Please Note: As mentioneed prior to the section on Speial Relativity, an e-book is availableh covering boith Theories - on Kindle (titled 'On Understanding Relativity', by the present author. Meanwhile, we ontinue as follows.
Introduction
In the theory of Special relativity as discussed in its own Section above, the motion of the moving bodies concerned and/or of their associated reference systems pertinent to that theory were considered always to be smooth, straight and uniform, not jerky, curved or accelerated - which forms of motion are as or more common in nature than uniform ones. Uniform motions and their associated special principle of relativity follows from the concept of inertia which applies in this respect only on such uniformly moving bodies and systems. We may appreciate that when someone on a uniformly moving train holds, say, a gun with a bullet in its chamber ready to be fired (without any criminal intent), that the person, the gun and the bullet are all sharing the same inertia as the train,all moving along smoothly together; there is no 'pressure' forcing any of these elements backwards, say (or in any other direction) as the train moves forward at its smooth and constant velocity (v) of, say, 100 mph. They are all in a sense 'at ease' with one another and 'unaware of' and unaffected by the train's steady motion relative to the outside environment, or 'terra firma'. So, when the bullet is fired down the aempty aisle towards a bull's eye target, purposely placed at the end of the carriage for such an experiment,its journey relative to the train proceeds according to the force arising from the gun powder alone (if we ignore any opposing air pressure) propelling the bullet for its particular mass; the on-going uniform backgroubd speed of the train has no effect on or relevance to the bullet's subsequent speed (w) as measured within the train; this is just as the special principle of relativity, relying on that 'equating' effect of shared inertia, would expect. It effectivley begins and continues its journey under zero acceleration (ie no change of speed) of its immediate environment - whatever that on-going uniform speed, just as it would if the train had een quite stationary (ie possessing only uniform 'stillness', albeit relative stillness.).
When Einstein first conveyed his thinking to us regarding his initial theory, in 1905, it was effectively described as a theory of 'the electrodynamics of moving bodies', not one 'of the relativity' of same per se, let alone of 'special relativity', nor of any other 'kind' of relativity. While the motion of the bodies concerned was stipulated as being necessarily uniform, as this was a requirement of its main postulate of the classical principle of Galileo and of Newton (based on the concept of inertia), this feature was not particularly emphasised at the time.. That the motions concerned might be similarly considered if they were, instead,non-uniform and thereby effectively broaden and generalise the theory, this was not mooted whatsoever in the 1905 paper. There was thus no intimation that the name of the theory then might require some appropriate qualification concerning its barely specified form of motion - that is, to differentiate it from any theory that might, one day, pertain to other forms of motion. There was, it seems, no such 'next theory' - of greater generality - 'waiting in the wings' as it were (at least that was ever mentioned at that time).
By late 1916, however, Einstein had written his book on 'Relativity - the Special and General Theory' - in which he was able to re-consider his approach to this subject area and in a sense rationalise his exposition of it - after having broadened and generalised his conception of it some time earlier. For at some point prior to writing that latter book, he had obviously decided that a more thorough analysis of the physics of moving bodies was required, one which was not restricted to the uniform motion of bodies - including those of electromagnetic waves (as light), but one which could considered all forms of motion of such bodies. At that point, he would likely have first appreciated that his 1905 subject was but 'a special case' - of a more general analysis of the electrodynamics of moving bodies - and hence the title of his initial theory would, only then, be usefully re-cast - not only as a theory of relativity - but one of '>i>special relativity' only - within which the electrodynamics of uniformly moving bodies (including light) would now be effectively implied or understood. It could thus be seen as an initial generalisation or extension of Galileo's and Newton's 'mechanics of uniformly moving bodies' - since it considered the implications of incorporating light (with itsconstant velocity) as one such body - something not included in earlier mechanical theories - albeit still of uniform motion).
Just when Einstein decided that his 1905 theory, even with its adaptation for the requirement of light, was however still only a special case - of an even broader conception of the physics of moving bodies seems uncertain. One couild imagibe thait it was around 1906/07, that head the thought bthat it waas virtually 'intolerable' to restrict his theory to uniform motions only (which idea he wlater exspressed in retrospect). At some point he mu ust have considered how one mighht apply its ideas to some form of non-uniform motion when, it occurred to himthat the laws of motion advanced by Newton, with their classicsal mechanics (which his own relativity theory had complemented), had required a seperate theory to account for the force (and mechanics) of gravity. There was indeed an excellent ezxample of a form of non-uniform motion - one that was ubiqitous Could his new theory not incorporate its characteristis and principles into reelativity theory ?? We can see that his shift in focus to this latter topic shortly after (ca 190709), can be seen as an interest in its non-uniformity per se rather than simply in gravitsation alone, albeit the latter conception and orientation soon overtook the former motibation in the eyes of his contemporaries. A year before writing the above-mentioned book on Relativity (by Dec 1916 - in German) in which the two 'levels' of relativity (special and general), were now clearly so presented and discussed, he had already read a paper (in Dec 1915) to his colleagues on which his new theory at a more general level would soon be published (in 1916 in vol. 49 of Annalen der Physik) - viz: wntitled 'The Foundations of The General Theory of Relativity'. One would assume that the material for this paper, read in Dec 1915 and published in early 1916, was however worked on and written up over the previous two or three year or so at least - ie during 1912-14, say. But, in the published version, he now refers to his theories of both special and general relativity without obvious introduction - as though hiis typical readers were already familiar with their prior description and existence, so named, and thus with the basis and justification of the effective re-naming of the 1905 theory in some later publication - quite prior to 1915. But when and where was this prior introduction to his new conception first published ?
We may usefully note that in 1911, he published an article on 'The Influence of Gravitation on the Propogation of Light' (Annalen der Physik, 35, 1911) in which he refers (in a footnote) to an even earlier 'memoir' on this same subject, written 4 years earlier (in Jahrbuch fur Radioakt. und Elektronik, 4, 1907, p 411.) Do either or both these articles include views that imply the beginnings of such a re-conception and generalisation of his 2905 views and thus the need and relevance of effectively re-naming tht earlier interpretation ? We do not (at this writing) possess the title nor the content of that earlier (19078) memoir but we do have that for the article of 1911. It is quite possible that they may provide the basis of his new, broader orientation and thinking on his subject - and that early. We thus examine next the 1911 article, at least, for such confirmation (or not) of this earlier possibility.
From Einstein's paper On the Influence of Gravitation on the Propagation of Light (1911).
He does indeed begin by pointing out that in 1907, he had already addressed the question of whether the propagation (and direction) of light was influenced by gravitation. [We have indicated just why hemay have done so; it was likely because t is a rather important and ubiquitous source of non-uniformn motions which he hoped coulkd be incorporated withinin his one theory, unlike Newton).] While he was not satisfied by this earlier treatment of the subject, he decided to now (ca 1911) return to the matter primarily because on reviewing the 1905 version (by ca 1908-09, say) he could see a means by which this idea could now be so incoprporated andbe tested experimentally. He wished to indicate this now but do so in conjunction with at least the beginnings of his more comprehensive theory within which this aspect (otherwise existing somewhat 'out in left field') could one day be more logically and fully 'fitted'. While such a 'jump' in the focus of his interests (from the electrodynamics of uniformly moving bodies, albeit including light) to the possible effect of one particular, if important source of non-uniform motion (ie gravity) may appear as something of a 'one-off' - with no general introductory 'lead-in' as to why he was now focusing on this particular broader area, or how it might prove to be but an extension of his existing areas of interest - it would seem that in terms of the analysis of this topic, Einstein would nevertheless gradually bring in many of the elements that would pertain to this more generalised version of his 'relativity theory' (as he was shortly to refer to it).
This would provide a basis for re-naming and re-conceptualising his earlier version of his theory (of moving bodies) and so better emphasise its identification with relativity per se - the greater generality of which he was apparently also now beginning to consider (ca 1908-11). He thus begins by showing that from this new theory (if centered initially on this narrower focus) it will somehow follow that light's path should be deflected (and thus its velocity necessarily varied in speed and direction rather than be soley uniform) by the gravitational field of the Sun. But, as just suggested, he points out that this particular consequence is but one of a complex of predictable outcomes that this incipient theory (of apparently broader inplications) may well account for, although the basis of these would (at present), he says, be too difficult to present and follow; they were apparently still formulating in his mind, but hopefully the present elementary reflections should, notes Einstein, allow the reader to gain 'a better idea of 'my evolving line of thought'.
This certainly appears therefore to foreshadow his general theory of relativity within which he would soon (by 1910-12, say) appreciate that his 1905 theory was but a special case (ie of motion) of a broader conception not previously appreciated. But, as mentioned, this would not be introduced (in the present 1911 paper) in terms of examining, say, what is involved when a more typical moving body - as our thrown ball or fired bullet, say - no longer moves uniformly but does so in, say, a continually accelerated way - relative to some seemingly 'still' or even uniformly-moving frame of reference. (Or, must such a reference system itself now be conceived as also moving non-uniformly (ie relatively) - being as it is generally seen as but an effective reciprocal of the body's motion being considered (with the relevant maths apparently being identical from whichever orientation it is examined)?
These complex considerations are thus left for now and instead we must focus on the effects on the the non-uniform motion not of material bodies only but on that of our most tenuous 'body' - light - as it is seemingly accelerated (vs its otherwise constant, and so uniform, motion) - by means of an applied force which acts to accelerate it rather than allow it to continue to move at its otherwise inherent constant velocity. The 'forces' by which our ball or bullet were kept moving uniformly (once we conveniently ignored their brief initial accelerations) were equally conveniently imagined or assumed. Can we generalise from that assumed situation to the one where the motions of bodies and/or of their reference systems (as the train, the station or the embankment) are considered instead to be non-uniform (as eg accelerated - by some imagined constant force) and do so in terms of any knowledge so gained firstly when considering (just) the case of light's acceleration - by constantly acting gravity ? We'll see.
The reasoning behind the expectation that light's path will indeed be so influenced by gravity (and possibly other bodies' motions comparably) is presented in the 1911 article in four numbered sections as follows:
1. A Hypothesis as to the Physical Nature of the Gravitational Field.
One imagines a stationary coordinate system K (frame of reference) in a gravitational field (eg as a square room set on the Earth say with its vertical axis parallel with the lines of force of that field. It would not itself 'fall' in that field as the Earth is of course too solid but any bodies somehow tenuously suspended within its space may well seek to so fall under that influence. One may thus imagine on a table therein a large bowl of whipped cream, vaseline (petroleum jelly) or a soft French cheese say - on the surtace of which small grains of lead buckshot are carefully placed. They would immediately seek to 'fall' deeper into that viscous material under the influence of the constant 'pull' of gravity. A second coordinate system K' is also imagined to exist somewhere in space not near any large body as the Earth and thus, unlike K, not under the influence of a significant gravitational field. One might consider for example a space vehicle on the surface of the Moon in this regard. If our bowl of 'semi-liquid' is placed on its floor with its buckshot and then the vehicle lifts off vertically with ever increasing speed (ie uniform acceleration conveniently equal in magnitude to that of gravity (g)), we would expect the shot gradually to sink into the bowl's content at the same rate as in the case on the Earth - if we conveniently assumed for a moment that the Moon's gravity was virtually zero. The inherent inertia of such small bodies would in fact be seeking to remain in their initial positions as the ever increasing non-uniform velocity seeks to force them upwards along with the vehicle, the bowl and its contents. They would thus effectively sink into such contents (actually, the contents would slowly rise above them). The motions concerned in either imagined situation can (at this stage) be considered just from the point of view of classical mechanics and kinematics, not (at this point) from, states Einstein, that of 'the theory of relativity'. [Note: this is the first example (ie as of 1911 and possibly 1907), of which I am aware, of the latter phrase and conception being used in print by Einstein.]
The motion (ie velocity) of any such material points (ie our very small bodies) along their respective horizontal axes, say x and y (relative to either system), if not affected by any larger bodies nearby, would amount to zero - in accord with the equations:
whereas that (as the buckshot) for the vertical axis (here taken as dimension z) would be:
With respect to the upwardly accelerated system K', this result follows directly from the principle of relativity (of Galileo and/or Newton) with its inertial basis, but for the stationary system K in its gravitational field, it follows, says Einstein, from our general experience that all bodies in such a field are equally and uniformly accelerated (ie whatever their composition, mass or form) - in this case 'downwards'. [Note: it is of course their acceleration that is uniform, not their velocities.] Einstein then stresses the fact that such experience is so consistent and reliable that this phenomenon is virtually a 'law' and yet, surprisingly, it has not been formally incoporated into "the foundations of our structure (natural laws) of the physical universe". This practical 'law', so taken for granted, may however be further analysed, notes Einstein, and so placed more firmly within our conception of physical reality with its associated relationships, causes and effects - especially if we can appreciate that the systems K and K' as described here, are physically exactly equivalent. For both may be conceived as equally free of any gravitational field if system K is, for the moment, also assumed to be uniformly accelerated at a value of g. As such, we can not describe the acceleration of either system (our frames of reference) as being in any sense 'absolute' (just as we could not when describing the uniform velocity of such systems within the strictures of the theory of relativity). [Again, the latter phrase is used here by Einstein as though we have been privy to it somewhere before.] This equivalence of the two reference systems thus holds in terms of Newton's mechanics. Einstein also notes that in the above conception, we may conclude that all bodies (apparently including light) would indeed be expected to fall equally in a gravitational field (whatever their form and composition) - which has seemingly the very same effects as the accelerations described temprorily for both K and K' when their inertias would now resist any upward thrust so applied.
He nexts points out that the equivalence of these two systems per se in regard to their gravitational- like fields (or conversely not) does not itself have major significance without considering what other physical processes such equivalence (eg of inertial and gravitational masses) may affect or determine. As with the 'ordinary' or 'usual' theory of relativity (Einstein's then descriptive terms for his 1905 theory - before he rationalised this with either of the terms 'relativity' or 'special') which showed that the laws of nature in any uniformly moving frame of reference K (or the bodies so moving in them) apply identically in any other such frames (K', K'', etc), one can address the possibility that all such laws also hold equally in both of our presently considered accelerated (ie non-uniformly moving) systems K and K'. If this can be shown, says Einstein, we may arrive at a principle which should have considerable heuristic value. For consideration of this situation equates to learning more about the way that a gravitational field itself likely affects all processes of nature which may operate within same. [This provides us with a kind of justification of entering this sphere from this one specific 'angle'; he apparently intends eventually to really generalise it after all.] He then approaches this specific analysis in terms of what he refers to as 'the ordinary theory of relativity' (ie not just in terms of classical mechanics as above). As mentioned, by 'ordinary' here, he means from the standpoint of what he would only later conceptualise as the 'special' theory of relativity (or otherwise from that of the core elements of relativity theory per se (as he was then viewing it) - before it was later differentiated into its two 'levels' of generality). This is covered (or led into) next - in section 2.
2. On the Gravitation of Energy.
The earlier (1905) theory of relativity led ultimately to an analysis of the relationship between energy and inertia (as analysed above in paragraphs..... in our account of what was later called special relativity, although it appears to have been submitted in a separate paper to the same journal as his 1905 paper some months after that main submission). It does form a useful bridge into the on-going development of his theories (as here). Thus the (special) theory yielded the result that the inertial mass of a body increased with any increase in energy it contains (see earlier account of this). So, where any such increase in energy amounted to a magnitude E, the increase in inertial mass equated to E/c2. Einstein then asks: "Is there a corresponding increase in gravitational mass (when similarly 'given' more energy)? " Our examples in section 1. above appears to supply a positive answer to this in that we can see that the effect of the gravitational field on the gravitational mass of small bodies in system K is very comparable to the effect on their inertial mass of a constant accelertion in the other (upward) direction where no effective gravitation applies. So we might well expect that the small gravitational masses would indeed increase as any energy increased within them to the exact same extent that occurs in the inertial masses under such energy increases. Otherwise, bodies could fall with varying accelerations in the same gravitational fields according to the energy they contained (which is apparently not found). And moreover, the other satisfying result of the theory of relativity (of 1905) by which the law of the conservation of mass was shown to merge with the law of the conservation of energy (which again may be reviewed) could not be maintained if the above equivalence of their differently conceived forms of mass as affected by energy was not the case. Rather, it would mean that we would have to abandon the law of the conservation of mass in its old form for just inertial masses but maintain it for the so called gravitational masses - and this, says Einstein, may be regarded as very improbable.
