that additional velocity rather than to the added radiation energy. And the total energy of such a moving body (of mass m) would thus become:
since, under the theory of relativity, the kinetic energy of any moving mass becomes:
- rather than just the value: m.v^{2}/c^{2}, as understood formerly.
[It should be pointed out here that the foregoing quoted (if paraphrased) passage indicates that for this demonstration at least, the derivation of the equation E = m.c^{2} does not proceed in terms of a tangible moving body (with its kinetic energy) alone but by way of, in addition, an introduced element of 'radiation energy' (as light, say) which thereby allows a comparison of mass and energy (and a possible conclusion of their equivalence (of effect) in any subsequent analysis) and thus a source of the parameter of c to be seen to so enter that relationship - neither of which may have been the case (seemingly) had the demonstration been confined to a tangible moving body of given mass alone, never mind a still one, even though it is exactly that latter entity which will allegedly possess the enormity of latent Energy. (Unless, any relativistic parameters so analysed always involve the v/c ratio and can thereby introduce the c values that way...sufficiently ??)]
616. In any case, the body (as described above) now has the energy of a mass of (m + E_{k}/v^{2}) which is moving with velocity v and so has thereby increased its inertial mass by the amount of E_{a}/c^{2}. That is, the inertial mass of a body is not a constant but depends on (varies with) its energy content - which we have attributed to its velocity and, in this present example at least, to any absorbed radiation (the added mass thereby providing the other ingredient of total energy (ie in addition to the portion of its magnitude increase due to its velocity). [Should these statements not be qualified by 'appears', etc ?] In like fashion, energy of a body may be emitted by an amount E_{e} and the converse of the above so results - with the kinetic energy of a moving mass becoming less than before - as it slows and/or loses any such radiation. Thus, we may say that the inertial mass of a system of moving bodies may be regarded as a measure of that system's energy and the law of the conservation of the mass of such a system becomes identical (unified) with the law of the conservation of energy (of that or any (?closed) system of moving bodies - which neither takes up nor sends out any energy beyond itself).
[Thus, possibly, the logic of the free-floating box example of #4 becomes clearer. In this latter regard, we may ask whether the absence of any reference in that latter derivation to any other coordinate system(s) beyond that of the Box (K) itself - ie as K', K'', etc moving uniformly at different velocities - nevertheless allows one to conclude that energy and mass are equivalent and transferable and thus that E = m.c^{2} - apparently without involving Lorentzian transformations. If so, it would suggest that the latter equation needn't necessarily therefore be a general result only of the special theory - for which differing coordinate systems and those transformations appear generally to be involved and even crucial. The relationship of concern must still apply within a given system K per se. [Can this really be the case?] Possibly utilising the mass and energy transfers by means of light photons allows the speed of light (c) to enter into the conclusions without having to do so by way of observing/measuring from a differently moving system and later applying the associated Lorentz transformations (with their v/c sources of c) as the means of confirming its validity ??]
617. In any case, for our present derivation, which (more expectedly) does entail differently moving coordinate systems, the expression for the total energy may be written in the form:
We may note that the term m.c^{2} (as referred to briefly above), is however now seen as nothing else than the energy possessed by the albeit moving body (but as judged from the reference system moving with it - which makes it effectively 'at rest' and unmoving therein and so becomes effectively 'energy possessed by (what is perceived by a fellow traveller to be) an unmoving body' (unlike the kinetic energy arising from the same body when viewed as moving, by those on a differently moving reference frame) before it absorbs the added (or emits the subtracted) energy - with effects as described above. That is, this energy is not the result of its velocity - as kinetic energy when measured as such but rather is (must be?) an inherent/latent energy possessed by all matter whatever its motion or speed (if any) ? If its mass increases with the energy it absorbs, it clearly now possesses greater mass gained thereby - ie even if it is not moving subsequently. [Its electron velocities and the cohesive energy somehow engendered therein may somehow add to its mass (in the form of new latent energy and not additional protons. Such increased energy (and thus mass) will however still increase the kinetic energy of that body when available as a moving body. [Might this explain why one used 'a moving body and its kinetic energy' in order to establish the existence in the first place of the latent energy of a body (whether moving or not) ??]
[While the conclusion (deduction) regarding the equivalence of Energy and Mass and the associated equation ('equality') of E = m.c^{2} only follows from the application of the 'v/c transformations' of special relativity, the underlying logic seems to depend on the understanding that "when an unmoving body of mass m is 'forced' into motion (to overcome its inertia) through the application of a force, some of the energy from that force (which it gives up thereby) becomes associated with the now moving body - as its kinetic energy - which may possibly be manifested as an increase in the body's mass, as its velocity increases, but/or some of it (either from that force's energy or from some extraneous source of radiation energy, say) seems to be taken up by the body as the better explanation of that increase in its mass and possibly vice versa as it slows down back to its resting (still) state. As mass and energy appear to be inter-changeable, it may be deduced that the resting mass (being greater than zero) must represent a source of energy in a sense already applied to (and integrated within) it the past (as radiation or?) - which could be extracted in the form of radiation energy, say, whether the body was moving or not. That this latent energy of a non-moving mass is apparently however of a much greater magnitude than would be released by simply setting fire to it or digesting it, say, it may well be the case that the analyses of these mass-energy conversions only in terms of the special theory of relativity is what is required to reach the conclusion regarding the enormous magnitude of energy represented by E = m.c^{2}. This was touched on above and will also be considered further below.]
618. Again, we may enquire if this somehow follows from Maxwell's earlier ideas ? A direct comparison of this relation with experiment was not possible at the time Einstein first formulated these conclusions, nor indeed until about the 1940s, I believe, when atomic elements could be bombarded with such as alpha particles, protons and neutrons and the equivalence of mass and energy (as per E = m.c^{2}) became amply confirmed. Previously, the fraction E_{a}/c2 was too small to be perceptibly measured in comparison with the mass m of the system to which the energy E_{a} was applied (or emitted). It was because of such imperceptible changes that classical mechanics (understandably) concluded wrongly that the law of the conservation of mass, while valid, was independent of that regarding energy. [Note: The conclusions of the immediately foregoing paragraphs may be augmented with those now described based on Einstein's later 1905 article on this subject and/or those shown in the later sections of his earlier (main) 1905 paper - in its Sections 6 to 10+.]
619. Einstein completes this Section (15) by noting that the success of the Faraday-Maxwell interpretation of electromagnetic action transpiring over a distance (by means of an electromagntic field) convinced physicists that such action at a distance for any other physical processes can also never transpire instantaneously (as with Newton's conception of gravity) but rather must always involve some degree of time-consuming transmission through an intermediary medium (active or otherwise) of some kind. The theory of special relativity thus replaced the idea of instantaneous action at a distance ('conveyed' by an almost infinitely fast signal where v could be much greater than c) with that of action which could occur only after a brief moment of time during which such as light or other electromagnetic waves can act and which, notes Einstein, plays such a fundamental role in his theory. [However, the conclusions of the general theory of relativity will apparently require that this view be modified to some extent, as considered further below.] But before that, we may seek to better comprehend just how the equivalence of energy and matter (as in a sense two forms of the same thing; although, in fact, energy comes in many forms so this shorthand phrase should probably be modified/qualified) and how they inter-relate (which discovery arises not so much within the development of the special theory but, as Einstein reports it, of that theory. Indeed, as he says, "the most important result...to which the theory has led concerns the conception of mass..." (ie seemingly as a repository and source of immense latent energy). And the inter-changability ('unification') of mass with energy is only realised in terms of this fundamental quantity c (or its more general upper velocity limit equivalent per se)- although does so by an amazing squaring of this already enormous number!
620. Thus, the magnitude of the Energy theoretically available in a small amount of matter (mass), typically measured in grams or kilograms, requires that amount to be multiplied by a (velocity?) value of 90,000,000,000 ! (?square kps) to make one fully aware of the amazing extent of that magnitude of energy (in joules or whatever) somehow stored therein. p>
[Note: Section 16 which would normally appear next is located prior to Sections 14 and 15 immediately above as it seemed at the time to fit the sequence better there; it may however be reviewed here also if felt that its content is better considered as in the original sequence. In any case, these 3 sections cover essentially the final (Electrodynamics) portions of the Special Theory, as covered in the 1920 book. Sections 17 and 18 can be seen (below) as providing a useful introduction to Part 2 - The General Theory - which then follows more specifically in Sections 19 to 32 of that book. Click as required further below].