On the other hand, we do not yet have any convincing theoretical evidence or argument from the usual (1905) theory of relativity that the weight (mass due to gravity) of a body does for certain depend of the energy it contains, as has been shown with respect to inertial mass. But, says Einstein, we shall show that our hypothesis of the equivalence of systems K and K' in the above regards will lead (more directly) to the conclusion that gravitational mass also reflects the energy it contains which implies that such energy and its mass is reactive to the gravitational field. On this basis, we may expect that even a very small body (as light even) should reflect the possession of such energy and so mass variations. To demonstrate this, we let two infinitely small 'systems' - S1 and S2, each provided with necessary (if equally minute) measuring instruments, be situated on the vertical z axis of a coordinate system K at a significant vertical distance h from each other such that the gravitational potential in S2 (situated the higher) is greater than that in S1 by the amount g.h (as this potential apparently increases with distance (height) from the field's source). [Note: the two 'sub-systems' S (contained within the positional coordinate systems K and/or K' are so described (as 'systems' themselves) in that they can imply more than simply a single body of certain mass therein but a rather more complex 'mini-system' - the energy of which may be distributed (and emitted or absorbed) amongst various mini-constituents (bodies) - probably including light.] We thus let a definite quantity of energy E be emitted from S2 as radiation (seemingly light or heat) towards S1. The levels of energy (in whatever form) in S1
and S2 are measurable on any occasion by appropriate instruments of identical form anywhere along axis z. We do not know what if any influence the gravitational field (in which system K at least is situated) may have on the transfer of such energy nor on the measuring instuments utilised. We may set system K' as described earlier (accelerated up an axis z) in place of system K (which otherwise is in a gravitaional field) as they are in other respects equivalent - as per the earlier postulate described above. The material 'mini-systems' S1 and S2 are rigidly connected to the z axis of coordinate system K.
We can judge the process of the transfer of the energy from S2 to S1 from the point of view of another coordinate system K0 which is free from acceleration. At the moment the energy is emitted from S2 (called here E2) towards S1, let the velocity of K' relative to K0 be zero (ie they are at rest with one another). The radiation energy E2 will arrive at S1 after the time h/c. But at that moment, the velocity of S1 relative to K0 is = g.h/c = v. Therefore, by the 'ordinary' (usual) theory of relativity (as introduced in 1905), the radiation arriving at S1 does not possess just the energy E2, but a greater amount: E1 - which is related to S2 (to a first approximation) by the equation
By our earlier assumption of equivalence, exactly the same relation will hold if the same process takes place in system K in its gravitational field (and so not accelerated upwards). Although in that case, we replace g.h by a potential theta of the gravitational vector in S1, if the arbitrary constant of theta in S1 is equated to zero. We would then have the equation
This equation expresses the 'law of energy' for the process under investigation. The energy E1 arriving at S1 is greater than the energy E2 which was emitted from S2, the excess being the potential energy of the mass E2/c2 in the gravitational field. This proves that the principle of energy of present concern can be established if we ascribe to the energy E, before its emission from S2, a potential energy due to gravity - one which corresponds to (is accounted for by) the gravitational mass E/c2. Our assumption of the equivalence of K and K' thus removes the difficulty mentioned above (see......) which was not solved by the ordinary theory of relativity. [It seemingly required the logic associated with an extension of that theory - namely......]
Einstein then summarises the foregoing in terms of 5 steps in its development which should make clearer the meaning of this outcome. [Note: Step 2 appears to introduce an aspect not mentioned above.]:
1. The energy E, as measured in system S2, is emitted from it in the form of radiation to be absorbed by system S1 in an amount equal to E(1 + g.h/c2, as shown above.
2. A body W in S2 of mass M is lowered from it to S1 which entails work being done equal to M(g.h).
3. The energy E was transferred from S1 to the body W when it is a part of system S1. We let the gravitational mass M be changed thereby so that it acquires the value M'.
4. We let W again be raised to S2, which will again entail work in the amount of M(g.h).
5. We let E be transferred from W back to S2.
In summary - the effect of this cycle is simply that S1 has undergone an increase in energy equal to e(g.h)/c2, which may be seen to be represented by the quantity of energy [M'(g.h) - M(g.h)] conveyed to that system in the form of mechanical work. By the above described principle or 'law' of energy, we thus have the equality:
That is, the increase in gravitational mass is equal to E/c2 and therefore is equal comparably with the increase in inertial mass as given by the (?extended or usual) theory of relativity.
This result emerges still more directly from the equivalence of reference systems K and K' with respect to the former's gravitational mass exactly matching the latter's inertial mass. Energy (including that of light seemingly) must therefore possess a gravitational mass which is equal to its inertial mass. To which term it is referred thus depends on the context - where the mass/energy concerned is either subjected to a gravitational pull or its inertia opposes an applied uniform acceleration. If a mass M0 is suspended on a spring balance in system K, the balance will indicate the apparent weight M0 x g - due to the inertia of M0. If a quantity of energy E is transferred to M0, the spring balance will indicate a weight of (M0 + E/c2)g (by the law of the inertia of energy). Exactly the same thing would happen if the experiment was repeated in system K with its gravitational field (vs the inertia effects due to the acceleration of K') - this too following from the fundamantal assumption of their equivalence in such respects. [It may be salutary to recall here that 'energy' is however not a 'thing' that comes in standard little 'globs' (as Feynman usefully reiterated) so it would not be the (added) energy per se that responded to a gravitional field (or to a comparable acceleration) but the mass of whatever was so increased that would so respond (I believe).]
3. Time and the Velocity of Light in a Gravitational Field.
One has always understood that one of the crucial tests of Einstein's general theory of relativity was eagerly awaited, not least by Einstein himself, when in 1921 Eddington sought to measure any deflection of the light from selected stars as it passed near to the Sun, on its way to the Earth. This prediction was inherent in the general theory which, by 1911, had however yet to be formulated in its essential form. But the previous section of the 1911 paper (being considered here) shows that as well as bodies of given mass being attracted to the source of any gravitational field in which they are located, that various forms of energy (and particlarly that of light radiation) (with their mass implications) should behave similarly. We would presume that this deduction was only carried forward into the logical fabric of the full general theory over the next few years (with the same essential reasoning) and thus provided the basis of Eddington's eventual test of same. The light energy (and its mass) from the stars concerned should thus be deflected by (and towards) the massive gravitational field of the Sun. The deflected path of the light would however be longer than otherwise and hence the unique status of the assumed constancy of its velocity would appear to have relinqished or compromised thereby. The reasoning behind Einstein's conclusions in regard to this, and to the gravitational deflection itself (as of 1911 at least), entailed a most precise analysis of the relationships over time and space of the various elements concerned (as introduced in section 2 above) and which we attempt now, as per section 3, to describe (if only approximately as it includes some calculus steps). [It is also a model of how difficult (and even tedious) it is for Einstein (or anyone else) to verbally describe precisely the complex interactions involved when they necessarily entail various relative subject-object relationships.] Thus:
'If the light radiation emmitted from S2 to S1 in the uniformly accelerated system K' had a vibration frequency v2 relative to a clock in S2, then at its arrival at S1 it would no longer have the frequency v2 relative to an identical clock in S1 but, rather, would have a greater frequency - v1 - which would relate to v2 thus:
This seeming assertion is explained in terns of the Doppler principle (which review). For if we again introduce the unaccelerated reference system K0 which has zero velocity relative to K' at the time when the light is emitted, then S1 has the velocity g.h/c relative to K0 - at the time the light arrives there. By Dopler's principle, the v1 to v2 relation as shown above results and is so accounted for. It also holds true for the stationary reference system K in its gravitaional field, as per our assumption of the equivalence of systems K' and K, assuming the light energy is transfered as described.
It then follows that a ray of light emitted in S2 with a frequency of v2 and a definite gravitational potential, will at its arrival in S1 possess a different frequency v1 when measured by a clock in S1 that is identical with the one by which the frequency was measured at its emission from in S2. For g.h we may substitute the gravitational potential theta of S2, with that of S1 being taken as zero, as so assume that the relation which we deduced for a homogeneous gravitational field also holds for other forms of field (as our acceleration 'field'...or...?). As such, the frequency v1 would then be the following function of v2 :
This result (valid to a 1st approximation as per the above deduction) permits the following predicted application: We let v0 be the vibration frequency of an elementary light generator as measured by a delicate clock situated at that generator. We imagine that they are both located within our system S2 which is situated on the surface of a benign Sun (ie not too hot). Of the light there emitted, a portion reaches the Earth where system S1 is located and where we measure that light's frequency by a clock identical with the delicate one mentioned. Then, by equation (2a) we will find that
where theta is the (negative) difference of gravitational potential between the surfaces of the Sun and Earth. Thus, according to the view expoused here, the spectral lines of sunlight must be displaced towards the red end when compared to the corresponding lines from terrestrial light and do so by the relative amount :
[I'm uncertain whether this is 2 x 10-6 or 2.1-6] In any case, Einstein points out that it was difficlut (in 1911) to confirm if this deduction regarding the influence of the gravitational potential concerned on such a red shirt was an accurate estimate as the confounding influences of pressure and temperature were still difficult to fully control for.
He then comments that equations (2) and (2a) as derived by the above line of resoning may each appear, on superficial consideration, to assert an absurdity. For if there is constant transmission of light from S2 to S1, how can any other number of periods per second (wave frequencies) arrive at S1 than are emitted from S2 ? But this may be easily resolved: For we cannot regard v2 or respectively, v1, simply as frequencies since we have not yet determined the time in system K. What v2 denotes is the frequency (number of periods per second) with reference to the clock of S2 (with its particular time units), while v1 denotes the number of such periods with reference to an otherwise identical clock in S1. But nothing compels us to assume that such identical clocks influenced by different gravitational potentials necessarily go at the same rate. [Equally, nothing compels us not to assume that they do (on past experience of timing events generallly). We must surely therefore have some hypothesis or 'inkling' that possibly they may not go at the same rate.]
Seemingly Einstein did in fact have such am 'inkling' for he then says: 'On the contrary, we must define the time in K such that the number of waves between S1 and S2 is independent of the (?idea of an) 'absolute' value of time. For the process under observation is by nature a stationary one. If we did not satisfy this condition, we would arrive at a definition of time the application of which would allow time to "merge explicitly into the laws of nature". [This latter quoted phrase by Einstein needs better analysis on my part; what exactly does it mean ?] And such a 'merger' (whatevr it is) "would certainly be unnatural and impractical", says Einstein. Therefore, the two clocks in S1 and S2 do not (?can not) both give the correct 'time'. If we measure time in S1 with the clock in S2, then we must measure time there with a clock which goes 1 + theta/c2 times more slowly than that clock - when compared with one at one and the same place. For when measured by such a clock, the frequency of the light at its emission at S2 is:
and is therefore, by (2a), equal to the frequency v1 of the same ray of light on its arrival at S1. [Do we need to better differentiate the different clocks?]
This result, says Einstein, has a consequence of fundamental importance for 'our' theory. [One is curious to know to which theory exactly he is referring here.] For, if we measure the velocity of light at different places in an accelerated but gravitation-free system K', employing clocks of identical constitution, we will obtain the same value c at all such places. By our assumption of equivalence, we should find the same result for system K as well. However, if the latter exists at a location with a differing gravitation potential, this will only hold true if we use a clock of a different constitution than as used for K'above. Where that potential equals theta, we must use a clock that goes (1 + theta/c2) times slower than the clock used at the coordinate origin with its zero potential. If the velocity of light at the origin is taken to be c0, then its velocity (c) where the potential is theta - will be:
Thus, to quote Einstein: "The principle of the constancy of the velocity of light holds good according to this [as yet unnamed] theory in a different form from that which usually underlies the 'ordinary' theory of relativity".
[One wonders how 'principles' per se could ever adopt different 'forms' and indeed what might be the range of such possible forms ?? In any case, the foregoing 1911 paper on light and gravitation, with its references to the relationship between inertia and energy (as given in his later 1905 paper), seems to have provided Einstein with a specific entry point into his newly evolving general theory around which the latter theory would eventually be fully constructed. As such, this new theory would seem to assume an orientation centred on that particular source of non-uniform motion rather than develop from more general principles that simply extended the limitations of the uniform motion to which his earlier theory was restriced. Such an unfocused, general extension into analysing the effects of non-uniform motion per se (ie generally) would presumably lead on (if only eventually) to an appreciation of the implications of the gravitational field - as one such source of that more general motion. Instead, it appeares to have dominated that theory's evolution along that initial orientation. What, we may wonder, motivated that narrower focus, from the start ?]
We continue with the final section of the present paper:
4. Bending of Light-Rays in the Gravitational Field.
From the above conclusion - that the velocity of light may be a function of the gravitational potential of the place where it is being propagated and measured - we may infer, says Einstein, that by Huygens principle (which review) light rays propagated across a gravitational field will undergo a defection - ie towards the source of that field. We have seen above (and elsewhere) that as light was an electromagnetic wave, it wouldn't have been surprising to learn that if it passed a large body which generated a strong magnetic field, its rays may well be deflected (one way or the other) by virtue of same. But a graviational field, on the other hand, would be expected to attract and/or deflect the paths only of bodies possessing mass. If the Sun so deflects light rays, we would assume that light must therefore possess mass - as has been established in theory above. The arguments to support this conclusion would thus be expected to revolve around evidence or reasoning which supported that conclusion. What de we find ?
Einstein begins (or continues) his line of reasoning in this regard thus: We let E be a wave front of a plane light-wave at the time t, and let points P1 and P2 be in that plane a unit distance apart. If shown graphically, the two separated points thus appear on a horizotal line on the plane of the paper. This plane is chosen "so that the differential coefficient of theta, taken in the direction of the normal to the plane, vanishes, and therefore also that of c". [Sadly, I have little idea what that latter sentence means. One wonders (1) what are the options for choosing such 'planes', (2) what is the differential coefficient of anything, including theta, (3) what is the direction of the normal to the plane (and what other directions are possible), (4) is it implied that the velocity of light c also has such a coefficient and (5) why would the coefficient(s) of both thus disappear ?
The remaining reasoning, in algebraic form, based on this uncertain platform is thus rather difficult to follow. We may nevertheless try to get some isea of what is indicates thereby. It thus appears that after time t + dt, the corresponding wave front of the light (across the plane of the paper) is determined by utilising circles drawn around the points P1 and P2 with respective radii c1dt and c2dt where c1 and c2 denote the velocity of light at those two succeeding points. The radius of the circle around P1 is greater than that of P2. By drawing the tangent which joins these two circles we find that it is not parallel with the original horizontal line of the light ray E but makes a slight angle to it which it would eventually meet. The angle of the path so deflected cdt is thus:
This assumes that the angle is positive when the ray is bent towards the Sun (denoted here by an ingreasing value of n' (which appears to be the converse of gravitational potential (theta) which increases the further away any body is from the source of the field concerned). [Note: the underlined values represent the true differentials of these parameters; cf such as dt.] From these considerations, it is possible to calculate the angle of the deflection according to the strength of the gravitational field. Again, these calculations can apply with equal validity to any uniformly accelerated system K', as the fully latter equate to such a gravitational field. The final equation representing such deflection (a) of a ray of light passing a large body as the Sun amounts to :
where k = the constant of gravity (due to the Sun), M = the mass of the Sun (in this example), and delta the distance of the ray from the centre of that large body. This yields a predicted deflection of 0.83 seconds of arc so that the star from which such light may derive would appear to be located at an angular distance from the centre of the Sun that much greater than it would otherwise be. To test this conclusion (and the important hypotheses on which it is based), it would be necessary to measure such an angle during a total eclipse of the Sun when any starlight passing near to the Sun, for example, would be visible and not be 'outshone' by it. [This was eventually accomplished (after equipment improvements transpired) by Eddington et al in 1919 and Einstein's prediction so verified. This verification provided support not primarily for his 1905 theory, I believe (although the reasoning behind it seems to have utilised considerable conclusions from same) but more for the later general theory as it gradually evolved in that expanded sphere (of gravitational effects). We have not seen reference to the mass or energy of the light ray as factors in this behaviour of light but then we understand that any body possessing mass (and/or the energy that contributes to same) should so behave and the 1919 evidence strongly supports this.]
On Developing a General Theory of Relativity from a Broader Perspective (than that of just Light being influenced by one particular force generating Non-Uniform motion).
[We return here to our earlier discussion]: In the sections on Special theory, we had conveniently ignored the prior period of acceleration of the train as it progressed from its 'standing start' of zero mph and gradually accelerated up to its steady 100 mph. The even briefer periods of acceleration of the thrown ball or the fired bullet were also purposely overlooked. During these initial periods of non-uniform motion, the expectations predicted under the application of the special theory would not have been seen or fully realised. Thus, a ball thrown straight upwards by a seated passenger as the train accelerated would, on its return journey, tend to fall behind the thrower and, if he stood up, instead of standing comfortably ('at ease'), he would experience a force 'pulling' him off balance backwards. His own inertia, and that of the ball, would now prevent the laws of mechanics applying as they normally would if the train was still or moving at a steady, uniform rate - as per the principle of relativity discussed previously. The extent of these disturbing effects on moving bodies would likely depend on the rate of acceleration - which could itself be quite steady or it could vary one way or another. However, just as in the case of the uniform velocity of the train once it achieves its smooth 100 mph, the effects of its earlier accelerating speed on such would still require some agreed reference frame by which they too could be precisely analysed and measured. One would assume that in our present examples, this necessary reference would again be such as the railway carriage, the station and/or the embankment, and that one or other would also be so utilised when measuring the converse effects during any decelerating phases - when it eventually comes gradually to a 'stop' somewhere. For the bullet or the ball, these brief acceleration phases were easily overcome by the initial forces applied and were essentially ignored. They would contrast with any situation where (and if) such accelerations continued indefinitely.
But how would one calculate the extent of any such body's on-going motion precisely - as its velocity and/or that of its reference frame similarly changes (reciprocally) moment by moment ? What are the determining forces at work and the resulting effects in such situations ? How do the different variables involved affect each other ? What are their inter-relationships ? Einstein and others argued that it was not very satisfactory to not understand the comparable principles presumably at work in such non-uniform (changing) forms of motion, as these can be just as common in nature as those acting within the rather special restictions of uniform motion. A more general principle of relativity of such non-uniformly moving bodies and/or their reference systems would seem to be applicable here - one that might provide a foundation for a more 'general theory of relativity' say - to apply to all such forms of motion - both uniform and accelerating (as due similarly to gravitation and rotation). And, as the core topic in this aspect of science is Motion, one must consider the possiblity that such investigations, as in the case of the special theory, may well reveal yet other unrealised truths and laws about the basic elements of motion (and velocity) - namely, about Time and Space, those fundamental attributes of nature, as well as about Light and Gravity - equally fundamental features of nature (but surely more than just the latter two).
We may thus consider how the 'firing' of bullets or light pulses may be affected within these same (accelerating) situations. Thus, if we imagine that both the observer on the moving train and the one in the station have, to assist them, a clock and ruler placed along the far wall of the train (effectively representing the time t and the x axis of space), then the motion of such missiles could, presumably, be measured against such a 'backdrop' - whether uniform or accelerated. (We give the observers and their measuring equipment miraculous visual acuity so that they can measure such changing motions accurately without difficulty.) Equally, one could measure the accelerating motion of the train itself relative to the station or embankment if they had such measuring aids available (and, again, their reciprocals).
If the motion/velocity of the moving systems concerned are not uniform but accelerating, one wonders what is 'experienced' - by a fired bullet or released/propelled light pulse for example ? Is there any sense of backward or forward 'pressure' compromising their speeds relative to the now varying speed of its moving reference M (as viewed by an equally variably moving or a stationery observer) ? If light (with its energy) has any mass and inertia itself, despite being so tenuous, one would assume that this must be the case. This may or may not be addressed below as we consider Einstein's important 'general theory of relativity' further. How might we arrange for such as the bullet or the light pulse to move in a continuously accelerating manner and/or their frames of reference ? We begin by re-considering the principle of relativity per se - as it applied to the special theory - and how it may have to adapt as a comparable basis for a more general approach to relativity. As in our earlier analysis, we will be guided in part by following the numbered Sections of Einstein's book on 'Relativity: the Special and General Theory' which, we may recall, was written for the non-mathematician in late 1916 and likely first published (in German) in 1917, and in English in 1920 (and generally referred to as 'the 1920 book').
Sections 1 to 18 were covered in our earlier account on Special Relativity. Section 17 - on Minkowski's new conception of the space-time continuum - has relevance in the present account on General relativity (as well as in the Special theory) and it and Section 18, also already touched on at the end of the coverage of the special theory, can also serve here as a convenient bridge into this Introductory account of the General Theory (for the non-mathematician). [We may also draw at times on the less mathematical portions of his original paper of 1915 - to augment certain points - after discussing its more general introductory remarks. We have, in addition, already analysed the 1905 paper on Energy and Inertia and the 1911 one on Light and Gravitation both of which would appear to be germane to our on-going snalysis of the General theory and may be quoted or discussed at times.] Thus:
Section 17 - Minkowski's 4 dimensional Space-Time.
[To be added]
Section 18 - The Principles of Relativity - Special and General.
In the theory of special relativity, the motions of concern (eg of the various 'missiles' discussed or of the moving train per se) were measured against a necessary backdrop of an agreed body of reference (or reference system). This was typically the uniformly moving train with the missiles' motion measured from both the point of view of an observer travelling in the Moving train (M) and of that of an observer in the unmoving Station (S) or on the embankment. It was pointed out that the two reference systems M and S could in fact adopt the same stance as the other such that the train can be considered to be in motion relative to the station (or to the embankment beside the train line) while the station or embankment can themselves be viewed as (effectively) being in motion relative to the train. In such a conception, the two bodies of reference (M and S) may be treated as simply bodies per se which, respectively, are in motion relative to the other (agreed) body. The motions of smaller bodies (as the missiles described above) can with equal validilty be considered relative to either of our equivalent reference bodies, or indeed, they with any of them. In some of the subsequent discussions, the smaller moving bodies may be dispensed with and the two other, reciprocal bodies be considered as either the moving body or the reference body according to context - as they are equivalent as far as the mathematics and physics are concerned for the important purpose of deriving and concluding valid laws of nature. While from the rather local, parochial point of view this may appear rather unrealistic, one should be aware that the station and embankment, being fixed to the surface of the Earth, are, with it, moving indeed - at a very great velocity - around the Sun such that the small additional velocity of the train from that point of view is hardly significant and may in fact even be in the negative direction. It is the relative difference in velocities between them that is relevant for the purposes pursued here.
The principle of relativity asserts that in addition to the motion of any moving body being validly measured relative to some such agreed reference body - ie to either the apparently moving one or to the apparently stationary one - that the laws of physics pertaining to such motion (of any body) thus have exactly the same form in both cases. As we have shown earlier, the (uniform) motion of the reference system (body) has no essential effect on the laws that pertain (determine) the motion of bodies moving relative to either reference or to each other. The laws so concluded have exactly the same form no matter which body is chosen or specified as which. It is, as mentioned, the 'relative motion' per se that is important. This conclusion does not arise by theoretical considerations alone, notes Einstein, but on the basis of our broadest empirical observations (ie 'experience') - first described by Galileo. Such bodies, whether of reference or not, must however (to have equal validity in confirming laws of motion within the orbit of the special theory) only move uniformly and without rotation in relation to each other. This restriction characterises the special principle of relativity on which, in part, the special theory of relativity is based. The laws that have established and confirmed on this basis are those that operate validly only in the special case of uniformly moving bodies and situations with their shared inertias.
[The foregoing section from the 1920 book on the two principles continues below but first we insert here the essence of Section 1 of Einstein's paper 'The Foundation of the General Theory of Relativity' (Annalen der Physik, 49, 1916) as it covers similar points but from a different (and slightly earlier) perspective - beginning with the special principle. Further sections from same (the 1916 paper) may also appear, as and where appropriate, to augment and amplify those written later for the layman from the 1920 version. Material from the later 1905 and the 1911 papers may also prove relevant.]:
1. Observations on the Special Theory of Relativity. (from the 1916 paper)
Einstein's Foundation paper on General Relativity published in 1916 was apparently based on a paper he had recently read on this subject in Dec 1915. He begins by pointing out that the 'theory of Special Relativity' (a term first used (by Einstein?) about 1912 seemingly; but where?) was based in part on the principle of relativity, a principle which in its essence at least also applied validly to classical mechanics. But, when combined with the law of the constancy of the velocity of light, that latter principle was necessarily modified and no longer applied validly to the classical mechanics of moving bodies - as described in the 1905 paper. Rather, it applied only to the latter subject, now as the electrodynamics of such bodies, in terms of his new theory of moving bodies (later termed as the theory of special relativity - a conception and terminology that was apparently introduced to the scientific literature at some unknown point around 1910-12 as mentioned. The 'special' nature of the principle (and of the theory on which it was partly based) concerns the restriction that the motion of the bodies involved must be uniform for the theory to hold; it does not extend to cases of the non-uniform motion of such bodies. The principle so restricted stated that physical laws which hold good in a system of coordinates K so chosen that they can take their simplest form (including that of light's constancy), will also hold good in relation to any other system of coordinates K' which moves (at whatever speed) in uniform translation relative to K. The equivalence of K and K' in this regard does not therefore extend to where system K' does not move (whether faster or slower) uniformly, relative to K. Because of the law of light's speed, the resultant theory fulfills this definition by virtue of the consequent relativity of simultaneity (and thus of time and space, so discovered), by means of the new (Lorentzian) transformations (which may be reviewed if required).
This modification of our understanding of the conceptions of time and space revealed by the special theory of relativity would have far reaching consequences. However, says Einstein, one important aspect appears to have remained unaffected (seemingly when that new theory should have addressed and affected it (somehow). This concerns those laws of geometry and of kinematics which are generally still interpreted as "describing the relations between measuring rods and clocks" (used to measure distance and time) in the same way as they were interpreted under classical mechanics. Thus a stationary rod of selected length is accepted as always measuring a distance of definite length of any body independently of its location, orientation and or the time concerned. Equally, two selected successive postions of the hand of a clock at rest are similarly accepted as measuring an interval of time of definite length that has transpired between such regular progression of such hands, independent of where and when this takes place. This was accepted to be the case, whether under classical mechanics or even under the new special theory of relativity - the law of the constancy of light's speed not being relevant in this regard where one is perceiving such from within a single, stationary frame. However, he will apparently show us subsequently that under the General theory of relativity (a phrase and conception to which, again, he seems to assume that the reader has, sometime before 1915, already been introduced), this simple interpretation of the laws by which space and time have typically been measured (with rods and clocks) independently (for just stationary reference frames) is not the case. While we may have reasonably assumed that the basis of the oversight (of this 'aspect' which should have been 'affected' as much as other aspects of comparable relevance) would be revealed, explained and, by means of the general theory of relativity (admittedly yet to be presented) corrected - in the next section of his paper. However, he does address an issue which may well be this same one on "an inherent epistemological defect" that is found in both classical mechanics and in the special theory of relativity, one that was first clearly pointed out by Ernst Mach (in...??). Thus:
2. The Need for an Extension of the Postulate/Principle of Relativity (ie from the Special to the General).
In both classical mechanics and in the Special theory of relativity, there was, notes Einstein, an 'epistemological defect' (possibly that aspect that had apparently remained 'unaffected' by the dictates of the principle of relativity); Einstein's choice of terms here is rather ambiguous), a defect that, in any case, was first pointed out clearly by Ernst Mach. [We may estimate that Einstein first began considering such matters around 1912 - after publishing his 1911 paper on Light and Gravitation.] [Conts...]
Section 18 - The Special and General Principles of Relativity (Conts...)
We return now to Section 18 of the 1920 book where these matters have been presented (in hindsight) in a slightly different guise for the lay reader]: 'The special principle of relativity was thus not valid for situations in which the motions of the bodies and/or references concerned were not uniform. We may ask therefore: 'what laws of motion (and nature) may be established and confirmed in terms of accelerated motion or, even better, in terms of any and all forms of motion - whether uniform or otherwise' ? A more general principle of relativity would (as suggested above) be required presumably - if we wish to find that all bodies of reference (K, K', K'', etc) - whatever may be their various states of motion - are equivalent when establishing/confirming all valid laws of motion. [Seemingly, this focus on 'bodies of reference' includes 'moving bodies' per se as well.] We would thus say that not only does uniform motion have no disturbing effects on the laws concerned but that the laws associated with all forms of motion may be equally predicted. But establishing such a general principle - on which a more general theory of relativity may be soundly built - requires considerable analysis of the complex physical interactions and relationships involved for any and all forms of motion. Special relativity relied upon the fundamental role of Galileo's inertia operating within and on uniformly moving bodies only - measured against relevant bodies of reference (and/or vice versa). On what might general relativity so rely - where the motion of the bodies concerned may be non-uniform - as, for example, on a train which suddenly accelerates and/or brakes, or indeed any moving body becomes affected by comparable accelerations - as, for example, of a rotational or a gravitational nature ?
When travelling on a smoothly moving train, one may barely be aware of its motion such that if it passed a stationary train, the latter may appear as though it was the one moving (in the opposite direction) while you were apparently relatively stationary. And such an interpretation is, according to Einstein, quite justifiable from a physical or mathementical point of view in terms of arriving at conclusions about the special principle of relativity and the laws so derived. The laws of mechanics and of light can be interpreted with equal validity in regard to either interpretation - as that could actually be the case. But, if the motion of the carriage is suddenly altered from uniform to a rapid deceleration by the application of brakes, the behaviour of any unfixed bodies in the carriage (as juggled balls, tea being poured or the passenger being unaware of the train's motion) no longer obeys the same laws that were assumed for the conditions of smooth, uniform motion. The effects/outcomes for our thrown ball, fired bullet or transmitted light would presumably also be different. Different laws would now apply and manifest themselves. The moving inertia of the unfixed bodies (provided by the formerly moving train) remains but is no longer shared with that of the train itself which alone has been opposed and altered (by the brakes). Free bodies would seek to continue moving forward due not only to any forces propelling them but to their various respective inertias as well. Conversely, such bodies in an accelerating train (before it achieves its steady 100 mph) would tend (seek) to continue staying 'still' or at least be less responsive to the forces propelling them forward. And without such forces, unattached bodies would appear to move backwards (if free to do so) relative to the train as it continues to accelerate forward (whether steadily or jerkily)
Galileo's law of relativity no longer holds with respect to such a non-uniformly moving bodies - whether seen as moving bodies or the reciprocal reference systems/bodies. The latter might then appear to possess "..a kind of absolute physical reality..", notes Einstein - as opposed to the relative motions by which any principle of relativity, whether special or general, should validly determine the motions of all moving bodies. But according to Einstein, this conclusion does not hold when we analyse matters further. There is (as concluded previously) no absolute motion; all laws of nature can be formulated to prove consistent only within a conception of relative motion (whether uniform or not) and not one based on any assumed 'absolutivity' - where no reference system is necessary. But the reference body to which any other body's motion is necessarily referred (ie is necessarily relative to) can itself be either uniform or non-uniform (or be perceived as such). If the train is still in its accelerating phase, is the station or embankment deemed to be uniformly 'still' (relatively) or, by being effectively a reciprocal of the non-uniformly-moving train (at that point), is its motional status now best conceived itself also to be (effectively and reciprocally) 'non-uniform' - as in the opopste direction ?