621. While Einstein has given us in the 1920 book (in Section 15 above) an account of how the above mentioned unification of the conservation laws for mass and energy came about (and why therefore it could be concluded that mass and energy are essentially two versions of the same thing - and, significantly, their quantitative equivalence given is by E = m.c^{2}) - we may find our latter account(s) of same (immediately above, which we have called account or version #3) while incomplete, may at least provide us with a useful platform for better understanding his original more technical treatment of it (or part of it?) as presented in his later 1905 article: 'Does the Inertia of a Body Depend upon its Energy Content?' submitted separately (in Sept) to the same journal (and volume) to which he had already submitted his main 1905 article on moving bodies, in June of that year. Our brief analysis of that later article, as shown earlier and referred to as 'account #2, was followed to some extent from what might be termed account #1 - the introductory treatment of certain relevant concepts as given in the last 3 (Electrodynamics) Sections (8 to 10) of Einstein's main paper of 1905, especially section 10.) We may still ask whether a complete derivation of E = m.c^{2} has been given in any of these first 3 accounts ? If not, they should however assist us in comprehending any subsequent and hopefully more complete interpretations of same to be presented below - in versions #4 to #7. That there have been a number of such presentations in later years (and likely others) seem to suggest that those provide by Einstein initially (as dealt with in accounts #1 to #3 above) must have been either incomplete or left certain ambiguities.
To facilitate reviewing any of the 7 versions of the derivation presented, we show here the paragraph numbers at which each begins: #1 (a preliminary basis from section 10 of Einstein main 1905 paper)- par. 317; #2 (from Einstein's later 1905 article on Inertia and Energy) - par 328; #3 (from section 15 of Einstein's 1920 Book) - par 597; #4 (by Adams) - par 622; #5 (by Doyle) - par 634; #6 (by Tao) - par 655; and #7 (by Cox and Forshaw) - (after par 660).
. We note that on one chat-site (of Prof Terence Tao) it was suggested that the derivation was indeed 'essentially' contained (implied?) within Einstein's later 1905 Inertia-Energy article - submitted on 27 Sept that year (and referred to above as #2) - but may need further interpretation. Before we consider this further, it may be helpful to reproduce first what appears to be a relatively simple and straight-forward derivation (here #4 - from Adams) of that famous equation and then seek to interpret Einstein's and any subsequent derivations of same in terms of this seemingly straightforward 'model'. It is the one that appears not to involve that very basis of special raltivity - the view of the relevant activities on system K from a differently moving system K' (with its information lags, and v/c transformations), although we may need to revise that view. (We note that Tao's derivation (#6) will, however, entail the determination of pertinent Lorentzian transformations during each of the 5 initial steps of his reasoning, although the last one involves deductions arising from that basic platform which, because of all those transformations, would imply the involvement of systems K and K' as generally defined in the varous treatments by Einstein).
On Adam's Derivation of Einstein's Famous Equation E = m.c^{2}. (#4).
622. This first full derivation to be considered (beyond those implied earlier by Einstein himself) is available from www.adamauton.com . As mentioned, it could possibly serve as a 'model' against which other derivations and/or introductions to the energy-mass relationship may be compared and hopefully clarified. Einstein himself apparently arrived at this version some years after his 1905 articles by way of one of his later 'thought experiments' (although whether before or after his 1920 book is not made clear). Nevertheless, he had worked out the essential basis for this surprising relationship during that summer of 1905, as described to some extent (or just implied?) in his later 1905 article on Inertia and Energy.
Having appreciated that the ratio v/c was required rather than the unqualified value of v alone when considering the motion of bodies from a differently moving reference frame, he saw that the measures of those more dynamic parameters associated with a moving body - such as mass, momentum and energy - had now to be considered in terms of this new adjustment. In pursuing these matters, he seems to have concluded once his relativistic interpretations were better clarified that there must be some kind of equivalence between mass and energy. They appeared to be governed by just one conservation law and to this extent he began manipulating these variables in his mind in conjunction with relevant thought experiments, to see if and how a more precise quantitative relationshoip between them may emerge. [Did it not include his usual systems K, K' since any equations involving v/c would presumably imply measurements/observations from such differently-moving reference systems - even if not always made explicit ? (Or am I weong here?)]
[Note: The following resume of this derivation is placed here as it seeks to set matters out in a way which I may better follow in arriving at its startling conclusion (ie than as I had set out previously and have now removed).
623. In his theory of special relativity, Einstein showed that our previous conceptions of space and time, as they apply to the velocity of moving bodies, had to adjust to the reality of there being a previously unrecognised fixed upper limit to the velocity of anything. This meant that the formulae pertaining to many parameters concerned with such motion (when viewed/measured from elsewhere) had effectively to replace the values previously used for velocity (v) when determining those parameters, with an adjusted one - based on the ratio of v to that upper limit (c) - ie on v/c. (It turns out that c happens to be the speed of light; 'happens', no doubt, necessarily - for some explicable reason that we may go into later but that c exists for more general reasons of its own, as it were.]
624. Three such important dynamic parameters of the motion of bodies, beyond their velocity per se (as dealt with essentially by Einstein's 'kinematics' alone), are thus those of mass (m), momentum (p) and energy (E). Thus momentum was known in pre-relativity times to be a (linear) function of the velocity at which a body (of mass m) moves. That is, that p = f(m,v) - the particular function being their product. [Note: some of this has been presented earlier when detailing Section 15 of the 1920 book] - ie:
which, to prove more consistent with the next step in the development of the present reasoning, may also usefully be shown as:
For, in relativity terms (ie as viewed/measured from a differently moving reference), a moving body's momentum is actually more accurately given by:
Thus, rather than being divided by 1 and so give a value for the magnitude of momentum of a moving body as the unaffected product of m.v itself (as viewed on its own reference frame, say), the latter equation produces instead a magnitude that is greater than that 'basic' (nominal) value to the extent that the velocity v of the body is a proportion of the ultimate possible velocity c. (And crucially, that latter velocity is the limiting factor in determining how quickly information about any such measuements can be obtained - there being no other source of such information and hence no other more accurate version of reality - from some distance.) If v is a very small part of the value of c, the fraction v/c is very small and hence so would be the amount to be subtracted from 1, leaving a value only marginally less than 1 by which the nominal momentum product is to be divided. Its value would thus be only slightly greater than 'normal' - ie than as always previously accepted as such in classical times.
[We may point out here that while the fraction v/c may be rather small, where v is small compared to c, and thus the unity divisor is reduced only marginally by the above subtraction, that it (that fraction) is still much larger than the value of v/~ (where ~ = infinity) - which was effectively assumed in classical physics when it was believed there was no upper limit to velocity and so v/~ = 0 which is tantamount to our second equation above (p= m.v / 1-0) = m.v). We may reasonably use this divisor therefore as a contrast to the more valid relativistic subtraction 1-v/c. In this regard, c may be seen, paradoxically, not so much as the enormity (it generally is) but as a restricted limitation which governs the magnitudes of the associated parameters !]
And where v becomes a large proportion of the value of c, the amount to be subtracted from unity is not zero or something a bit more but approaches 1/1 = 1 itself (vs the 0 subtracted in theory in the pre-relativity formula) so leaving a value for 1-v/c as 1-1 = 0, or somrthing very near that, say, rather than the full value of unity) to be divided into the momentum product (m.v). This would produce a much inflated value for that fast moving body's momentum of given mass. And with such velocity (v>>0), the other parameters of mass and energy - would also increase greatly, as will be shown below.