On the basis of the descriptions given above, we would presumably conclude, again, that the laws of motion pertaining to such situations should be equally and validly derivable from either (reciprocal/symmetrical) points of view. One would hope to derive thereby general laws which explain (and associated equations which predict) the behaviour of bodies moved by all degrees of accelerating force (including that of gravitation), from maximal to minimal, then through those of zero acceleration (and thus uniform motion) - to all similar degrees of deceleration; that is, as a single generality through a 'continuum' of such accelerations/decelerations, including none. The resultant equations would presumably include a term which could accommodate the effects of variations in the magnitudes of such accelerations - with that calculated for gravity (somewhere in the middle) being able to be generalised to all other magnitudes (including zero) - that is, to handle the outcomes for both non-uniform and uniform motions of whatever kind as they may affect all relevant processes of nature concerning moving bodies. We may note here that in our former considerations of special relativity we were not particularly concerned with the sources of the forces/power/energy which actually caused the bodies concerned to move. We seem to be more concerned in our present focus.]
While the physical representations of the non-uniformly moving bodies concerned (especially in the role of reference body) may prove awkward to appreciate, it is by means of the appropriate mathematics arising therefrom which should allow the derivation or discovery of any new laws of motion which apply in this (and all) sphere(s) being, ideally, applicable generally - to both unifrom and non-uniform motion. And would any general principle of relativity so concluded still have to prove compatible with the constancy of the speed of light ? It was, after all, this demand which required modification of Galileo's original principle of relativity and so forced recognition of the apparent variability of time and space. Does this stll apply when seeking a further generalisation of this principle and its associated theory ? While laws may, in theory, be formluated for reliable application in quite jerky non-uniform motion (where it is alternately accelerated and decelerated, say, in no apparent pattern), a much quicker and fruitful analysis of the relevant laws of non-uniform (and uniform) motion would probably be expected for motions which are more consistently (ie uniformly) accelerated (or decelerated), for example, so that while the velocity of any body being considered would indeed not be uniform, the acceleration (rate of change) of the velocity concerned at least would be so. Thus we have seen above that Einstein proceeded in developing his more general principle of relativity (and subsequent theory based in part on that postulate) in terms primarily of the more 'consistently accelerated motion due to gravity'. He thus begins (in his 1920 book) by describing gravity and in particular the role of the gravitational field when seeking to extend his theory (and derivation of pertinent general laws) from the restrictions of consistently uniform motion alone to the more general one that can incorporate in particular such consistently accelerated motion.
Presumably, this would be a more efficient way to establish relevant laws of non-uniform motion generally if they were derived in the first instance by way of analysing those more consistently accelerated forms of motion (as with gravity) than by any irregular forms - with varying rates of acceleration or of motion generally. Moreover, this particular form of non-uniform motion occupies a very basic place in the behaviour of the universe and of motion in general and it may be quite understandable therefore for Einstein (or any physicist) to focus on this particular form primarily as an important topic in its own right and not just (or even at all) as one, albeit efficient, approach to generalising his relativity theory to incorporate (any and all) non-uniform (and uniform) motions generally. Besides, the gravitational form of accelerated, non-uniform motion conveniently lends itself (via the solar system) to certain reliable means of testing the theory, in contrast, one would imagine, to other, more irregular or less predictable forms. [Note: We must mention at this point that in 1907 and 1911, Einstein appears to have anticipated this focus on the relevance of gravitation to his basic concerns when he published two articles on same (as now described earlier). To what extent, if any, these anticipated his intention to seek a generalisation of his special theory, or even if he had then yet characterised his 1905 theory of moving bodies by this term (with the implication that a general theory was thus conceivable and almost inevitable), is still to be determined (here).]
[But before we consider mainly this rather specific form of (gravitational) acceleration and the theory which would apparently develop in conjunction with (or out of) same, we may note that for any form of acceleration (as our train building up to its steady 100 mph), equations will presumably be concluded (from that focus on the gravitational form) that will also account for the effects of such gradually reducing (or increasing) magnitudes of acceleration at any given moment until (in the reducing case) it just becomes nil and the 'steady-state' (of uniform motion) is, at that point, achieved. One would expect that the equation at that point would be but a special case that was a direct continuity of that for more general accelerations of different prior degrees of such non-uniformity (of velocity). One would then have a general principle that applied to any form of motion. Do we achieve this ? And more (as with respect to space-time ? We shall see.]
Section 19 - The Gravitational Field.
If an apple falls from a tree (as famously observed by Newton (allegedly) or we drop a stone from a shoulder high position, each drops to the ground. When we ask ourselves why they do this, and not move in any other direction instead or simply remain floating where they are, the answer is usually along the lines that they are 'attracted downwards', towards the ground, by the action of gravity arising from the much larger body of the Earth. But modern physics, notes Einstein, finds it more useful to note that this may imply a kind of direct action (at a distance) of the Earth acting on the apple or stone (or equally of the Sun acting directly on the Earth). But such direct actions have been reasonably concluded not to be possible - as based, for example, on considerations of how electromagnetic forces operate. Thus, when a magnet attracts a small piece of iron over a distance of, say, 5 inches, even within a vacuum, we do not regard this as implying that the magnet somehow acts directly on the iron across an intermediate empty space. Rather, we are persuaded - after the manner of Faraday - that the magnet must somehow 'call into being' (ie engender) something physically real (ie a 'force') in the space immediately around it which is conveniently called its 'magnetic field'. It is then this field which, in its turn, operates directly on ('influences') the piece of iron which is thereby somehow induced to move towards the magnet - by this indirect means (in reference to the magnet at least).
Thus, the piece of iron does not have the magnet in its 'sights' (as it were) - as its ultimate goal - but rather is only influenced moment by moment and micron by micron by the immediate and more directly-acting magnetic field/force, the strength of which increases nearer the magnet and so determines the direction and speed that the iron, quite ignorant of its ultimate 'goal', moves. It has been suggested that the construct of the field in such cases is somewhat arbitrary in that it (and other apparently indirectly acting phenomena) were formerly represented by a very similar, almost magically-operating, construct called 'the ether', although it had generally claimed relevance over much broader spheres of phenomena. Einstein acknowledges this but notes that with the aid of more specifically focused 'field' constructs, such phenomena as described as electromagnetic (or gravitational) can be represented theoretically and mathematically much more satisfactorily than without it - as in the transmission of light waves, for example, and within Maxwell's famous equations. And more importantly in our present discussion, the effects of a gravitational field may be usefully regarded in an analogous, specific manner to that of the magnetic field.
Thus, the attractive action of the Earth on a stone also acts indirectly in this same sense. For, like the magnet, the Earth too can be assumed to produce a (kind of) field in its surroundings - namely, the 'gravitational field'. And, again, it is that field, not the Earth itself, that is seen as acting directly on the stone (or any other body (mass)) and so draws/pulls it towards the Earth - and does so with steadily increasing velocity (ie acceleration) the longer it acts and nearer the Earth it becomes. That is, the effect of the field's attractive strength increases nearer the Earth, and likewise decreases with distance from the Earth; it is not the Earth's 'strength' itself that so alters, although its mass (and that of any other very large body, as the Sun), does directly affect their respective field's strengths around them. It's rather like a division of labour; each would seem to have its sequential role. But what accounts for any field's 'strength' ? What is the actual mechanism at work here ? We may recall that a body continues to move at a uniform speed after given a single impulse of force (as per Galileo's law) and doesn't require any on-going continuation of that applied force so to do. If the force was so continued, the consequent velocity (of any body) would be accelerated (ie increasingly). One may assume therefore that the acceleration due to the gravitational field is the result of such an on-going steady force arising therein/thereby, the constant (unincreasing) magnitude of which is determined by the mass of the body that engenders that particular unchanging field (force) strength.
While the question to be answered by the special theory of relativity concerned the anomaly of a limit to possible velocity (via the 'law' of light) was explained by 'discovering' that time and space must be perceived as variable, there was no attention paid in particular to how the 'strength' of the source of the uniform motion concerned came about. That is, to how the train's engine, for example, produced just the right amount of power (force) to first accelerate the train to a steady 100 mph and then maintain that uniform motion with just the exact amount of same to continuously offset its mass and the opposing and on-going track friction and air pressure, etc. But, by focusing on just a particular form of non-uniform motion (due to gravity), the prime concern of the general theory of relativity seems to centre more (or wholely?) on such questions as how to account for the strength of the force responsible for the consistent acceleration (not just velocity) of this concern - (as well as on how our conceptions of space and time, for example, or the constancy of light's velocity, may also be so affected). What, we may reasonably ask, is the specific question which the theory of general relativity was formulated to answer ?? [We may recall that a theory is a suggested answer to a question.]
It would appear to have been formulated in order to answer the same question as the special theory was intended to do - that is, to account for certain anomalies and inaccutacies in the measurements of uniformly moving bodies (as advanced by Galileo and Newton) by finding a way by which the principle of relativity could incorporate the limitations imposed by the motion of one particular body, namely light, but to do so now under even more general conditions: that of both uniform and non-uniform motion. By adapting to the demands of the constancy of light in the former case, it was discoverd that the elements of the velocity (of such moving bodies) - time and space - were found, somewhat surprisingly, to be variable. By adjusting, in addition, to the demands of non-uniform motion, it may be found that some other unrealised aspect of nature is the case. Such motion as provoked by the 'force' of gravity and its consequent accelerating effects can apparently be most effectively utilised in this quest. Does, for example, the all-pervasive (universal) influence of gravitation (as the particlular form of non-unifrom motion chosen by which to investigate this matter) make apparent (or require recognition of) significant effects on our understanding of nature that might otherwise remain unrecognised ? And would these effects be manifested if other forms of non-uniform motion were utilised instead and if not, are we really only investigating this albeit important universal 'force' and its consequences rather than those of non-uniform motion per se (our ostensible initial concern) ?
The increase and decrease in the intensity of gravitational attraction (acceleration) of a body to the Earth the nearer and farther, respectively, it is from the Earth has been calculated from various measures taken in the past. It was believed to obey a quite definite law, called the inverse square law, which operates in terms of consistent properties of the gravitational field itself, not those of the Earth - even though it is the latter which firstly 'engenders' this subsequently independent field. Unlike the electric and magnetic fields, which also operate independently of the specific materials which first engender them, the gravitational field operates equally on bodies of any kind - be it metal, wood or even (we might anticipate) light waves or particles. They are all attracted (eg 'fall') with an acceleration which does not depend on their material or physical nature but only on their mass, while a magnetic field, for example, only operates on certain metals as iron, cobalt and nickel (*) which possess appropriately 'responsive' electromagnetic characteristics (ie 'responsive' to the magnetic field).
[(*) Possibly the responsive materials have themselves much weaker magnetic 'fields' which become attracted by the stronger field of the magnet - specially if both such fields are comprised of oppositely rotating hollow spirals of such 'field forces' so that when they are near enough to 'engage' they are inexorably pulled together - as two oppositely turning cork screws. In the case of gravitational fields, might there be some comparable spiral attractions engendered by both the larger and smaller masses?? But Einstein would (?later) describe gravity not as a force so engendered but rather as an 'influence' which any large mass has on the structure of the 4-dimensional space-time continuum around it (which seems very much like the gravitational field just described); gravity thus becomes a curved property of that continuum such that other masses (as planets) move preferentially (are guided) along such 'mass induced' curved paths accordingly. Of course, empty space is generally considered to be comprised of absolutely nothing so what property or characteristic of it (or of a gravitational field?) is it that becomes or accounts for it becoming 'curved' and how that (form of its 'strength'?) acts to change the path of any moving body we hope to discover.]
The inverse square law of the gravitational field (of the curved space-time continuum...and its 'strength'?) can with advantage, notes Einstein, be expressed in a form that can be derived from considerations of Newton's more fundamental law of motion, where
That is, a body of given mass when (somehow) accelerated becomes itself the source of a force whose stength is a function of that mass and the magnitude of its acceleration. Einstein describes the above mass as 'inertial' when, as typically, it is considered to be "a characteristic constant of the body then accelerated". If the 'force' which causes the mass's acceleration is however known to be gravity, the equation can be more specifically expressed as:
where 'Gravitational Mass' is similarly "a characteristic constant of the body (so) accelerated". From these two relations (equations) it follows that, in regard to gravity:
{Again, one wonders if the latter 'Intensity' is due to the extent of the curvature arising in the continuum (as a function of the mass concerned.) As the acceleration due to such 'gravity' is accepted as being independent of the nature of the body concerned and always the same for a given gravitational field, the ratio of the two forms (conceptions) of any body's mass must likewise, notes Einstein, "be the same for all bodies". By a suitable choice of units, we can then make this consistent ratio equal to unity. We would than have the following law:
This law had been described previously in mechanics but its significance, if any, had apparently not been further interpreted. Measurements of two different phenomena may prove equal in value but don't necessarily described essentially the same thing. But Einstein points out that it is in fact the same quality of any body - ie its mass - which so manifests itself in terms either of 'inertia' (reluctance to move or change motion - typically in a horizontal direction) in response to an applied force, or of 'weight' (ie heaviness) - in response specifically to being pulled doen vertically by a gravitaional field. That is, any mass may be so qualified according to the prevailing circumstance or context. Just when and where this differentiation is made and, importantly, how this is connected with (and so helps reveal) the more general principle of relativity which we seek is addressed next. We may recall that the special principle of relativity declares that the uniform motion of any body can be validly measured in terms of an agreed reference frame and that the latter's motion (if also uniform) has no effect on the expression of the relevant laws governing that body's uniform motion. That this is the case is a reflection of the property of inertia of the body's mass (and that of its reference body). We are apparently now seeking to find a more general principle which informs us that this same outcome (but possibly with new, as yet, unrealised, laws of nature) should be possible even where the motion of the body and/or its reference system is not uniform but accelerated (probably at a given rate). The existence and validity of such a principle is, says Einstein, (somehow) indicated by the effective equality of inertial and gravitational mass. For their equality argues for accelerated gravitational mass having the same significane as accelerated inertial mass on the motion of bodies.
If inertial and gravitational mass of any body are effectively equal (in both quantitaive value and in the underlying quality responsible for same), this, says Einstein, argues for a general principle of relativity - which therefore should apply equally to uniform and non-uniform motion of both moving bodies and their necessary reference bodies (with the case of uniform motion becomimg simply a special case of this more general principle - represented by an equation that ceases to apply for a declining accelerated motion just at the point it becomes (and remains) uniform). Such a principle will thus be described as one of the postulates on which his developing general theory of relativity will be based (just as the special principle was so postulated as one of the two postulates of the special theory; the other being the constancy of the velocity of light). Again, we might assume that the problem which such a general theory is advanced to resolve may be similar to that for which the special theory was so advanced - but probably less restricted in its objective. The problem resolved by the special theory would then represent a special case of this more general category of such problems; one that was likely at the limit of some relevant dimension of that category. For example (to repeat oneself!), it could be at the limit of decreasing acceleration - when velocity of the moving body or reference frame concerned actually ceases to be accelerated at all but becomes constant and steady - ie with an acceleration of zero - which happens to be but a special case of motion that is otherwise more positively accelerated (or even any thing?). It is just that one (ie Einstein) happened to address that latter problem (of constant motion) first - but with an eye more on the then current problems implied by light's constancy. Additional discussion on this matter is given in the next section.
Section 20 - The Equality of Inertial and Gravitational Mass as an Argument For the General Principle of Relativity (ie as a Postulate in the General Theory of Relativity).
Einstein then reasons as follows: We may imagine a portion of empty space with no large bodies (stars, planets, etc) near it. A given reference system is then assumed in respect of which smaller bodies may, if no force is applied to them, remain at rest therein and any such bodies already moving there will, for the same reason, continue to do so at their original velocity; that is, Galileo's law regarding inertia would apply where no additional force is applied. We may imagine the reference body to be a large enclosed room containing an observer with necessary measuring equipment. As he and the room are not near any large planets, etc, they do not experience any sense of gravity (or other force) in any direction so that he could not, for example, readily stand on the floor without attaching himself to it. Rather, he may just float gently about the room unless he somehow attached himself to the floor, wall or ceiling.
But, the free-floating room has a large hook attached to the centre of its roof outside - on which a very long rope is connected and stretched vertically above it which allows a powerful unseen being to pull the room in that 'upward' direction at an ever accelerating velocity. Thus, in the course of time, they would reach some immense speed (relative to observers situated elsewhere not subject to that same constant pull). But how would the man in the room experience this situation ? He would soon find that he moves naturally to the floor of the room and would experience an increasing pressure of the rising floor onto his feet and legs (if in a standing position) since his own inertia would resist such movement. He would no longer be required to attached himself to the floor as he would now feel very similar to anyone standing up in a room in a house on the Earth - that is, subject to the usual continual force of gravity, via a gravitational field, pulling him steadily (and constantly) downwards. He wouldn't keep falling - with increasing speed (be accelerated) - as the floor would of course prevent this.
If he had been holding a small body (as a stone) in his hand and now let it go so that it becomes free to move, it too would be drawn towards the floor (ie it would 'drop'), just as on the Earth. When previously held, it was subject to the same upward force (acceleration) as the room and observer. But when freed of this connection, it then approaches the floor (ie in the opposite direction) with the same accelerated motion - its inherent inertia effectively resulting in the floor coming up to meet it (as, in a sense, it doesn't 'want to' be pulled upwards by the rising handgrip of the observer but, like him, 'wants' not to be moved or accerelated upwards (or in any direction) - due to their inherent inertia (as a function of their respective masses). Again, Einstein stresss that the acceleration towards the floor (of both the free occupant and stone) would be of the same magnitude for their given masses whatever may be the material of such bodies so 'falling' (or apparently being pulled downwards as in the case of 'real' gravity.
The observer would likely soon conclude that he and the room must now be in some kind of constantly acting gravitational field. But why, he may think, isn't the room itself falling downwards due to this force ? Maybe it is but if so, he wouldn't himself be experiencing pressure against the floor - just as one wouldn't if in a lift that was in free-fall. He then notices the hook and long vertical rope above the roof and so reasonably concludes that the room must be suspended at rest in that gravitational field, so it can't fall. Einstein then suggests that although this conclusion by the observer might be thought unwarranted, there is no reason why the situation should not be interpreted any less as being a suspended room in a gravitational field than as a room being pulled upwards at an equally accelerating rate. This, he suggests, provides one with reasonable grounds to extend the principle of relativity to apply to two or more reference systems which are not only moving uniformly with respect to each other (as in the special principle of relativity) but to those that are accelerated with respect to each other. This, he says, provides a powerful argument for the validity of such a generalised principle of relativity (as a possible postulate in the formulation of a general theory of relativity) - one which applies equally to uniform and non-uniform motions of whatever sort.
The rationale of this conclusion depends, says Einstein, on the fundamental property of a gravitational field to give all bodies the same continuing acceleration whatever their nature (unlike magnetic fields, say) since it depends only on their masses. This means equally that it depends on the law of the equality of inertial mass and gravitationl mass as this law (apparently) implies that particular property. Thus, if a rope was suspended from the roof but inside the room - with a weight attached to its free end, it would be thus placed into a state of vertical tension which the man in the room could reasonably attribute to the downward force experienced by the suspended body (due to the action of the gravitational field below the room that he assumes applies) - thus placing the rope under tension. It would be the gravitational mass of the body in that field that would account for this tension of the rope. On the other hand, an observer who is poised freely in space nearby, may interpret the situation differently, for he will take the rope effectively as but an extension of the rope above the room and its constant upward pull will necessarily accelerate the suspended body via similar tension in the inside rope as it does on the external one used to pull up (accelerate) the room itself. The tension in the internal rope may now be seen as determined by the inertial mass of that same suspended body. The general principle of relativity implies that this equality of mass according to context or perceptual stance must be the case and, moreover, the law of same gains a physical interpretation thereby. The two forms of mass are completely equivalent.
If the (general) principle of relativity can be construed to prove equally applicable to both uniform and accelerated motion, then there would seem to be scope, says Einstein, to realise a more general theory of relativity based at least partly on same. In the special theory, the bodies to which this uniform motion validly applied should always have included light such that all moving bodies fell within a single category which (as then recognised) had a maximum possible velocity and allowed for light's velocity (which happens to be at that maximum) to remain constant. The theory thus required that time and space had to be recognised as being variable - as a function of the ratio of the velocity (v) of any relevant uniformly moving reference system to that ultimate constant speed of light (c) which meant that receipt of the information about space and time (and thus velocity) was inevitably delayed (and so altered or varied) to that extent. The special theory was premised not only on the constancy of the velocity of light (with the implications so arising) but on the compatability of that premise with the (special) principle of relativity which, due to the property of inertia, required that differing speeds of reference systems, if uniform, had no effect on the outcomes of the laws of motion (whether mechanical or electrodynamic).
We may assume that a more general theory would incorporate similar constraints (or determinants) but would apply to all forms of motion - whether uniform or accelerated - and, again, whichever it may be would have no effect on outcomes of the laws applying. As such, some adjustment would seem necessary for the latter motion - where inertia would likely prove to be of a different nature or form. This would presumably be incorporated within the construction of a general principle (and theory) of relativity. [But, again, see the Comment at the end which questions whether there has in fact been any true 'generalization' of the special principle (or theory) to this end (of non-uniform motion generally) rather than simply the construction specifically of an excellent theory of gravitation per se (with its one particular form of constant and pervasive accelerating influences) which has been effectively 'called' a theory of general relativity, rather than that gravitation theory having necessarily arisen as but part of a more general theory - which should (ideally) be able to generalise to all non-uniform and uniform motion).
In this regard, we may quote Einstein verbatim here - in the event that his sometimes obscure remarks may, with further analysis, clarify this matter: Thus, he says "From our considerations of the accelerated room, we see that a general theory of relativity must yield important results 'on' (with respect to?) the laws of gravitation. In point of fact, the systematic pursuit 'of the general idea of relativity' (ie of relativity applying more generally than just to uniform motion) has supplied the laws 'satisfied by'(?) the gravitational field." [I'm not sure what 'satisfied by' means here; I would have thought it should have read '...the laws by which the gravitational field operates'; subject and object seem often to be obscured in some of his phrasing.] In any case, he warns the reader not to conclude from the earlier remarks concerning the man in the room that the existence of a gravitational field is necessarily always 'only apparent'. Also that we can not always choose a reference system such that no gravitational field exists with reference to it, for it depends on the 'kind' of gravitational field present; there are special ones where this may be possible but others - as that for the Earth - where it is not. [One recalls my own example of activity on the Moon where it was convenient to assume its gravitational pull was effectively zero. All this qualification seems a little too general and abstract and as yet untied to physical or concrete specifics - that is, to constitute an efficient learning or explanatory vehicle. Does the theory of gravitation arise out of the general theory of relativity of is it the other way round ?
The general principle of relativity (if such exists) has been described thus far as requiring an equality between the two forms of any body's mass - inertial and gravitational - the validity of which Einstein's examples above have supported. The acceleration induced in a body's motion by a gravitational field can thus be construed as the same as that produced by any constant force acting on a body - of whatever mass. Its velocity would increase constantly. Einstein seems to point out that a general theory of relativity based on such conclusions will help formulate and/or confirm the laws of gravitation arising from the effects of the gravitational field. He warns the reader however that while the man in the room in space may have concluded that his sense of weight on the room's floor reasonably implied that he and the room must be in a gravitational field (rather than being in a room which was being pulled upwards with increasing velocity by some other unknown 'force' which their joint inertia was resisting), we mustn't assume that it is virtually arbitrary whether we consider that any such force may or may not be gravitational depending on what reference system we choose to refer our accelerated motion. In certain cases, he points out, we have no such choice and so the relevant field must be real.
The foregoing analysis of the equality of the two forms of accelerated (vs uniform) motion - inertial and gravitational - provides a counter-argument, claims Einstein, to the suggestion made earlier (in Section 18) that the sudden deceleration (or, equally, acceleration) of the train in which a passenger's inertia (as a relatively unfixed 'body' seated in the carriage) had been shared with the uniformly moving train, could be interpreted as supporting the existence of some such absolute ('real') change of velocity of the train - rather than as simply a change of that motion relative to ots specified reference frame. For we may recall that the moving train can with legitimacy be construed as being effectively at rest - with the station, embankment and indeed the Earth (or a stationary train being passed) perceived as the reference body (effectively) moving in the other (backwards) direction. Some brakes are applied (or, contrari-wise, some increase in forward acceleration effected) and the passenger may legitimately relate the effect of such braking (or acceleration) to a gravitational field instead - which has suddenly occured either in front or behind him and his stationary reference (the carriage) so that, relative to them, the embankment or previously stationary train now begins suddenly to steadily reduce (or increase) its apparent velocity from its former uniform character in that other direction - thus becoming suddenly non-uniform in response to a forward- (or backward-) pulling gravitational field, while it exists. But, might we not reasonably ask: 'Is construing these various relative (vs absolute), motions in this rather contrived manner really legitimate and realistic ? Einstein seeks to answer this affirmatively in the next section. [I believe we have already touched on this above and indicated that this type of contrived interpretation may however lead to a productive mathematical analysis even if the physical representations are rather obscure or contrived.]
Section 21 - In What Respects are the Foundations of Classical Mechanics and of the Special Theory of Relativity [arising from same] Unsatisfactory?
He begins by recalling the most fundamental law of classical mechanics, as first advanced by Galileo and confirmed within his basic laws of motion by Newton. This states that any material body sufficiently far removed from other such bodies - having been given a brief impetus by some force - will continue to move in a straight line therefrom at a steady velocity - indefinitely (unless subjected to some additional or counter force) or, if not so impelled initially, will continue in its relative state of (apparent) rest, again indefinitely. These outcomes are considered to be the result of the property of Inertia that every such body of mass possesses. In both cases, such relative motion or stillness must be assessed in relation to agreed bodies of reference which themselves must only move in a uniform manner relative to each other and to any moving body being so assessed. This law of maintained steady motion or stillness so determined is not however valid for bodies moving non-uniformly relative to reference bodies whose motion is or may also be of this form. Both in the classical mechanics of Galileo and Newton, and in the more encompassing sphere to which Einstein's special theory of relativity applies (ie including light and its constant velocity implications), one must differentiate between uniformly-moving reference bodies relative to which such recognised laws of nature (of both mechanics and electrodynamics), as the law of inertia, can be said to hold true and, on the other hand, reference bodies which move non-uniformly (as accelerated - in a gravitational field) relative to which such laws (formulated only for uniformly moving references) do not so hold. Seemingly, other laws of nature may well be formulated which can however, apply validly in such as gravitational fields, with their non-uniform (acceleration) effects on both bodies and reference frames (which are now seen as effectively equivalent and apparently needn't be differentiated). [Note: there may be some inevitability that any non-uniformly moving body's reference body (these two always being effectively reciprocal) must itself also be perceived as moving non-uniformly (by definition?]
Although uniform motion of moving bodies and their reference systems prove consistent with the fundamental basis of mechanics and electrodynamics - that is, with the concept of inertia on which these depend (as reflected in the special principle of relativity) - various physicists such as Newton, Mach and latterly Einstein felt that this particular type of motion needn't be the only kind on which laws of nature pertaining to motion may be discovered or based. There may well be other consistent laws (and possibly more general ones at that, which could subsume any derived on less general grounds) - which prove relevant to both uniform and non-uniform motion (such as that accelerated motion (of potential reference systems and bodies) produced by gravitationsl fields or, equally, by rotational motion?). If different effects are produced on otherwise similarly moving bodies when referred respectively to reference bodies which are moving uniformly or non-uniformly (as accelerated or rotational), to what causes or laws can these differences be attributed ? If they can be identified, then there would likely be discovered a wider range of natural laws and thus a broader understanding of nature generally. The physicists mentioned all objected to accepting the prior status quo (of simply exluding non-uniformly moving reference systems from their analysis). A new 'mechanics' (or electrodynamics?) was probably required to include the latter category - one which went beyond the confines of both classical mechanics and the electrodynamics of special relativity - with its uniform motion restrictions. This objection could, said Einstein, only be removed by extending relativity from that prior restricred orientation to a more general approach in which a broader principle of relativity may apply to all forms of motion and lead to a more general theory (of relativity) whose equations will hold true for every moving body and their chosen references body - whatever may be their states of motion, and however caused. As implied above, this broader unrestricted sphere of motion could place constant velocity as but a special case of a more general domain of variable, accelerated motions.
Section 22 - A Few Inferences from the General Principle of Relativity.
The special principle of relativity makes a statement regarding the predictable motion of any body - as in a uniformly moving reference system (eg a train), when referred either to that system or to one that is, relatively, not moving or moving at some different but still constant velocity. To generalize this principle, it was found useful to justify that any constantly increasing velocity (ie acceleration) of the reference body (or the body itself?) can be equated with the acceleration produced in a gravitational field. One could then compare or contrast the motion of the body concerned as seen from this same reference system moving at a constant velocity (or from the relatively stationary position) with the situation in which the body's motion was seen as occuring on (and/or from??) a passing train that was actually accelerating continuously. How would the assumed gravitational field (or any other constant inertial force) so accelerating the passing train influence the perception and measurements of the moving body ? That is, what is the comparable statement that a general principle of relativity would make regarding the motion of the body concerned as viewed on (or from?) the accelerating position ? How does a gravitational field (or any inertial equivalent engendering accelerations of a less consistent, known magnitude) influence the body's perceived motion ? Presumably, the latter must be seen as accelerating or decelerating itself. And are time and space similarly affected ?
For most (?all) material bodies so viewed, past experience has already provided the answer - in that the body is known to not continue at its one constant (uniform) velocity (represented graphically by a straight line) but would be seen rather to move at a constantly increasing (accelerating) velocity - now represented graphically by a curvilinear line. [One assumes that this same outcome would be the case had the situation been represented instead by a controlled vertical descent of a lift/elevator (in place of the uniformly and horizontally moving train) passing which was a similar lift in free fall in a gravitationsl field in either or both of which a moving body is similarly viewed (through glass walls) from either lift or from a stationary floor past which they are so moving.] But it appears (according to Einstein) that how a pulse of light would compare in the two situations (ie on a uniformly moving reference system where it is transmitted linearly with constant velocity c) and on one when it is considered with reference to an accelerating reference, is not so generally understood. However, it seems that in the latter case, the path of the light may be reasonably predicted not to follow a straight line but also to take a curvilinear one - so reflecting a comparable accelerating status of any mass in such a field. At least, this is what his new theory of general relativity (and its associated general principle), would apparently predict. That is, it may be reasonably inferred (ie on such theoretical grounds) that 'light has mass and, despite its velocity being assumed to be constant, is propagated curvilinearly in a gravitational field'. This alteration of direction, if not of speed, equates to a change of velocity (as normally defined) and thus an infringement of its sacrosanct constancy of this parameter.
Thus, just as the special theory predicted that light could be validly incoporated within a single conception of uniformly moving bodies (despite the unique constancy of its velocity) if it could be shown that time and space were, as concomitants, actually variable, so the general theory now predicts that light responds to a gravitational field by adopting a curved (accelerated) path through it. In both cases, there was presumably a prior disinclination to expect such behaviours - as they opposed existing understandings. The subsequent verification that time and space were indeed variable, provided evidence to support the special theory in this and its other regards, while any verification that light's path was indeed deflected (curved) by the Sun's gravitational field should likewise provide support for a general theory of relativity whose validity might otherwise have remained in doubt. Just how and why this result would have this significance, we hope will be revealed shortly. Any actual result obtained would then serve as an important test of what, to this point, was just a theory (and a still developing and, thus far, little described one at that).