625. [It seems somewhat paradoxical that the existence of an upper velocity limit which would (one imagines) prove to be a 'constraint' on the magnitude of any lesser velocities being thus calculated - ie to better 'fit into' that now realized limited scope of possible velocities - and yet the effect of applying this proportional v/c ratio, rather than utilising v alone (effectively in ratio with infinity), on such parameters as discussed here, results in them (with velocity as part of their structure and calculation) becoming not smaller than otherwise (as velocity itself effectively becomes) but larger !? Thus, while the value of momentum in classical (pre-relativity) terms does increase with increasing velocity (for a given mass), in relativity terms this increase with velocity (for a given mass) becomes even greater (rather than less) - as does the mass itself! (But how to compared like with like if we can't stipulate such as 'for a given mass'??) We shall have to consider this point further. Do the expanded values of mass and energy with velocity - due to.............(?) also so arise and/or play a significant part in this ?]
626a. In any case, when a body's velocity is increased, as from a state of rest relative to its surroundings, or even just from some slower speed, that acceleration of its mass requires a force applied to it to produce that increase. Such a force would imply an application of energy to so effect that acceleration and the consequent transfer of same to the body being accelerated - up to some new uniform velocity say (that we shall assume here). The greater the mass and the greater the velocity sought, the greater would be the energy required (and transferred) to effect same. That is, the body moving at this velocity would possess not only momentum (p) as a function of its mass and velocity but would also possess/acquire kinetic energy (E_{k}) as a function of the magnitudes of those same two variables. If it struck some other moveable body, such momentum and energy would thus be transferred to same accordingly - after some 'impact'.
626b. [Note: This next paragraph had been 'parachuted in' from an earlier account of Adam's derivation. Anything useful may be integrated into this present version about here.] "How then does light itself - as a body of possible concern - attain that maximum possible speed of c (its own inherent velocity) ? Because particles of light (photons), which move at velocity c, were known to possess momentum (and energy), it suggested that they must possess virtually no mass (otherwise they could not, as they invariably do, attain that enormous (indeed maximum possible) speed of c; the normal mass increase becoming too great to further increase its velocity. [This seeming masslessness of light was apparently determined by Maxwell in the 1850s but in terms of electromagnetic waves.] But how can a photon (or a packet of such waves) have such amounts of (?kinetic) energy and momentum (relative to their size) and yet have little or no mass ? It was Einstein's great insight that the momentum and energy of a photon must (in their impact effects at least) be somehow equivalent to the effects of an equivalent quantity of mass. The energy need not actually be convertible to the same quantity of mass per se even if the effects of the momentum thus accounted for prove the same. [See also article by Philip Gibbs about here. He seeks to answer the question: 'Does Light Have Mass?' (Relativity FAQ - 5 Aug 1997). His opening remark is "The short answer is 'no', but it is a qualified 'no', because...etc."] For more massive bodies, a different but comparable relationship would presumably emerge.]
627. As with momentum, the energy of a moving body had also been calculated in pre-relativistic times to be a similar, if non-linear, function of both its mass and velocity (ie (E_{k})= f(m,v) - with the similar formula:
This had been determined empirically in the 18th century after various experiments. [We may review our discussion of this in the analysis of Einstein's 1920 paper above (as #3).] That the energy acquired by an accelerated body (provided by an application of force and thus energy) increases according not only to ( 1/2 of) its mass per se (the greater one half the mass, the greater the energy needed to so accelerate ir (and be thus acquired by it) but to its subsequent velocity squared. Its increase in kinetic energy so effected was not therefore just linear but exponential to this (squared) extent and thus would increase markedly with ever greater velocity for that given mass (which, again, would in any case apparently increase itself thereby(tbc)). And while momentum does now effectively increase at some exponential rate when calculated in relativity terms (as described above), this pre-relativity energy value for a moving body, being non-linear/exponential, is even more apparent when also calculated in relativity terms - when the value of velocity v is effectively 'adjusted' in terms of v/c. The increasingly larger denominator subtracted from unity thereby results in Energy also increasing exponentially by this additional factor) - ie as well as doing so more directly due to the squaring of its velocity.
628. While one may well appreciate why the momentum and energy of any moving body of mass m would increase with an increase in velocity per se, and do so as a multiple of its mass, it seems less obvious why, in the case of energy at least, this increase should be at a rate determined by the squaring of that velocity value rather than by simply that velocity per se (or even just 1/2 it). For its velocity is that at which it actually moves (and not that value squared - even if divided by 2). Thus if a body moves at 100 mph, it doesn't suddenly do so at 10,000 mph, or even at 5,000 mph, when its kinetic energy is being calculated. Yet its energy, if not its momentum, does for some reason increase at this very impressive (squared) rate with such increases in its own velocity. Who first discovered this empirically and when ? But, importantly, is there a simple physical explanation for this empirical reality ? Do we just accept it as a fact and not analyse it for its underlying mechanics ? For knowing that may provide us with some insight into why energy (even if latent in a still body) is, again, and possibly unsurprisingly, a function of 'mass' or its equivalent and velocity (albeit that of light) but with the latter again....squared! Why ?
As our interest is in a still body's latent energy, any velocity somehow involved must be that of the trillions of apparently unmoving, but potentialy extremely fast-moving, photons (or equivalent) and their associated protons, neutrons, electrons and gluons presumably) of which any still body is ultimately composed. And hence, its energy E would equal their total (atomic) mass times the latter's (latent or unrealised) but 'freeable' velocity - squared !! Is there a comparable basis for the squaring element in the derivation of these two forms of Energy (latent and kinetic) ? How does it come about in the case of E = mc^{2} for any still body (where the respective unsquared c elements seemingly arise from different 'sub-components' in the derivation; although ultimately they must both come from the same one source - namely, the velocity of any photon/radiation) involved in the explanation/derivation) as compared with where the squaring of v arises in the case of an empirically-derived equation for kinetic energy (E_{k} = 1/2 mv^{2}) of a moving tangible body ? And how does any still body's analysis entail even one never mind two 'sub-sources' of the velocity of....anything, especially of light (c) ? One must assume that tiny 'bits' of that still body's mass are allowed to escape (or can be actively released) in the form of photons and it is their (?potential) velocities that are somehow shown to be aspects of the mass of the otherwise still body concerned, and hence a part of its total latent energy content - somehow bound up in or between its total constituents (ie its mass).
629. In any case, we would again not utilise the pre-relativity equation for kinetic energy of a moving body thus:
but, rather, should use:
And, as above with momentum, this value of a moving body's energy is only slightly greater when its velocity is slow compared to that of c, but when it is a larger proportion of same (and v/c approaches 1), the resultant kinetic energy (as with momentum) would be very much greater.
630. Now, the bodies of concern in the foregoing discussion have been considered to have a gven mass and to have moved at either a relatively slow or fast uniform velocity - compared with velocity c. We may enquire, as apparently Einstein did at some point, what may be the momentum and energy associated with a moving body (again in relativistic terms) when that 'body' is but a tenuous photon of light (as anticipated at the end of the preceding paragraph (628) ! ? In such a case, its velocity is represented not by v but instead by the maximum possible velocity (c) as, for whatever reason, light moves (in a vacuum, through its electro-magnetic field) - or can or must do so if massless - only at or near that particular maximum possible velocity (depending on certain factors such as its frequency, intensity, etc). Thus, its momentum (relativistically) would presumably be represented by:
and its kinetic energy similarly by:
The problem with these formulae is that there is considerable doubt as to whether light and its photons have any mass, when in 'full flight'; that is, in addition to its more obvious momentum and energy when in that (its typical) state. As stated here, these parameters would approximate to magnitudes near infinity (and would actually equate to same had we replaced v with the full value of c - which we probably should have). This apparent quandary is resolved by appreciating that a photon of light very likely (therefore) has no 'inertial mass' so, unlike all other bodies, doesn't require an infinity of energy to accelerate it to a velocity of c, nor does its kinetic energy and momentum approach infinite values therefore. To attain a velocity of c, light, as a 'body' of some kind, must be an exception; it doesn't require an infinity of energy to accelerate it to c, nor does it acquire thereby an infinite level of momentum, or kinetic energy, because, 'body' or not, it must have no mass, as generally understood. But it does have momentum - in its stead.