Because of the exceedingly small mass of each element of a light ray (beam or pulse), the magnitude of this curvature of its path is calculated to be rather slight in most gravitional fields available in practice. But with respect to the powerful field engendered by the Sun, a deflection of 1.7 seconds of arc should, predicted Einstein, be measurable and confirmed in the following convenient manner: As seen from the Earth, certain fixed stars appear in the neighbourhood of the Sun (at least as perceived in two dimensions; they are of course millions of miles further out). While the light from them and thus their existence is not differentiated (visible) to us in that position due to the strong ambient light of the Sun, they may be seen during a total eclipse of the Sun. Their positions should be displaced 1.7 sec of arc ourwards according to Einstein's calcualtions based on his general theory - as compared to their positions when measured when the Sun is in a different position relative to them (eg at night). This test was finally accomplished by Eddington at the solar eclipse of 29 May 1919 and the results did indeed confirm the prediction of the theory (published just 3 years before).
But a second important implication of the result was that the motion of light in a gravitational field, by being thus displaced into a curvilinear path, proves contrary to one requirement of the special theory of relativity which relies in part upon the constancy of light's velocity. For that constancy of velocity is negated (as mentioned) not only by any alteration in its speed element but also of its directional element. Hence, we must re-define the specific domain of motion in which the special theory holds true to be limited to that of uniform motion; it is seemingly not valid within the unlimited domain that includes non-uniform motion and particularly that due to the gravitational field - where a more general theory of relativity will (it seems) prove the more valid. The special theory is, however, not 'overthrown' by this newly realised reality; rather, it can now be seen as a special limiting case within that more comprehensive theory. The latter theory has allowed us to derive theoretically the influence of a gravitational field on the course of a natural process - namely the character of the velocity of light - the laws of which were already known for the more specific and limited situation lacking a gravitational field. But of equal value, this discovery provides an opportunity to learn more about the laws 'satisfied by' the gravitational field itself. [Surely there must be some other equally valid phrase to express this - than 'satisfied by' ?? In particular, this seems to abscure the fundamental matter of cause and effect.]
[We may note here that while we began by seeking to extent the generality of the principle and theory of relativity to incorporate the domain of non-uniform motion into a general theory, the focus of our attention has (as already mentioned) gradually shifted somewhat to that of accelerated motion (that is always a feature of such motion) and specifically to that of the gravitational field per se, not only as representative of this extension but increasingly as the specific object of interest in its own (important) right - being as it is a major 'force' of universal significance. In this regard, we may note that in his actual definitive paper on the General Theory (1916), he more explicitly states that "It will be seen from these rsflections (as detailed earlier in his account) that in pursuing the general theory of relativity, we shall be led to a theory of gravitation...". {But, it almost seems that it is the other way round!?] Relevant to this conclusion, he notes that this persual tends also to follow from the fact that we are able to 'produce' a gravitational field simply by changing the system of coordinates' (presumably appropriately). [This requires expansion, exemplification and detail.] One might suggest that he was going to pursue this as a main objective in any case in that he had already in 1911 written an important paper specifcally on gravitation. Again, therfore, we may ask: 'which theory led to the other' ?? ]
We may recall that when the room floating in space was suddenly being lifted continuously, the occupant perceived this (with the now felt pressure of his feet and legs onto the floor) as being due instead to a gravitational field below him. This perception is considered valid - as justified within the general theory of relativity. Also, any laws which characterise gravitational fields must, according to the general theory of relativity, be equally satisfied for other such 'special' gravitational fields realised in this same (somewhat contrived?) manner. While such fields can and do occur by other (more ?natural) means, it was to be hoped, says Einstein, that any general law of gravitation - which would apply equally to all such fields - could be derivable nevertheless from such 'special' fields alone. Happily, this hope was indeed so realised but to explain how, it was necessary, says Einstein, to overcome a rather serious difficulty that stood in the way of a more direct explanation (and thereby confirm its validity). This explanatory detour entails extending further our ideas of the space-time continuum (from their variability as revealed via the uniform motion associated with the special theory) to that within the domain of non-uniform motion and curvilinearity associated with the general theory of relativity. Would the nature of that (apparent?) variabilty be somehow qualified thereby ?
Our perception of time and space, as considered in terms of the special theory of relativity and its domain of uniform velocity, were affected by inevitable delays inherent in obtaining information about these elements of velocity - by means of light - the nature of any moving body's velocity being associated with delayed transmission of such time and space data. We have seen that in a non-uniform, gravitational field, this restriction would be increased by virtue of a resulting curvature of light's path; its constancy was affected with respect to the spatial (distance) element of its otherwise constant velocity. In such a field, where it was thus accelerated, its transmission of time and space data would presumably take even longer (ie by having to travel over a (?slightly) longer path...or ?).
The frame of reference of the uniformly moving reference frame (K) utilised when developing the special theory was based upon the accepted Euclidian (straight line) geometry and Cartesian coordinate system applicable thereby. Thus, the motion of a body relative to this system could move along the x axis and the postion of same could be determined in such terms using also the orthogonal vertical and depth dimensions of the y and z dimensions - all mutually independent at 90 degrees to each other. The principle of special relativity as based on these Galilean domains was developed in such idealised conditions in order to determine and confirm the fundamental principles involved. Onto this framework was then added the implications of light's special velocity characteristics by which it became apparent that time and space measures had to adapt (or rather, we had ro recognise their existing adaptations) as described earlier. The Cartesian coordinates could then include a 4th orthogonal dimension to represent time (all 4 later representing the space-time continuum). But if such linear, uniform motion turns out to be but a special case of a much more general (and widespread?) domain of motion - namedly that of acceleration (changing velocity) with its curvature implications - can we rely on using a linear model of defining such space and time and their combined continuum? Einstein points out that we can not (see below). The effect of gravity in particular - on the path of light waves - would have implications for any analysis with a geometric component if that geometry required those effects to be taken into consideration. A straight line geometry would have to be replaced by one which is effectively based on curved motion.
Section 23 - Behaviour of Clocks and Measuring Rods on a Rotating Body of Reference. [Or, comparably: 'On the Physical Interpretation of Space and Time in the Case of General Relativity (with its non-uniform, accelerating, non-linear motion - however arranged, affecting the path of their information transmission (by light).]
We may recall that in developing the special theory in terms of that idealised coordinate system, we had to ignore the period of acceleration until the moving reference system (the train) got up to its steady state of 100 mph and we assumed similarly that the various moving bodies in that train travelled at their different velocities also in a uniform manner - again conveniently ignoring how they got up to or would later slow down from those steady speeds (as the train, ball or bullet). This was justified on the grounds that we were seeking the basic interactions underlying this idealised special form of relativity. But it appears that the more common and general situation is more one of variable and changing velocities - of both the typically smaller moving bodies and of their necessary reference frames. And thus the most characteristic form of such motion - acceleration (always involved in such changes)- may well be best analysed, again under relatively idealised conditions, in order to develop the basic principles determining this important and more general motion. And, as we have stated above, the acceleration due to the gravitational field presents itself as probably the most convenient and consistent means or model by which to do this. This phenomenon is in any case of great importance in its own right, being a fundamental characteristic of the universe and major determinant of the motion of all bodies. This being the case, it would appear necessary to utilise a more relevant method of establishing the albeit still idealised framework of coordinates in which the resulting curved (non-unifiorm, non-linear) motion of any body, including light, may be best analysed and understood.
The values of space and time as calculated for a moving body in terms of the special theory of relativity have been shown to depend upon the limitations of the velocity of light (c) in ratio with the uniform velocity (v) of the reference system (K) to which the body's motion is referred. No non-uniform (eg gravitational) influences affect the uniform motion of the reference K operating within this so-called Galilean 'space-time' domain. If we refer this same domain instead to a second reference body K', which is not moving uniformly, but rotationally, relative to K, we may consider how this (source of curvature) might affect the measurement of time and space: We let K' be a circular disc some miles across which is rotating horizontally about its centre in its own plane. An observer sitting somewhere between the center and the edge would experience a centrifugal force acting on him (and his own inertia) outwards - making him lean towards the periphery. From the point of view of an observer on the uniformly moving reference K, this would appear to account for that effect. But the observer on the rotating disc could himself equally feel (as is warranted by the general principle of relativity which seems to treat all accelerating motion as equivalent) that the disc was his local reference body 'at rest' (not rotating) and the force he is experiencing is a gravitational one - pulling him continuously outwards (rather than downwards as is more usual as per Newton's conception of gravity); with the field's 'horizontal' strength increasing as one moves futher from the centre of the disc. If we assume that the observer there accepts the validity of a general theory of relativity, he will have no difficulty in accepting that a general law of gravitation could be formulated on the basis of such equivalences and so explain not only the motion of stars and planets but also the field of force as experienced by himself - that is, a complerely general law.
The observer then performs certain experiments on his circular disc with clocks and measuring rods as he is curious to know how this particular non-uniform motion might affect the measurement of time and space (eg compared to how they were affected when measured for bodies moving relative to a uniformly moving (rather than accelerating) reference. What does he conclude when so measured in this environment ? To begin, he places one of his two identical clocks at the center of the disc and the other one near the edge - both then being 'at rest' relative to it. The question is then considered: do the two clocks run at the same rate as viewed from the non-rotating reference K ? The centrally placed clock is seen from there as having no velocity whereas the one on the periphery is seen as moving in its rotational (accelerating) manner. As show in an earlier section, a moving clock runs at a slower rate than one in the non-moving centre when they are observed/measured from a relatively still position (due to a lag in the time it takes to receive the information about the position of the clock's hands (on which alone the time may be validly adjudged)). This basis for explaining the alteration in both time and space measures would seem to assume that the moving clock (or measuring rods) are moving directly away from the observer concerned. This may be difficult to accept in the case of the varying distance of such clock and rod positions that must occur on the periphery of a rotating or revolving platform - where the distances concerned would necessarily increase and then decrease from any fixed viewing point - cyclically. [Possibly this is accounted for by the use of trigonometry?] In any case, we would assume also that the clock mechanism doesn't actually run at a different rate but, rather, the time its hands subsequently show only 'appears' to have moved at a slower rate - with such appearance nevertheless proving to be the only available means of validly assessing the time (or distance, and hence velocity) concerned.
Einsten then generalises this conclusion by stating that whether on a rotating disc or in a gravitational field (with their equivalent accelerations), a clock will (?appear to) run (ie advance the perceived time) more or less quickly according to the position in which the clock is situated (at rest) vis a vis the acceleration field concerned. We cannot then, says Einstein, obtain a reasonable definition of time with the use of clocks which are at rest with respect to such a body of reference (in such a field?). The appropriate coordination point for time in the domain of relevance cannot be reliably chosen (defined). This insurmountable difficulty is also the case, says Einstein, when seeking to establish/define valid coordination points for space and distance measures when determining the velocity of a moving body (always with respect to an agreed reference). The length of a measuring rod (as viewed from the non-accelerating reference body K) when placed tangentially on the edge of the rotating disc (its length in line with the observer and thus moving away from him (momentarily?) in that orientation will be less than its unit length (of 1) since moving bodies generallly undergo a shortening (apparent shortening?) in the direction of motion (away from the observer?). This would contrast with the perception of the measuring rod from K when it is set out instead along any radius (vs the circumference) of the rotating disc - say perpendicular to the observer. [But again, would it not maintain this position only momentarily as it would assume continuously changing positions with each rotation ?]
Nevertheless, if the observer then measured the circumference of the disc with the rod and then its diameter (ie radius x 2), then, on dividing the one by the other, he will not obtain the usually expected value of pi - at 3.14... (as per Euclidean geometry) but rather, a larger number than that. This differs from the true value of pi that would be expected, as per Euclid, for a disc which is at rest with respect to K. In other words, Euclidean geometry does not hold true on a rotationg disc or, more generally, in a gravitational field (where we attribute the length 1 to the rod in all positions and orientations. As such, the idea of a straight line also loses its meaning here, says Einstein. [This again recalls the curved path of light in a gravitational field (and the comparable disruption of straight Euclidean lines of a Cartesian coordinate system when some such disturbance proves to influence such an arrangement.]
Section 26 - The Space-Time Continuum of the Special Theory Considered (as it traditionally was) as a Euclidean Continuum (analysed in terms of Cartesian coordinates).
Although brought forward before considering the topics in Sections 24 - on Euclidean and Non-Euclidean Continuua - and in 25 on Gaussian coordinates, we might first usefully review at this point the earlier discussion in Section 17 regarding Minkowski's conception of the 4 dimensional space-time 'continuum' - of such Cartesian coordinates: If space is the 3 dimensional continuum in which the exact central point of a body's position at rest may be described - in terms of the orthogonal coordinates x, y, z - then we may describe any number of similar positional points in its immediate neighbourhood by such coordinates as x1, y1, z1 - which may be as near to x,y,z as we choose. It is because of this latter property (of such points being an indefinitely small distance from each other) that we may call the area across such neighbouring points a 3 dimensional continuum where, effectively, a body's central point (or in purely geometric terms, we may speak of the point per se) and hence the body as a whole - may pass from one such positional point to the next without making a discrete 'jump'. But, as soon as one speaks of something 'passing' from one such point to the next, there is an implication of movement through such space - which always requires time. Such motion from one point to another taking time (and thus over an additional dimension) may now be considered not as just a position in space (nor even as a moment of time in eternity) but as a 4-dimentional 'event' (ie a body-moving 'happening' over space and time). Each such individual event may then be described, as regards where and when it so happened, by 4 numbers - namely, by the 3 spatial coordinates x,y,z plus the one time value coordinate t. This 'region' in which all events (moving bodies) occur over space and time (ie space-time) may thus also be appropriately considered a continuum - of 4 dimensions - as for every such event there are as many immediately neighbouring events occuring in such space-time as we care to choose. The 4 coordinates of these - x1, y1, z1, t1 (or whatever) - being an indefinitely small amount from those of the original x,y,z,t event considered. Through such a continuum, the whole body thus moves (as a continuous sequence of 'mini-events') over space-time.
As events which consist of bodies moving uniformly through such space-time regions typically do so in regular and predictable directions, in straight lines, the form and character of those motions may be adequately analysed and described without too much concern as regards such minute 'point to point' spatial distances over temporal durations as were introduced and considered above. Rather, they may be so analysed in terms of the more usual meters or miles per second, minute or hour, as is relevant, at a uniform (and thus straight) rate. It is probably for this reason that the subject of the Special theory of relativity, as discussed thus far, has had little or no need to introduce the concept of the 'continuum' (of the straight line). But when such movment is influenced by forces which result in motions which are not uniform and straight but rather are accelerated, for example (and hence curved when represented graphically/geometrically), then the analysis may well require consideration of much less predictable and more subtle variations of velocity. In that case, changes in the underlying elements of velocity - as time, distance and direction - can depend on a much closer analysis of such point by point and moment by moment changes in direction. These fall within the purvue of the mathematics of the calculus - where those 'indefinably small' changes (typically abbreviated as ds and dt), must now be so considered and probably best in a non-Euclidean manner. Analysis of continuous rates of change entail an understanding of the logic and reasoning of this branch of mathematics. It is for this reason that in the following sections of Einstein's account - of the General theory of relativity - there is necessarily much greater reference to this concept - ie of the 'space-time continuum' - although his present treatment has, thus far, avoided significant reliance upon the calculus or other 'higher' mathematics in dealing with same.
In these latter considerations, the role of time becomes less independent as a factor in the motion of such moving bodies. Its seeming independence was not questioned in the classical mechanics of Galileo and Newton, but in relativity it was recognised as being bound up with the differences in motion (over such time) of the reference systems pertaining to the motions studied. Were the motions of such reference bodies both uniform - or were either of them non-uniform ? In the special cases of uniform motion in the Galilean sense, time in reference K was shown to be equal to time in a differently moving reference body K' (ie t' = t). But as interpreted within the more general relativity model, where both the implications of light's constant and limited velocity and the more general form of non-uniform motion is appreciated, we find that time in the latter case (t') is a function of both the velocity of K' and that of light (c), as well as the time t. Ie: t' = t-(v/c2).x / sq rt 1-(v2/c2), where x is the distance travelled by the body of concern along the x axis. This, the 4th equation of the new transformation equations of relativity theory, proves consistent with the now accepted dependence of time within its single 4-dimentional space-time continuum. Its former conception as possessing independence has thus had to give way to recognition that it has lost that assumed status. It is always a function of (dependent on) motion (of moving bodies) - whether uniform or otherwise - in ratio with that of light. It is just that its 4 dimensional continuum status, while still extant in the uniform case (as a special case), becomes much more relevant when considering non-uniform motion - as in gravitational fields. The geometry concerned may be influenced by the visual medium of light causing a curved 'reality'.