631. Einstein reasoned that the considerable energy and momentum possessed by the virtually massless photon must therefore represent or be the equivalent of a quantity of mass (see also par 623). The relationship between a photon's energy (rather than its mass) and momentum in relativistic terms does then simplify to:
where values for mass m and velocity v in the usual formula for any mass-possessing body's kinetic energy (see above) are replaced by those for the massless photon's momentum p and velocity c. But, does E not = 1/2 m.v^{2}/1-v/c ? To answer this latter query, we may usefully show how the above relation is derived, thus:
That is, the kinetic energy of a photon, travelling as it does 'extremely fast' (at velocity c), is in relativistic terms equal to its momentum (vs its mass) times that velocity c. And its momentum is therefore similarly:
632. We have thus answered our earlier query (and possibly Einstein's) as to 'what would be the values of these two parmeters (in relativistic terms) for a massless moving body that happened to be a photon moving at velocity c' ? With this latter factor entering into calculations of these parameters, we may appreciate that energy, mass and momentum can be calculated in terms of each other with an inevitable role for the velocity of light proving to be the case not only when relativistic measures are made (where the ratio v/c is inevitably involved) but where any comparison may be made as between the actions of moving bodies which, for 'normal' ones, have some mass and for other ones (as photons) have no mass, but do have their own velocity c to enter into the relevant equations. In this regard, we may seek to examine, for example, what the precise relationship may be as between energy per se (rather than just kinetic energy in particular) and mass. Such a relationship is certainly suggested in terms of the shared relevance of the now relativistic measures of momentum amd energy as shown above (for both light photons and a body of mass m), in which the value c appears as shown.
633. Thus, it turns out that after Einstein considered further the two relationships (1) and (2) shown above, he somehow came to the startling conclusion that the (latent) energy in any body - whether moving (when it would accompany its kinetic energy) or not (when it would effectively manifest its 'static' energy only) - would turn out to be:
The subsequent explicit derivation of this initially implied relationship - between a body's latent energy and its mass - (for a relatively still body) - via an enormous 'exchange rate' of c^{2} (rather than 'just' c itself, say, which would seem large enough) - apparently followed from the implications of the special theory of relativity in regard to the relationship between momentum and energy (ie in such relativistic terms) where velocity v is necessarily adjusted in terms of the ratio v/c (even for light). While the kinetic energy of a moving mass was a function of its velocity - squared, the static energy of a non-moving mass apparently turns out to be a function of the (potential) velocity of its constituent if incipient photons (c) - again squared. [We may note here that if the basis of this relationship was investigated only in terms of the classical equations for such energy, mass and momentum (of any moving (mass-possessing) body), there would no source for the parameter of light's speed (or, more correctly, for the value c) and hence latent energy would not be concluded to be a function of mass and of c squared.] This relationship was first alluded to or implied in Einstein's article of late 1905 (when the equivalence between energy and mass was first indicated and subsequently made more explicit by himself and others.
There appears to be two rather different approaches in describing its derivation. We consider firstly here the one presented by Adams based on a thought experiment by Einstein which conveniently utilises an arrangement whereby the actions (and interactions!) of our two entities (types of 'bodies)', one with mass and the other without, can be compared and their effective equivalence regarding their momenta and energies utilised to confirm not only an equivalence between (unmoving) mass and energy in general but that crucial 'exchange rate' between them! This approach thus focuses on a more immediate or direct role for c (in terms of a photon's momentum) in contrast to other approaches which seem to do so more indirectly - in terms which focus more on the relativistic motion and the eventual non-motion of the total mass-possessing body itself, albeit still in terms of momentum and energy; that is, without considering in addition the emmision or absorption of tiny (photon) 'bits' of it. The latter approaches also seem to require more use of both calculus and reference to contrasting frames of reference K and K' and thus the inclusion of the relativistic source of c - in the value 1 - v/c. We'll see; maybe some reference to either absorption or emission of radiation or photons is inevitable(?).
Adam's Derivation.
634a. Thus, in the present thought experiment which seems to fall into the former, more direct category, we imagine a stationary hollow box of mass m floating freely in deep space uninfluenced by any forces. It has a covering which we may usefully assume has a fluorescent capacity from which a single photon is readily emitted from its left inner wall into the interior of the box (at the central point of which we may assume is its centre of mass C). It travels across to the opposite wall, which is distance L away, at the velocity of light - c. Since the momentum of the system as a whole (box and photon) must be conserved, the box would recoil to the left as a reaction to the emission of the photon (which has its own momentum and energy, if no mass). However, as momentum is a function of a moving body's mass and velocity, we may enquire how light photons can have any such momentum ? Einstein concluded that light's energy must somehow be the equivalent of the quantity of mass it would otherwise possess - to so account for its momentum and thus its impact upon that opposite wall of the box. It would strikes the right wall after time dt (which equals L/c) when its momentum and energy would be effectively transferred back to the box. This neutralises the motion of the box leftward and conserves the momentum of the system as a whole.
Being a fee-floating mass, the box's centre of mass must somehow remain in the same position throughout the photon's journey and change of position even though the box has moved leftward a very short distance dx during the photon's extremely fast journey across rather longer distance L. How can the movement of the box be reconciled with its centre of mass remaining fixed at C ? Again, the answer indicates that there must be an effective equivalence as between the energy of the photon and the mass that would otherwise maintain the centre of the box's mass at its original position as it travelled from left to right across the distance L of the box - and once it had been re-absorbed into the box's right wall - effectively as the same amount of mass from whence it originated - albeit now re-positioned over a wider width of the box. If the location of that centre of mass at C can be shown not to have changed under these circumstances, it argues for the equivalence posited - with a mass equivalence large enough to exactly maintain that position of C.
The foregoing elements of this thought experiment may now be represented amd analyzed mathematically as follows:
If the photon's energy is E and its velocity c then its momentum (p_{ph} is (as shown above):
That is, the photon's momentum (if not its mass, as it has none) is a function of its energy and velocity - ie p_{ph} = f(E,c); specifically = E/c and thus = a very small proportion (1/c) of its otherwise relatively large amount of energy (and vice versa with respect to its energy E being a f(p,c) - namely = p.c, or = an enormous multiple of that tiny momentum, rather than, as indicated, being a function of its mass and velocity (ie f(m,c) - as it has no mass and its energy (in terms of its momentum) serves in its place. [As with its momentum, its energy E and velocity c in the foregoing discussion could equally be shown with sub-scripts 'ph' for photon, and energy E with the sub-script k as well.]
634c. We may now compare these parameters of the single (massless) photon with those of the (mass-possesing) box from which it was emitted. The latter recoils to the left over the unknown distance dx at unknown velocity v (depending on its total net mass, also unknown) so the magnitude of its momentum (p_{bx}) - (briefly possessed for time dt = L/c) is a function of that mass and velocity - ie f(m,v) - specifically:
[While this may be the original classical equation for momentum, it may be used in this initial development of the reasoning used here since it will, I believe, fulfill the relativistic requirement subsequently. In any case, it proves to be a very important element in the case being made.]