There is an important sense in which Minkowski's 4-dimensional space-time continuum as it pertains to general relativity can relate to the 3-dimensional space continuum of Euclidean geometry. This is that the coordinate for the 4th dimension (of time) can be shown (at least as it functions in regard to the special theory of relativity, with its uniform velocities) to assume exactly the same role as the 3 spatial coordinates - as far as the natural laws of that theory are concerned. [Just what that 'role' (of all 4 coordinates) is in such circumstances may need to be further exemplified on my part. And how does this relate the the general theory? (see below)] In a formal sense, the 4 coordinates correspond exactly to the space coordinates of Euclidean geometry. But to show this is the case, it is apparently necessary to replace the usual time coordinate t by an imaginary value with a magnitude equal to: sq rt of -1.ct - which is proportional to it. When so replaced by this imaginary term (which nevertheless retains its correct temporal value in the calculations), the manner in which the general theory handles its subject is likely to be more readily appreciated (ie clearer) than otherwise, according to Einstein. [See also Appendix II.]
We may continue the previous topic by noting that in terms of the special theory of relativity, the Galilean coordinate system by which the uniform velocity of a body may be analysed can be treated as a 4-dimensional space-time continuum - ie as an 'event' (of a moving body). As mentioned, the character of this coordinate system -as a continuum - did not seem to be a particular consideration in the analysis of such motions (events) in our earlier discussions but this quality (and its relevance to the newly stressed conception of the moving body as an event) will now enter more into our discussions. [We may recall here that such a model of space is but an idealised, if invisible, framework imagined to exist as part of some (any) portion of the actual physical environment and utilised simply to better analyse the essential principles involved unfettered by the 'messiness' and exceptions of reality.] Thus, when an event (at a point in such an imagined continuum x,y,z,t) is in transition from one such Galileian system to another which is moving uniformly relative to the first, the equations of Einstein's new transformations are known to be valid. That is, they allow valid predictions as to outcomes according to the special theory - which essentially takes full account of the 'laws' of inertia and of the constancy of the velocity of light as it affects the measurements of the events as perceived from the two systems.
Minkowski found that the new transformations 'satisfy' certain simple conditions (ie they are consistent with known relationships or equalities) which, when better revealed, would help confirm the validity of those transformations. Thus, if we consider two neighbouring events in a 4 dimensioned Galileian reference system K - located at points x,y,z,t and at a very near neighbouring point - then their relative positions one to the other (wherever and whenever they occur) will be given by the differences (between the coordinate positions/times) - ie: dx,dy,dz (for space), and dt (for time). That is, the 2 events occur at slightly different places and/or times or both. If then referred to a second (differently moving) Galileian system, such differences between the 2 near events (as so perceived) would similarly equal dx', dy', dz', and dt'. The magnitudes of these two sets of distances/times when repectively summed will, said Minkowski, always fulfill (accord with) the following condition (equality) - with squaring to remove negative linear values:
Thus, the role of light's limiting velocity may be seen to accord with the relevant new (and valid) transformations - as they take account of the variations in the perception of distance and time values (as detailed earlier). This may be expressed as:
where 'ds' represents the indefinably small(d) 'space'(s) (distance) between the two adjacent events - as represented geometrically by the two idealised 4-dimensional points described. It has the same value for all selected Galileian reference systems (bodies). [Note that tha relations between the variables concerned are valid for the coordinate differences and, hence, for their differentials as well. And, as touched on above, if we replace the real time variable t with the imaginary variable sq rt [-1.ct], we can apparently more readily appreciate that the space-time continuum as it accords with the special theory of relativity, may be treated as a Euclidean (straight line) 4-dimensional continuum. And when the point (ie location and time) of an event at x,y,z and sq rt [-1.ct] are represented instead by x(1), x(2), x(3) x(4) we find that the result (ds2 = the sum of the squares of the foregoing) is independent of the choice of the body of reference. That is, if we ignore any negative values, the very small distance between essentially adjacent events equals the sum of the values of the x coordinates thus shown whatever the reference body utilised (eg whether K or K'). [This construction of the relevant continua thus lends itself to any calculus treatment that may be required when considering the motion of bodies influenced by more variable forces than we have considered thus far (ie than under conditions of uniform motion.] We 'return' now ro Sections 24 and 25:
Section 24 - The Euclidean and Non-Euclidean Continuum.
Einstein begins his next step by suggesting a model firstly of this Cartesian coordinate system as used for that more restricted Galilean domain - of undisturbed uniform motion (as just reviewed above in Section 26, brought forward). He thus imagines a flat, square marble table on which it is possible to 'pin point' any particular point (at x and y) on that plane surface by means of dividing it into an almost infinity of identical very small squares. The outline (sides) of each square is imagined to be composed of separate, exceedingly short rods of some fine material (lying on that surface) which would be shared by neighbouring squares. By counting the sides along any linear direction (axis) from any corner of the table to any required point (on that, say, x axis) and then along any direction at 90 degees to same (as on a y axis), one could arrive at a precise point where x and y cross on this flat 2-dimensional surface. These represent Cartesian coordinates of a 2-dimensioned Euclidean continuum. This model could easily be extended into 3 such dimensions in order to arrive at a point x, y, z in this still linear but now cubic continuum. A 4th orthogonal dimension could be similarly added - to represent time for example (which, pro tem, could replace the z dimension, if required, to better envisage the seeming independence of that time variable.
Einstein then asks what would occur if the marble table is heated (eg from beneath) in just one small portion of the table ? That is, if some such (any) disrupting variable could affect a portion of this rather idealised domain. Certain of the very small squares' material sides could, for example, be slightly expanded by such an influence (indirectly via the heated marble beneath). This would disrupt the precise straight line plan of the orthagonal coordinate system (laid onto the table's surface) and thus the method of locating valid meaningful points (at least for 2 dimensional uniform motion in space) on this Galilean domain of interest. And what if the disrupting variable was instead a gravitational field and the linear, uniform assumptions of the little squares had to be replaced by curvilinear ones ? Motion (and velocity) between them could not be assessed as uniform. Both space and time measures could be altered (assuming further dimensions were also considered).
Einstein notes that if the geometry of any spatial continuum is somehow disrupted by a known influence this would provide a means of better understanding how that influence 'works'. For we could assign a value (as 1) for the distance represented by the chosen 'little rods' for whatever size or orientaton they may assume under a particular influence. Otherwise, we have no real, non-arbitrary measure of the latter. If the influence of interest (for reasons described earlier) is a gravitational field, for example, this approach should provide an objective means of revealing its effects on any relevant continuum in which a moving body may be affected. Conveniently, notes Einstein, certain mathematicians had already calculated the nature of just such an altered continuum. It appears that if the 'little rods' (representing the very small distances and directions between the coordinate points of a continuum) are imagined to lay not ''on' but 'in' the surface of a 3-dimensional system of reference (K'), then the surface at least of that coordinate system may be treated in terms of 2-dimensional geometry (thus ignoring for the present its association with the assumed Euclidean 3-dimensional continuum). The surface itself, however, would not be considered as part of that Euclidean continuum with respect to the rods now assumed to be a part of (embedded in) it and we would no longer define the coordinate points (at the ends of the rods) as Cartesian (or necessarily orthogonal).
In such a case, the influence would presumably affect the continuum (and its integral rods) more directly. As such, we should find that different laws (influences) hold with respect to the architecture (geometric relationships) so constructed than would apply in the case of Euclidean continua of whatever dimensionality. It was the mathematician Gauss who calculated the way such relationships in (vs 'on'?) the surface may be so treated geometrically (as affected by whatever influences) and Reimann then extended this to the relationships (of moving bodies) affected in other non-Euclidean, multi-dimensional continua. [Note: I likely require some other version of these matters in order to better understand them. Are we 'in' the table's surface now (vs 'on') because of a 3rd depth dimension and/or 4th one of time...or...?]
Section 25 - Gaussian Coordinates.
Cartesian coordinates and Euclidean geometry were thus not the only (or best) way to analyse how moving bodies are affected in the most general circumstances. The distances and times concerned within the appropriate continua (as so influenced) could not always be correctly determined through that earlier methodology. This problem area was foreshadowed by the results depicted when noting the affect that the gravitational field (or as in the example of the rotating disc) had on length and time, as anticipated by a general principle of relativity. The geometry and analysis of Gauss and Reimann appear to overcome this problem and to lead to improved results. Einstein next describes the thinking underlying the Gauss-Reimann approach by showing how the orthogonal lattice work of the Cartesian coordinate system (applicable for the special case of uniform motion) can be seen as a special (limiting) case (*) of a broader range of otherwise non-orthogonal, multi-dimensional networks of space, time and point (body) direction and path of motion. That is, by the use of 'Gaussian coordinates'. Reimann arrived at this method of handling the problem of non-uniformity as follows: He imagined a set of roughly parallel arbitrary curves drawn onto the surface of the same type of table as described above, with another set drawn roughly orthogonal to these. Between the drawn curves of each set (referred to a u and v lines respectively) we imagine an infinite number of similar curves drawn parallel to the ones actually drawn such that every theoretical point on the surface has two such lines (one of u and one of v) through it. Such lines are uniquely numbered so that every point can be identified with its unique pair of numbers. These two numbers constitute the Gaussian coordinates referred to above - which totally cover the surface. As such, the lines so represented are but 4 of an infinity of possible curved and straight lines that can exist and be represented thereon - although, at the limit, particular directions of the lines can have validity one assumes.
A given point P may thus have the coordinate numbers u3 and v1, say, and an immediately neighbouring point P' could have numbers u3+du and v1+dv - where du and dv are exceedingly small numbers. The distance between such neighbours, measured (in theory) by equally short measuring 'rods', would be symbolized by ds. Then , according to Gauss, we have:
where the 3 'g' terms are all magnitudes which depend in a precise way on (some property of) u and v and such dependent magnitudes of same "determine the behaviour of the rods relative to the u and v curves and thus also their behaviour relative to the surface of the table". [NB I have no clear idea what this phrase in quotes (by Einstein) signifies. Possibly, this is how we account for a particular path of any given line - determined at the sub-micron level by some such influence (of the 'field?). But we shall press on in the event that what follows may provide further insight.] Thus...for the case in which the points of the surface here considered form a Euclidean continuum with reference to the measuring rods (but only in this case), it is possible to draw the u and v curves, each numbered, in such a manner that we have, simply:
Under these conditions, the curves become (or 'are') straight lines (as per Euclidean geometry) and are perpendicular to each other - in just the two dimensions. That is, we have the limiting case where the otherwise differing Gaussian coordinates thus determined are simply the same as Cartesian ones. On this basis, it appears that these two dimensional Gaussian coordinates are nothing more than an association of two sets of numbers with neighbouring (?orthogonal) points on the table surface, the values of which differ very slightly from each other. The same method can also be applied to a continuum of 3 or more dimensions. Thus, for a 4-dimensioned orthogonal continuum, every point of the continuum (as x,y,z,t) may be represented arbitrarily by 4 numbers - x1,x2,x3,x4 - which are referred to as coordinates. Adjacent (?orthogonal) points correspond to adjacent values of such coordinates. If the distance (ds) between adjacent points (say P and P') is measurable and well defined physically, then the following equation holds:
where the magnitudes of the g terms (whatever they represent) have values which vary with the position of the points in the continuum. "Only when the continuum is a Euclidean one, is it possible to associate the coordinates x1,x2 etc with the points of the continuum - when we would then have":
In this case, says Einstein, relations hold in the 4-dimensional continuum which are analogous to those holding in our 3-dimensional measurements. However, this manner by which Gauss treated the problem is not always possible - it being appropriate only when there is a sufficiently small region of the continuum concerned that may be regarded as Euclidean. For example, where only a small area of the marble table is warmed, the geometric behaviour of the rods composing the small squares is almost as it ought to be according to the rules of Euclidean geometry. The imperfections of the overall matrix of the squares are not apparent unless and until such a disturbance is more extreme and/or in evidence 'over' (?throughout) most of the table (where it presumably becomes a problem in non-Euclidean geometry).
Einstein sums up Gauss's method for a mathemetical treatment of continua in general as one in which the 'distances' between neighbouring points of a continnum ('size-relations') (now of a cubic form, say) are numerically defined. Each such point is assigned as many number values (Gaussian coordinates) as the continuum has dimensions - therby identifying their positions therein - each such unique identification differing from adjacent points by an indefinitely samll amount. Such a (Gaussian) coordinate system applied to a continuum is a logical generalisation of a Cartesian coordinate system. It is also applicable to non-Euclidean continua but only to those small parts of same that behave nearly as a Euclidean system - with respect to the defined small 'distances' between the points therein.
We may next recall (see Section 17 above) that Minkowski showed how the 4 dimensions of relevance to the uniform motion of bodies (events) may be represented, per point of the continuum, by the Galiliean coordinates x,y,z,t and that if the time variable t is replaced by the imaginary value sq rt 1 times ct, thus effectively incorporating the role of the important ratio of the velocity (v) to that of light(c), we may regard the space-time continuum - as per the special theory of relativity - as a Euclidean 4-dimensional continuum (as shown in the preceding section). In the measurement of the perception of such uniform motion, the new transformation equations thus provide valid results which take account of the limitations occasioned by the constancy of the velocity of light. The very small magnitude of the distance between adjacent Galileian coordinate points (of the sequence of 'events' of the uniformly moving body concerned) is represented by the symbol ds. Its value is independent of the choice of the body of reference (as K or K'). [We hope eventually to better understand this latter statement. It seems to relate to the necessity of treating any eventual analysis of non-uniform motion (with Gaussian coordinates?) by means of calculus - to handle curved paths. We'll see.]
Einstein points out that understanding the laws of motion as based on uniform motion alone is too restrictive - as, for example, the fundamental law that states that (due to inertia) any body will continues to move uniformly in a straight line or remain in a state of rest unless subjected to (another) force - at least when measured relative to a reference K which is in uniform motion. But relative to other references (K'), which move non-uniformly, that law is not valid. Such different reference systems or bodies are thereby differentiated according to this character of their motion and whether or not laws of nature hold with respect to them. The question soon arose as to why there should be this diffentiation. Ernst Mach in particular recognised this difficulty and suggestd that mechanics had to be placed on a new basis - one which would conform to a more general principle - of relativity and a theory whose equations and laws would hold true not only for uniform motion (as in the special theory) but for any form of motion (where, conceivably, inertia plays a different role or is interpreted differently?). In pracice, this broader scope amounted to seeking rules consistent in particular with accelerated motion such as rotational - where the direction at least (if not the speed) of the velocity concerned deviates from the straight line strictures of uniform motion, or by that motion engendered in a gravitational field - where both the straightness of any body's path and a constant change of velocity can result. [Note Section 26 was covered above, before 24 and 25; 27 thus now follows]:
Section 27 - The Space-Time Continuum of the General Theory of Relativity is Not a Euclidean Continuum.
When considering the development of the special theory of relativity - within its uniform motion limitations, we made use of an imagined coordinate system within which any moving body could be imagined to travel (uniformly) along its x axis, but with its y and z axes also available for any movement required into those dimensions as well. Moreover, at the origin of these 3 axes (or elsewhere in the system), one or more clocks could be assumed by which the time of any such motion could be similarly measured - again in relation to an agreed reference body K. With these 3 spatial coordinates and one for time (ie x,y,z,t) a simple and direct physical interpretation of an actual moving body could be thus described - as in our earlier examples of the train, the missiles or walking man and the station. And, as indicated in Section 26, these 'space-time coordinates' can normally be regarded as 4-dimensional Cartesian coordinates of a Euclidean continuum. That interpretation leading to the special theory included the space and time effects which the constancy of the velocity of light of that theory accounted for. But, as described in Section 21 where classical mechanics and the special theory are shown to be unsatisfactory in more general circumstances, a general theory (which allows for non-uniform motion) can not retain (accept) this law of constant velocity - for any gravitaional field would disturb the direction and thus speed (ie constancy) of the light path concerned. It would, as Einstein points out, "always depend on the coordinates". [This needs expansion.] For the presence of a gravitional field invalidates the definition of the 4 coordinates (regarding uniformity) on which the special theory was based. If not the speed then the direction element of velocity becomes variable. Or, it could be both. The geometry need generalising as well - to take account of the variations in shape (space) which would arise where the lines of same (as determined by now 'non-straight' shafts of light say) may often be curved.