The photon will take a very short time dt to reach the right wall - during which time the box will have moved its very short distance dx leftwards (when the photon strikes that wall) at which point it will stop. The velocity of the box over its short movement would thus be:
By the conservation of momentum, we can say that the box's momentum p_{bx}, which = m_{bx} . v, can be shown to equal
If, as shown above, m_{bx} / L/c = E/c and this is substituted into equation (1.4) and re-arranged, we obtain :
If we suppose 'for the moment' that the photon has a mass 'm' (rather than just energy), we may calculate the position of the centre of mass of the whole system - at the start, during, and at the end of (ie throughout) the experiment. We identify the position of the box as x_{1} and that of the photon as x_{2}, so that that for the whole system (c_{1+2}) will be:
For this centre not to change over the time of the experiment (despite the movement of the box to the left and that of the photon to the right (which latterly becomes embedded into the box's right wall), we must find that the effects of these changes (in mass and energy distribution) exactly balance with respect to the centre of the system's overall mass throughout the period of such activity; that is, that:
As the photon starts at the left of the box, its position (x_{2}) then will = 0. Then, by rearranging the above equation for the equality (balance) of the centre of mass before and after the experiment, we will get:
so that substituting equation (1.4) into this will show that:
This conclusion was our apparent goal but we may also follow the same or similar arguments further below (at *** - as set out earlier) in case they may reveal this as well or even 'more validly' - for the above appears to have focused on the 'supposed mass' of the photon ('m') - which of course in reality has no mass !? ['for the moment' doesn't appear to have been clearly rescinded.] We would wish, rather, to find that E = m_{bx}.c^{2} - for any and all masses - including that of the box. In the meantime, we may indicate that in the foregoing derivation, Adams had shown us that both 'm'.L and m.dx equal E.L / c^{2} - so that 'm'.L equals m.dx which, in any case, he also shows when showing that the centre of mass of the whole system doesn't change during the experiment. But when he considers 'm'.L = E.L / c^{2} and can thereby at least show that E = 'm'. c^{2} (as shown above), he can't do the same for m.dx = E.L / c^{2} - as he would be able to show only that E = m.c^{2} times dx/L (cf...'times L/L' which allows these distance measures to cancel out) thus showing again that Energy is equal to this tiny (dx/L) fraction only, of the full mass m of the box; that is, that it equals the mass equivalent in the box of a single photon ('m'). If however this applied equally to any and all of the trillions of individual masses that are potential photons (of the box or of any mass) then we would indeed have the sought for generality that
We may point out here also that the elements of the experiment as so arranged serve well to underline tha fact that the immense energy and velocity of the single photon (and of all light per se) stands in contrast to the miniscule magnitude of the single element of mass of the box from whence that photon derived. For this reason, the two measures of light's speed (c) which enter into the quantitative relationship of concern (ie of E : m) - as pertain to the motion of the box and of the photon, respectively - even when necessarily multiplied (squared) during the derivation, may be better appreciated as entirely fitting. That is, that the minute 'bit' of mass (m) must be multiplied by the enormity of c^{2} in order to equate to the immensity of Energy possessed by a single photon relative to the small mass from whence it came - and thus likewise by any totality of such mass (bits) that one may care to consider. This particular ratio - m : c^{2} was what had to emerge when the criterion of an unchanged centre of mass for the particular (and very contrasting) moving mass ans energy elements involved was accepted as necessary and appropriate.
[We return now to our earlier attempt to arrive at the above conclusion regarding Energy and mass:
If the 'momentua' of the box (p_{bx}) and of the photon p_{ph} are seen as equal, then the center of mass of the box should be remain at exactly the same position as was the case prior to the emission of the photon. This will be assumed to be proved so subseqently. We may then state that, as the momentum for a photon (p_{ph}) has been shown above (parag 631) to = E_{k(ph)}/c then:
E_{k(ph)} (times distance L) = m_{bx}.c^{2} (times the shorter distance dx).
634d. The foregoing analysis thus equates (the box's) mass with (the photon's) energy - with either parameter being a function of the other - the particular function entailing the enormous quantity c^{2}. We may usefully consider just what this equation is saying - at least in its present form (before being even further 'arranged' and simplified). It tells us that m.dx or the box's mass moving over (ie times) its short recoil distance dx (with the time for same initially ignored) is equal to or balanced by a very small fraction (1/c^{2}) of the kinetic energy possessed by the photon as it covers (ie times) the distance L (at velocity c). Thus the mass of the box as a whole (with its trillions of internal potential photons (still as mass) moving over brief distance dx has been thus equated with the much divided Energy of a single photon moving at speed over its longer distance L. It is then necessary to show how the equation for this relationship can be arranged such that the (latent) energy component of the still box replaces the kinetic energy for the photon (ie as the energy side of the equation) - and so equates to its own (the box's) mass - times the photon's immense velocity, squared - on the other side (of the equals sign). That is, that:
Rather than simply accepting this latter re-arrangement as an inevitable fait accompli, we should seek to show how this relationship arises - in which the box's mass (or indeed any body's still mass) will equate to its own latent energy - if multiplied by the enormity that is c squared. This will hopefully be shown below when we further analyse the interactions (of box and photon) within our present thought experiment - keeping in mind the source(s) therein - not only of the two c values that combine (multiply) to provide that c squared value but that of the particular still body (the box) by which its mass m must be thus multiplied - to equate to the latent Energy of that (or any) still body.
We may note that the box appears to determine (or pertain more to) the distances moved (dx and L) - by the box and the photon - while the photon appears to determine more the times (dt = L/c) and thus velocities involved - again for both box and photon. These are of course the two requisite elements of the velocities concerned in the experiment - in which the respective momentums and energies are crucial in determining the latent Energy of the box (E_{lat(bx)}) in terms of its mass and light's speed c, squared.
635. To this end, we consider next in more detail the effect of the emitted and re-absorbed photon on the position of the box's centre of mass - that is, when the mass and/or energy of the box and the photon together are considered as a total 'system' (of both structure and activity). This seems to imply that the two elements (box and photon - with their respective mass and momentum, velocities and energy) must somehow 'balance out' with an effective equivalence and inter-changeability as between mass and energy. Latent energy is held in the still box - including that from the now still photon once it has been re-absorbed back into the box. We may identify the position of the box's centre of mass as position x_{1} and that of the equivalent 'mass' of the photon (represented ultimately by its momentum but utilised here 'as though' it possessed 'mass') as position x_{2} (in the box's left wall initially but soon released and re-absorbed). The centre of mass (M_{syscent}) of the whole system may then be calculated thus:
Accepting that the system's centre of mass (M_{syscent}) does not (indeed can not) change in any permanent sense (due to the law of the conservation of mass) but is quickly re-established at its original position, the following equivalence nust be the case:
- noting that the photon started its journey across L from position x_{2} (ie position 'zero'). By re-arranging and simplifying, we get from the latter equation :
(which we may recall arises thus):
E_{k(ph)} (times distance L) = m_{bx}.c^{2} (times the shorter distance dx)
which is equivalent to the (non-kinetic) latent energy within the now STILL body of mass M (which we may again refer to by the symbol m) :
636. This seems to indicate that the latent energy (E_{lat}) of the total unmoving box (with its re-absorbed photon and its centre of mass re-established), and hence, generally, of any still body, is equal to its total mass times c squared. It would thus apply equally to each such element alone (ie to the box both with and without that photon and to each of the trillions of potential photons of which any body is composed (as seemingly concluded above at (1.10). That is, each has latent energy equal to its mass (or mass equivalent) x c^{2} and this energy may be manifested as kinetic energy possessed by any such potential photon that is released - with that energy again of a magnitude per photon that equals its momentum (in place of its mass) times its velocity - again squared. We may recall that kinetic energy of any tangible mass is similarly a function of its mass times its velocity - squared. One might well wonder therefore whether the sources (see **) of the two measures of light's speed (c) which necessarily become multiplied to give the (enormous) c squared value in the derivation of the equation for any still body's latent energy (as per Adam's description of Einstein's derivation of that famous equation - as provided in terms of his hollow box example) have a comparable source within the much earlier derivation of the similar equation for any tangible body's kinetic energy ? Or was that equation never mathematically 'derived' in terms of prior physical processes but only concluded empirically and thus accepted without further demur ?
(**) As suggested earlier, one might reasonably enquire why it is that the energy (E) and distance (L) component (as multiplied) need be divided by (be in ratio with) such an enormous value as c^{2} ? We may note that each aspect of this complex component (of E and L) were each previously in ratio with the unsquared value of c itself and thus their subsequent multiple must also entail both these maximum velocity values also being multiplied by themselves - ie as:
- with the activity within the total system involving the motion of both box and photon seeking a return of its centre of mass, as in its original still state. The c^{2} denominator then becomes part of the numerator in its multiple of the still mass - to equate to the latent energy E_{latent} thereof. That is, if the box's mass moves over its short recoil distance dx at velocity dx/dt (where dt = L/c), it effectively equals the photon's 'mass' (ie momentum) as it moves across its distance L over the same brief period of time (dt) at its immense velocity c (L/dt) - where, again, dt = L/c (and p_{bx} = E/c). We may then say that:
We may also note that that part of the box that was emitted as a photon did itself possess mass but this was (?instantly) converted to the photon's momentum and energy and if every such portion of the box was so converted, thus losing all its mass, it would produce an immense amount of energy - much more than if it was simply burnt to ashes, say - due to the effect of multiplying by c squared.
637. In the foregoing analysis, it seems to be the case that the role of the box's momentum - p_{bx} (a function of it mass and brief velocity) plays the key role in that it is exactly 'countered' by the momentum of the photon (p_{ph}) so that the box's center of mass remains precisely at its initial position. The lost mass of the box (on the emmision of the photon) and its subsequent mass and momentum must therefore equal that of the photon (which serves as its mass) which totally arrests the former's brief momentum.
In the event that the foregoing has not completely revealed the derivation sought, we may set out here again this part of the reasoning - to help further comprehend its relevance. [In this, it may be necessary to add furthe subscripts as each element should ideally be properly identified as each has an important role and the final 'logic' could be confused without same.] Thus, we see that the momentum of the box (p_{bx}) is a function of its mass m (which generally suffices for m as there is no other body involved having mass) which moves over distance dx for time dt (being the time the photon takes to cross distance L at its speed c (ie dt = L/c). That is (as shown above):
and from earlier (see parag ......) we recall that:
indicating (again as above) an equality as between a momentum component based on Energy (E_{k}) and another momentum component based on Mass which both entail a ratio with the speed of light (c). And when re-arranged, we find that:
That is, the box's mass (m) moving over its short recoil distance dx (at some velocity) is the equivalent to the photon's kinetic Energy E_{k(ph)} moving across the box's relatively longer distance L - with the 'exchange rate' between these two forms of essentially the same thing being the square of the velocity c of the photon (which arises as shown by the multiplication of the individual c values associated with those respective components of mass and energy.
We may recall that kinetic energy of any moving tangible body too was a function of its mass and its velocity squared. In this case, the latent mass of any still body must be the equivalent of the summation of the potential kinetic energy of all of its photons. In our thought experiment, we showed that the mass equivalent of the emitted photon (its momentum) was enough to moved the box in its recoil the short distance dx and that momentum just enough when it struck the opposite wall to re-establish the box's total mass and the position of its original centre of that mass.
To elaborate further on this: 'The box's momentum is a function of its mass times its velocity - the latter being determined by the distance (dx) it moves during its recoil - being the time it takes for a photon to travel at its velocity c across the distance (L - for Length) of the box until it strikes the opposite wall and thus halts and reverses the box's recoil motion - that is, after time dt. Its momentum, equal to m x its velocity (dx/dt), may thus be related to the photon's kinetic energy - as a (very small) fraction of its velocity c. That is, p = m.v = E_{k}/c . (This latter measure of momentum (derived earlier) takes full account of relativity, I believe.)
638. On this basis, it should be possible to relate the Mass of the box as its contribution to its momentum (according to its own velocity as determined by the velocity c of the photon) to the Energy of the photon - once, that is, the box's centre of mass has been re-established at its same previous STILL position, thereby confirming that it hasn't been affected permanently by the emmision of the photon and its subsequent impact with its momentum p_{ph} and the box's consequent reverse recoil. The box's Mass is thus a function of the photon's Energy (which was needed to effectively cancel that momentum - of a mass moving at velocity v), and vice versa. So latent energy of a still body (as the box) is a function of that (still) mass - namely as enormously multiplied by c^{2} ! Thus:
which conclusion emerges by means of the clarification of the relationship between the above mass and energy involved in the foregoing 'experiment' - which entails important roles for the speed of light (c) in regard to both the momentum of the box (to determine the time dt relevant to both the box's and the photon's subsequent motions) and the amount of energy entailed in the latter's momentum (if not its mass). Thus, there is an effective equivalence between the 'mass' and energy of both the portion of the box emitted and the subsequent photon (and the various equations could probably be equally well interpreted without the use of so many subscripts). We recall that the photon came from the mass of the still box and returned its 'mass equivalence' back to it on impact (and, in ideal conditions, totally so). The photon's temporary kinetic energy thus becomes, again, a part of the box (sytem)'s total still mass along with its trillions of other identical 'mass-resevoirs' of latent energy. Certainly, as atomic bomb releases enormous energy....and light.
639. The above derivation of E = m.c^{2} thus relies upon some degree of equivalence between the Energy and Mass of both the box and the photon such that one might be able to interpret the results (and their derivations) in terms of either or both the box and the photon - whether in motion or still. The 'experiment' was indeed so 'arranged' by Einstein we may assume. It was rather cleverly contrived to allow a comparison and contrast between the two concepts of concern - mass (of the box) and energy (of the photon) - and how they can affect one another when required to interact as described - with no extraneous elements involved - and so demonstrate their equivalence.
In short, the latent energy of the still box (or any mass-possessing body) is revealed (ie determine to be the case at a particulr value per amount of its mass) by means of determining the extent that the momentum of any photon derived from that body (which = E_{ph}/c) balances the momentum of that box (which also = E_{ph}/c) when forced into its temporary motion by the photon's emmission and then impact of its momentum.
But could this relationship have been equally well demonstrated by means of a less contrived and delimited 'experiment' ? Some earlier clue or suspicion about this relationship must have preceded the imagined experiment designed to come to that very conclusion. One in a sense likely already knew what it was that one was seeking to so find and support. When was it first mooted ?
Einstein appears to have considered that quite early on that the conservation laws - of mass, energy and momentum - seemed to indicate that such parameters of motion must be closely related and even two sides of the same coin and thus related - if in some uncertain proportions. By having such a conception in his mind, he was able to devise an imagined arrangement of two sets of interacting moving parts (of the box and the photon) which were likley to optimise the confirmation of the suspected effective equivalence of mass (from the box only) and energy (from the photon) and do so in terms of the exact function between them - viz: c^{2}. Their effects on one another could be compared and any equivalence concluded. Such equivalence was thus concluded and shown to result in the equation of concern. And if thus shown by later confirming experiment to be a true feature of nature, then one might reasonably expect to be able to find that this relationship should manifest itself in other predictable ways and so allow other forms of its derivation to be demonstrated. Thus, those identified here as methods #1,2,3,5,6 and 7 often focus on the increasing kinetic energy (and mass) manifested by a body of initial mass m as it moves at increasing velocity and then reasons that, as the velocity is reversed (decreased) back to zero, its original still mass must imply a similar repository of (latent) energy which was presumably initially instilled therein in the same way that they (mass and energy) increased with increasing velocity - with the square of the speed of light again determining the magnitude of the equivalence. [This likely requires a more precise explanation - which see below.] We may note that in the 'hollow box' derivation, the important re-establishment of its centre of mass - back to the still state - may prove comparable to returning the moving mass utilised in these other derivations back to its still state - of some original mass. One may well deduce that such an unmoving mass held its 'massive', if not increasing, store of energy comparably - somehow instilled as it was first created.
640. We may now compare the foregoing derivation (#4) by Adams - initially with our accounts #1 and #2 of that given originally by Einstein (in late 1905) and then with that provided again by him in 1920 (#3), and finally all these in turn with those also gratefully provided into the public domain by such as Doyle (#5), Tao (#6) and Cox and Forshaw (#7) - each of whom appear to approach the matter slightly differently. Thus Doyle's approach [see: www.btinternet.com~j.doyle/SR/Emc"/Derive.htm] is said to follow directly from Einstein's 1905 article while Tao proceeds by way of 5 steps which develop the ideas right through from Einstein's initial basis of special relativity to that submitted by Einstein in his later 1905 article - which he also identifies as the source of the original derivation. Finally, and we hope usefully, Cox and Forshaw, have produced a small book on the famous equation itself entitled "Why does E = mc^{2}?", although their approach makes less reference to Einstein's articles and relies on more contemporary ideas on the space-time continuum (after Minkowski) and the concepts of invariance and the relevant conservation laws generally - ie a more contemporary, abstract, generalised (vs historic, practical, specific) approach - albeit still based ultimately on Einstein's original work. [Derivations #1, 2 and 3 above may first be reviewed and compared with that of Adam's (#4), if required, befoe that of Doyle next.]
Doyle's derivation of E = m.c^{2} (#5)
641. The validity of the relationship between energy and matter was indeed, according to Einstein, concluded to be an implication (or result) of his special theory of relativity (as arose out of his main paper on Moving Bodies completed just weeks before the Inertia-Energy paper in which the derivation of this particular relationship is said to be shown, if implicitly. Thus Doyle begins by re-stating those two fundamental postulates on which special relativity itself is based - as these would (as already indicated) prove just as relevant to this associated/generalized field of mass and energy (since velocity and thus time and space are fundamental in this area as well). He does avoid certain detailed aspects where the mathemetics are deemed too complex but othewise his explanation seems as helpful and clear as Adam's (#4). Possibly Tao's version incorporates the mathematical aspects more thoroughly, although this would likely prove beyond my comprehension. (However, articles on kinetic energy and momentum in Wikepedia now also provide further insights into these aspects and perusing same (which we do below) may allow a better understanding of the various derivations than otherwise.) But, firstly:
642. Doyle begins, as mentioned, by noting that this relationship between energy and mass is derived directly from the theory of special relativity (as Einstein confirms) and that its validity and thus that of the special theory itself are both witnessed, for example, in the existence of the atomic bomb and of nuclear power. How does this important (and undoubted) relationship derive from the principles of special relativity therefore ? The crux of special relativity in this regard seems to centre on the importance of the qualification that the velocity of any moving body (v) is not unlimited but rather is restricted according to the ratio of v to c. The nearer that ratio is to unity (1), the greater will that resultant effects become. It can't exceed that value, that is, v can never exceed c - something not appreciated before Einstein (although seemingly considered by Poincare). One of those measures affected by this new reality concerns the Mass of such fast moving bodies. We shall consider this first followed later by Energy (par 646).
As Doyle states it "The faster an object moves, the 'heavier' it seems to get." That is, "..its mass appears to increase with speed." But this effect isn't noticeable until the speeds concerned become an appreciable proportion of the speed of light (where v/c approaches unity). [These statements could be interpreted as indicating that where such speeds are only approaching the required 'appreciable' levels, the heaviness and mass increases occuring are not quite fully noticeable, although that may 'appear' or 'seem' to be the case - only becoming more clearly noticeable and definite (actual) once sufficient speeds are the case. On the other hand, these two qualifying terms (as we have often mentioned already) may have to be explained otherwise; see below. For why were these descriptions not expressed without those two easily over-looked qualifying terms ? The reader would very likely have accepted them just as well if they had been unqualified in that regard - to be followed by some explantion as to the basis of such claims.] To whit:
643. The extent which a body's mass increases with its speed (note: not 'appears to increase', I believe, but this should be checked) is, says Doyle, given by the (relativistic) equation:
where m = the mass of the moving body concerned and m_{0} = the body's 'rest mass'.
[NB. The derivation of this equation should itself be described here as it is clearly quite germane to our quest. Does it appear earlier?] We may note in this regard that the mass being so calculated does not arise by dividing the rest mass by 1 - v/c (the latter ratio reflecting the proportional factor by which the object's velocity is necessarily qualified in special relativity) but is divided rather by the square root of this value - after it has first been squared. While there is no doubt a good mathematical (if not physical) reason for this, why isn't this explained ?? Is it simply to eliminate any negative values?] Shown graphically, the (?apparent) increase in a body's mass with its speed begins to be noticeable at about 40% of the speed of light. By 70% of same, its mass would have doubled and at 80% trebled. Beyond 95%, its mass would quickly seek to increase towards infinity although this could never be achieved as it would be too 'massive' for its speed to be further increased sufficiently - as it would require an infinitely (and thus impossibly) large amount of additional energy. [Note: should the terms 'appear' or 'seems' also be assumed here...or ? See next paragraph!] In the other direction, Doyle points out that it is important to note that when brought back to its resting status, the body's mass could never become less than the unity (ie 100% of what its 'resting mass' was initially - whether 10 kgs, 100 kgs or whatever) which, as noted, would double by the time it was accelerated from same to reach 70% of the speed of light. This point proves essential (in the present derivation) in understanding how the equation E = m.c^{2} is derived. {We may contrast the above with what was assumed to occur before the relevance of relativity was appreciated.]
644. We may, suggests Doyle, enquire just how 'physicaly real' (vs appearance of same?) is the increase in mass being considered. Would the object itself be aware of ('feel') a 'real' increasing density (per unit of its size - as an electron say) - if, that is, it could somehow experience such 'awareness' - as its speed increased ? Doyle states that it wouldn't 'feel' this, just as it wouldn't experience (or 'feel') any distance contraction or time dilation for its motion (velocity) under the conditions described for special relativity. These effects are only apparent to (observed or perceived by) 'an external/neutral observer' for the reason that they are relative to (depend upon) (the speed of) the frames of reference concerned (and the consequent lags in receiving the relevant information regarding same). [See later Tao's derivation of E = m.c^{2} which arises by a series of 5 calculations which entail determining how various phenomena 'transform under changes of reference frame'. Such changes and the transformations required are the very basis of the theory and this matter of perception seems again implied .] Thus, a ball thrown into the air by a passenger on a train travelling at 100 mph possesses little or no forward kinetic energy as experienced within the train itself (reference K). A glass mirror place in front of the ball by someone on the moving train and towards which it is given a slight nudge when at its top-most point wouldn't cause the mirror any problem. But if the observer in the station through which the fast train passed (reference frame K') could magically place a hand-held mirror in front of the ball just before its arrival at that top-most point, it would of course be smashed to smithereens since, to him, the ball did possess considerable kinetic energy (and mass). And, to such an observer, "it appears that the faster an object is to move, the more energy is needed to move it and even more to move it even faster (compared to what was needed to move it a given amount at lower speeds); ie to accelerate it. And once so accelerated by applications of such energy (force), it then itself possesses that kinetic energy and, as it would then appears to 'resist further acceleration' (or deceleration!), it may be inferred that its mass must (?appears to) be increasing with its increasing speed - this (mass) and its cohort (inertia) being what resists such acceleration (and deceleration) - by definition - from its initial resting and later moving status.
645. [Note: One would assume that the need for more energy being needed to move a faster moving body than a slower one of equal mass would be a matter of proven experience rather than something reliant upon 'appearance'. This seems to tie in with our earlier reference to the terms 'appears' and 'seems', and also to the term 'apparent' as italicised above. We may return to this possible concern later but then again, it may be but a mole hill. We may still however reasonably enquire what accounts for any such appearance of the need for more energy to further move an already moving body or of that body's increasing mass as a result - ie as opposed to these effects being seen as actual, not just as appearing to be so. On what are such appearances based (determined) as opposed to there being no need for such a basis ? They seem to be based on the perceptual stance of the observer.]
646. In any case, we need to look secondly, says Doyle, at the Energy so 'involved' in very high speed motion. As an object moves faster, its mass increases if measured from a neutral point (as shown above) and thus the greater the energy (?force) required to move it even faster. The fast moving body was assumed (under classical mechanics) to possess kinetic energy (as transferred to it) according to its mass and the velocity at which it is moving - ie which equals:
this often called Newtonian kinetic energy or energy of (a body's) motion. [Review derivation of this relationship/equation: That 'energy' of motion (initially called 'Vis Viva' or 'living force') was considered to be some function of a body's mass and the velocity at which it moved, was first theorised by Liebnitz and Bernoulli in the early 1700s. Later, Gravesande in Holland (ca 1800) provided experimental evidence of this by dropping round weights from different heights into soft clay and found that their penetration depths were indeed proportional to the square of their impact speed. Chatelet in France worked further on these studies and conclude that the correct equation was in fact E = 1/2 m.v^{2}. In 1829, Coriolis added more to our understanding of the mathematics of such 'kinetic energy' (this term first used in about 1850 - by William Thomson). We may note that because the velocity term is squared, there isn't a linear relationship between the energy needed (or acquired) and the velocity concerned. It takes much more energy to move an object (as an airplane) at twice any given speed than simply double the amount for the original speed (rather, its nearer 4 times as much). And having such a greater amount applied to so move it, it will have acquired that much greater a magnitude of energy as well. This contrasts with measures of the similar property of a moving mass - its momentum (p) - which increases only linearly with increase in its unit velocity. Ie p = m.v . [Might the energy of a still mass 'require' the squaring of the velocity of light (as c_{2}) similarly to properly represent that outcome ?]
647. The above equation for kinetic energy is quite suitable for bodies moving at 'everyday' (slow) speeds although in fact it is only approximately correct. But this discrepancy between the results it generates and the exact result is (as in the case of momentum) so miniscule (and probably not measurable) that is of no practical significance. However, we may nevertheless again note that the reason why such a result is not quite accurate is because the value of the velocity (v) concerned, even if quite slow, cannot in effect be taken at its face value since there is an upper limit to possible velocities - as explained in the sections on special relativity. Velocity can never be unlimited (infinite) and thus no body could move from a to b 'instantaneously' (nor indeed at any similar velocities seeking to approach same - at the speed of light (c). Rather, any given velocity is always only a proportion of that upper limit of velocity (c) - which happens also to be the speed of light. Therefore, the value of v (and the constituent elements of same - ie distance and time) should be effectively replaced in all formula in which they appear (if precision is desired) by one calculated in terms of the ratio of v to that upper limit - ie to v/c (within which the variablility of both time and space become subsumed). Where this is a very small fraction (to be subtracted from the implied value of unity in the denominator of the equation relevant to its calculation), the discrepancy from an unaltered value of v is barely measurable. (This is shown by the appropriate (ie non-classical, Galilean) transformation equations.) But, if the value of v is quite large (ie a much larger proportion of the speed of light), we find the ratio concerned to be much closer to unity, the resultant denominator much smaller and hence the greater the values of the quantities (as mass and energy) being calculated. The correct net values are then calculated by means of the improved (Lorentzian) transformations of special relativity.
648. This essential basis of special relativity will thus affect any of the physical laws that were based on some involvement of the variable of velocity (and/or its constituent elements of time and space). Some of these have been described by Einstein in Sections 10 to 14 of his 1920 book (see above). And it applies again here - in respect of the relationship between mass and energy (as touched on in his Section 15). Thus, as shown above, we note that a body's mass m increases with the speed of its motion to equal its non-moving rest mass m_{0} divided by a value of less than 1 (ie by 1 - v/c). Now, if any of the measures in such a calculation happened to be in the negative direction, it is possible to end up with an answer that is in error. Thus, one must subtract the square of the fraction v/c (ie v^{2}/c^{2}) from the square of 1 (1^{2} (which equals 1) and then take the square root of same; this eliminates any negative values. It may be seen that the greater the value of v, the smaller would be the denominator and hence the greater increase in mass m would result when the resting mass m_{0} was thus divided (ie when it subsequently moves at that greater velocity). The effect on Energy increase would be comparable.
649. With Einstein (and Doyle), we may then ask what would be the situation where the velocity v is set to be very slow ? In any such calculation, it is the case that the equation (for kinetic energy and for mass) is actually more complicated than presented here - whatever the actual velocity considered. It is actually made up of a series of components which are additive. Thus while the classical version of kinetic energy E_{k} may well = 1/2m.v^{2} as a good approximation when the value of v is small, this formulation may be more precisely analysed in terms of a binomial expansion of the relativistic formulation - which shows it to be comprised of a series of increasingly smaller components which soon become quite insignificant, while the 1st of which ironically provides nothing with respect to the kinetic aspect of the energy concerned. [The physics (vs mathematics) behind this should be further clarified.] This 1st component (which only emerges by means of the relativistic formualation) in fact represents the value of the (latent) energy of the 'rest mass' of the body - before it has moved and hence didn't (in some unspecified time in the past) enter into the (classical) calculations which sought only a close approximation of a subsequently moving body's now kinetic energy.
650. We attempt to show next how that 1st energy component of a body's resting mass enters into the calculations and what it signifies: For slower velocities, the equation for kinetic energy of classical mechanics provides, as mentioned, a good approximation for determining that value, ie with:
We may usefully note here that just as a moving body's energy (E
A moving body possesses both momentum and kinetic energy therefore - according to its mass and velocity, although the latter (E_{k} increases by the square of that velocity (and thus grows more quickly) and not just linearly as does its momentum. But both of these properties rely on both mass and velocity and, as noted, this measure alone does not provide precise results when treated with classical mechanics and require instead analysis in terms of special relativity. This entails re-calculating the classical formulae and replacing the values for velocity by means of a division by the sq rt of 1 - (v/c)^{2}. I believe this is also represented below (at times) by (1 - v^{2}/c^{2})^{- 1/2}. Before we consider this, we may note that the energy (E_{k}) of a moving body (in classical terms?) is related to its momentum (p) by the equation:
651. Some focus here on the property of momentum is relevant in that it will apparently enter the reasoning regarding the derivation of the value for the energy of a resting mass (our central concern). Thus, the energy required (or the work done) to accelerate a body from rest is a function of the force applied to it (F) times its 'displacement' (distance) in a given time (ie its velocity). When analysed by the relevant calculus (seemingly because any acceleration from a resting position must entail a changing rate of motion), the integration of the momentum differentials so arising (ie S v.dp) can provide a useful measure of kinetic energy E_{k} (with S here representing the integration sign and the double dot (..) signifying the 'dot product') in terms of the momentum (p) - ie:
This equation assumes that such a body has no kinetic energy before it begins to move; ie while it was 'at rest'. In special relativity, the expression for linear momentum (normally p = m.v) must (as with kinetic energy for fast velocities) be changed (to m.v/sq rt 1-(v/c)^{2}. The relationship bewteen the energy and momentum of a moving body so considered would thus differ from that shown above at **. By integrating by parts, we apparently proceed towards this relationship thus:
On the above basis, kinetic energy becomes :
Re-arranging, we find that E_{k} = m (1-v^{2}/c^{2})^{- 1/2})(v^{2} + c^{2} ) + C = m (1-v^{2}/c^{2})^{- 1/2})(v^{2} + c^{2} - v^{2} + C = m (1-v^{2}/c^{2})^{- 1/2}) c^{2} + C
652. We may note that during the course of these 4 (mysterious) steps of calculus reasoning (left out of Doyle's account), the component m.c^{2} has magically appeared and within which, for the first time in our analysis, the enormously large denominator of the fraction v^{2}/c^{2} - ie c^{2} - has somehow become separated from its numerator and is not returned instantly to its physically meaningful, unsquared value (of just c - the upper velocity limit in nature) - by having its square root taken simultaneously - as had always been the case (with regard to c^{2}) to this point in our analyses of relativity. Why is this ? Or, is the source of ^{2} not from that squared ratio at all ? And if not, what other physical understanding has guided this particular mathematical procedure ? For while a mathematical (ie algebraic) manipulation may reveal a physical reality - such as this m.c^{2} entity (that has, we presume, always been 'out there' somewhere as part of physical rality), it can of course, never create such an entity - as a simple mathematical 'by-product' say (one would suppose). We note also that the value of c^{2} on its own then also appears (three times) in the 're-arranged' equation that follows (above). In both cases, one would assume that there must be some physical explanation as to how the physical realities (with their enormous values) so represented actually come about within any step by step physical analysis of the evolving formulae (and, crucially, of the physical realities so represented). It isn't 'just numbers', surely - but actual physics.
653. In any case, when v = 0 (ie when the body concerned is taken to be at rest), then (1-v^{2}/c^{2})^{- 1/2}) will apparently = 1 and E_{k} = 0. The constant of integration may apparently then be found [this needs clarification} so that we can (?then)obtain the formula:
We see that we now have 4 of the 'isolated' c^{2} components (shown in bold) referred to above in this final form of the equation. As mentioned, where the body's velocity is a significant proportion of the speed of light (c), it is necessary for accuracy to use such relativistic mechanics in order to obtain the correct value for kinetic energy. Similarly, we find that the momentum for the body so concerned (which somehow enters into this calculation) becomes under relativistic mechanics:
This can be compared to the formula for kinetic energy (or work expended) when accelerating a body from rest to a very fast (relativistic) speed:
which is the same as the formula so described