We were thus led to the conclusion that for any general principle of relativity that might validly apply (ie for non-uniform motion), the space-time continuum through which a body may so move (and the characteristics validly measured) can no longer be treated as a simple straight Euclidean one. For this more general case, we may usefully consider again the physical example of the marble tabletop with its local variations of temperature across its 2-dimensional continuum (of adjacent points). A Cartesian coordinate system could not be constructed thereon with rigid equal-length rods, due to the expanding effects of the disturbance (heat) applied to some local parts. Similarly, with a more general non-Euclidean continuum - as where the disturbance is a (more fundamental?) source of non-uniform motion - as a gravitational field - it is equally impossible to construct (mentally?) a 2, 3 or 4-dimensioned reference system (body) in or on which rigid measuring rods and clocks can validly indicate position and time (and thus curved path of the motion) directly. The rigid, straight line Cartesian/Euclidean continuum does not in that latter case correctly represent the actual 'field'/continuum through which bodies necessarily move, nor would inflexible measuring devices be appropriate for same.
This was the difficulty described in Section 23 concerning the use of clocks and rods on the rotating disc. But fortunately, the discussions of Sections 25 on the use of Gaussian coordinates showed us the way to overcome this problem. Thus, we may refer the 4-dimensional space-time continuum to Gaussian coordinates instead - and do so in an arbitrary manner. We assign to every point of the continuum (ie to every moment by moment position and time of the continuous sequence of the event(s) of interest) 4 numbers - x1,x2,x3,x4 (coordinates) which have not the least direct physical significance but only serve the purpose of numbering the points of the continuum in a definite if arbitrary manner. Nor are we even required to differentiate such coordinates as either space or time coordinates (since velocity entails both equally). We are thus dealing with an abstract, theoretical conception that will hopefully allow a valid analysis of actual behaviour of bodies not necessarily moving (at whatever velocity) uniformly (and/or whose reference bodies can be viewed similarly). But is this abstract description of our real world adequate for this purpose ? How can such arbitrarily numbered coordinates be of meaningful value for a representation of real moving objects over real space and time ?
We may try to answer this reasonable question, says Einstein, thus: We imagine a material point in motion (somewhere). If it had only a momentary existence without duration (just as its position and magnitude is virtually without occupying space), it could be described in space-time by a single system (?set) of Gaussian coordinate values - x1,x2,x3,x4. Its actual permanent existence would have to be characterised as a miniscule part of an infinitely large number of such sets of coordinates which are so close together as to constitute a continuity (of the point's motion over time and space). [We seem to be entering here the mysterious world of the infinitely small in which the calculus is magically utilised in order to give greater accuracy to very gradually changing rates of motion.] We thus have, to this point, a uni-dimensional line in a 4-dimensional continuum - which can thus correspond to many points in (non-uniform) motion - with their speed and/or direction not remaining constant. The line could even be straight (uniform) - as a special case of the generality of otherwise curved (non-uniform) motion with which we are mainly concerned here.
How can we utilise this abstract conception so that some contact with real, physical motion can be analysed ? Einstein points out that the only statement regarding such points which can represent a physical existence are those concerning their 'mutual encounters' - with points from other such lines - where they would necessarily share the very same coordinate values x1,x2,x3,x4. Such encounters constitute the only actual evidence of a space-time nature that we can represent with physical statements. When we described the motion of a material point of a body relative to a body of reference, we stated nothing more than 'the encounters' of this point with particular points of the reference body. We may also determine the corresponding values of the time by the comparable observation of encounters of the points of the body with relevant points of the clock's hands and dial. Positions in space are measured similarly by means of measuring rods.
Based on the foregoing, Einstein suggests that the following statements hold generally: Every physical description about a moving body resolves itself into a number of statements which refer to the space-time coincidence of the two events concerning A (the moving body or specified points of same) and B (what was formerly called the reference system or body but now as another 'event') and do so in terms of the 'agreement of their respective Gaussian ccordinates x1,x2,x3,x4' (that is, their above-described 'encounters'). This 'Gaussian' description of the space-time continuum (x, y, z, t) completely replaces the former description which utilised a more definite or concrete body of reference - without suffering the inadequacies of that latter mode of description. That is, it is not tied down to the Euclidean character of the continuum that is so represented when seeking to analyse the non-uniform (variable) motion of bodies in such as gravitational fields.
Section 28 - An Exact Formulation of the General Principle of Relativity.
We are now in a position to replace the provisional formulation of the general principle of relativity (as given in Section 18) by a more exact one. The provisional form was: " All bodies of reference K, K', etc are equivalent for the description of natural phenomena and the formulation of the most general laws of nature - whatever may be their state of motion". This had assumed that such bodies of reference were of the same rigid form as applied in the fomulation of the special theory. But where the motion of concern is of a non-uniform character (as on a rotating reference or in a gravitational field), the reference body in relation to which such motion is manifested, is more accurately represented, says Einstein, by a Gaussian coordinate system - which is not rigid but more malleable in nature. This may be expressed by a more formal statement of the general principle of relativity thus; "All Gaussian coordinate systems are essentially equivalent for the formulation of general laws of nature". This can also be expressed in a more general way - one that is less modelled on an extension of the special principle; this should prove to be even clearer. Thus, according to the special theory, the equations which express the general laws of nature 'pass over' into equations of the same form when, by making use of the Lorentz transformation, we replace the space-time variables x, y, z, t of a Galilean reference body K by the space-time variables x', y', z', t' of a new reference body K'. But according to the general theory of relativity, substituting instead arbitrary Guassian variables x1, x2, x3, x4, is what allows the equations to pass over into equations of the same form (more directly?). This is the case, as every transformation (not only Lorentz's) corresponds to the transition of one Guassian coordinate system into another. The special theory of relativity deals with a Galileian world of uniform motion of bodies (with their 'isolated' material points) being referred to rigid reference bodies. Non-uniform motion, as in a gravitational field, is not normally a consideration in that world (or domain). But we may examine the situation in which a moving body of a Galileian domain (of otherwise uniform motion) is referred to a non-Galiliean reference body - ie one moving in a non-uniform manner (as in a gravitational field). A gravitational field is then effectively present with respect to such a reference body. [Sections 20 - on how the equality of inertial and gravitational mass supports the idea of a general theory of relativity - and 23 - on the behaviour of clocks and rods on the rotating disc - may be reviewed in this regard.] That is, we can consider the special case where the degree of non-uniformity of the motion concerned has been reduced until it is at that limit of such motion - namely, has become effectively uniform but is to be treated by the same analysis as all the more usual non-uniform motions in nature - as that in a gravitational firel. But in such fields, there are effectively no such things as rigid reference bodies with Euclidean properties - which are therfore not available for analysing the motions of bodies considered under the general theory of relativity - even those of (just barely) uniform motion. For their reference systems must essentially reflect the reciprocals of even the almost uniform motion so considered and so be non-rigid in form. The motion of clocks (and thus the timing and velocity of all such moving bodies) is also influenced by gravitational fields and therefore would also be best analysed in this more general way; that is, in terms of the general theory. The definition of time within the latter theory may thus necessarily appear less plausible than when handled within the more restricted special theory with its known, uniform motions and rigid (unmalleable) reference systems.
The reference body or system to which non-uniformly moving bodies may be referred when analysed within the dictates of the general theory of relativity are treated as non-rigid and as moving (ie relatively or reciprocally at least with respect to the moving body for which it has been ostensibly chosen as the agreed reference). If the body for which it is serving as reference is moving non-uniformly, then it follows that its own reciprocal motion (ie as viewed from the point of the otherwise moving body) would also be non-uniform and that the latter's theoretical motion less predictable; it would also suffer alterations in its non-rigid form during such relative, unpredictable motion. And the definition of time as based on the motion of a clock's hands which can follow no rigid laws of motion would also become more malleable. Any clock utilised to determine the time dimension (and velocity) of such movement may be placed at some agreed location but on a non-rigid reference body; its accuracy would thus require special means of analysis. If the readings observed simultaneously on adjacent clocks differ from each other by an indefinitely small amount, such clocks satisfy the condition necessary to allow an analysis (by means of the general theory) of the velocity of any body moving non-uniformly. Whether as space or as time, this approach is suited to the continually changing rate of velocity (ie non-uniform) as depicted by curved rather than by straight lines; these rates of change must be analysed by means of calculus principles. And the reference body chosen must itself reflect these changing rates.
Such a non-rigid reference body is essentially the same as an arbitrarily chosen 4-dimensional Gaussian coordinate system but may with advantage be viewed as though it were something more like a 4-armed octopus (a ?quadropus) of quite variable form - each arm representing a different dimension of space or time. The conception of the reference system utilised in the general theory as such a malleable 4-armed 'quadropus', rather than as a typical rigid Gaussian coordinate system, allows one to better visualise three spatial coordinates independent of a time coordinate (even though such independence need no longer be adhered to). For every point of this malleable 4-dimensional reference system may be validly treated as equivalent to a space point - with which every material point of any body referred to it can be conceived as being at rest.
On the above basis, Einstein then makes the following relevant assertions (paraphrased) with respect to which we will hopefully gain further insight as to their exact meaning and significance: 'The general principle of relativity requires that all such 'quadropus' conceptions (of relevant reference systems) can be used with equal right and success in the formulation of general laws of nature - with the laws themselves being quite independent of the particular malleable reference system of this type chosen (of which there must be many conceptions, seemingly). Its as though he is saying that 'any tetrapodic octopus (our quadropus) will do'! The great power of the general principle of relativity lies in the comprehensive limitation which is imposed on the laws of nature in consequence of what is implied within the conception of the reference system described above'. [The exact character of those limitations and how they imbue that principle with such power will hopefully be more fully expounded (with examples) below.]
Having introduced our subject in terms of the early topics presented in the 1920 book as above, as well as two sections from Part A of the 1915 Foundation paper, we now address certain other topics covered in Part A. We shall attempt to discrern any common thread in these two differing approaches to the basic kinematics of general relativity.
He begins by....... (conts)
Section 29 - The Solution of the Problem of Gravitation on the Basis of the Principle of Relativity.
The title of this section seems to imply that we are already aware that there is 'a problem with gravitation' and that it requires a solution. Are we ? And what is it ? Seemingly, it is essentially the problem of how to deal with the motion of bodies undergoing non-uniform motion generally (as compared to that handled by the special theory of relativity) but the problem is to be focused on and answered through the specific topic of gravitation, which has been effectively 'parachuted in' to represent that more general problem - to be resolved thereby presumably by a general theory of relativity. We might note that while Einstein began his work on this more general theory (from about 1910) on the basis that our understanding of the nature of motion should probably not be restricted to just the special case of uniform motion (as he published in 1905), one of the examples of such non-uniform motion which such a new theory should be able to analyse - that due to the effects of a gravitational field - seems to have pushed aside any attention that might have been addressed to that initial more general problem; rather, it has itself, within the orbit of general relativity, become the main focus of enquiry in its own, albeit important, right. Be that as it may, we apparently now proceed solely along that more restricted track - of gravitation.
Thus, the considerations of the foregoing Sections may be seen as providing essentially the means (methods) by which we are now able to solve the 'problem of gravitation' (seemingly, and ?necessarily, within the orbit of general relativity - as a means to better elaborate that theory). These methods would appear to consist primarily of measuring the motion of bodies moving non-uniformly relative to a given reference system which is (apparently necessarily) conceived as possessing (almost magically) a (?theoretical) non-rigid, malleable quality. Unfortunately, this latter conception is thus not exemplified (to this point) by any actual such reference system with respect to which any actual body's motion is so measured. It appears to have been introduced simply in order to provide a relevant structure on which the mathematics required to solve the problems referred to (regarding gravitation) may be generated and justified. To develop this approach, Einstein begins again by having us consider the original Galileian domain (of no gravitational field affecting the relevant reference system/body K). In such a situation, we can analyse the motion of a straight and uniformly-moving body relative to a chosen reference body - as the train passing a station, or a bullet fired down the aisle of that moving train - where we can validly utilise an imagined rigid frame of orthogonal coordinate axes (with attached measuring rods and clocks) superimposed upon either the station or the interior of the train (as required) - by which we can measure the distance/direction and the time (and thus the velocity) of the uniformly-moving body concerned as the delayed information about these (and the resultant variable, not constant, magnitudes of time and space is thus affected - as explained by the special theory of relativity.
This situation may now be re-considered in which the moving body is instead referred to the type of 'random Gaussian coordinate system' described above (ie our 'quadropus'?) - serving as the (?theoretical) reference body K'. In the first situation, the body is moving uniformly by virtue of the train's engine exerting a steady uniform force arranged such that it exactly overcomes whatever mass, air pressure and track friction (partly contributed to by gravity) would oppose that constant forward motion. The gun powder in the case of the bullet similarly provides such force that it too exactly overcomes opposing mass, air pressure and gravity such that the bullet moves at its assumed steady velocity. In the present case, the motion of the undescribed moving body concerned (ie whether uniform or non-uniform) is not (at this point) described. However, we might reasonably assume that 'it's motion (whatever 'it' is) is non-uniform, as this motion is the subject of present interest. If that non-uniform motion was in fact one of steadily increased velocity (ie steadily maintained acceleration) - as arises in a gravitational field for a free-falling body, for example, the direction of that body's motion would normally be assume to be vertical and downward rather than horizontal and forward, as pertained in the case of the uniform motion examples described earlier.
In any case, Einstein then asserts that "with respect to our malleable reference body K' (at least), there is an associated gravitational field (G) of a particular kind" (emanting from or acting on it (?), and/or on our moving body (?). And, in this case, we learn of the 'behaviour of measuring rods, clocks and the freely-moving body of concern (or its represenative 'point') - as they now relate to the malleable reference body K' - 'simply by mathematical transformations'. [Which will have to be more fully described.] This contrasts with the distances and times more directly measured with such rods and clocks attached to the albeit imagined rigid coordinate axes in the case of uniform motion - where the receipt of the relevant information about these (described here by Einstein as 'the behaviour' of such measuring devices and of the moving material body concerned (or its 'point') was delayed according to the v to c ratio. Presumably, such a (significant) delay is still the case with respect to their 'behaviour' in the gravitational domain ?
This influence by the gravitational field G on such 'behaviour' (measurement outcomes?) is interpreted by Einstein by means of an hypothesis he then introduces: 'this influence acts according to the same laws even when the gravitational field is not derivable in terms of the special theory (of uniform motion) - simply by a transformation of coordinates' (seemingly from rigid to malleable ones). [This would benefit from further explanation.] When that field is derived from the special case by such transformations, we may investigate the effects on the measurement of time and space for a moving body as a comparison. The outcome may be formulated, says Einstein, as a valid, if not fully general, law (concerning gravitation, I believe) no matter 'how' the malleable reference body (K') used in the description (or analysis?) is chosen. [It would be helpful in this regard if he had given us some idea of just what are the possible different ways ('hows') such malleable reference bodies can apparently be chosen - albeit it not mattering which! Another kind of Alice-in-Wonderland scenario here it seems.] The law concerned is thus not (yet) 'the general law of the gravitational field' (the necessity for which seems not yet to have been explained but the implication seems to be that such a law is apparently required or will be useful to some desirable if as yet undescribed end (as how to predict outcomes) - since the gravitational field under consideration is of a special rather than general kind. This more general law may however be determined, says Einstein, by stipulating certain demands which the more restricted law mentioned above should meet. These are that: (a) it must satisfy the general principle of relativity; ie that.... (This principle permits us to determine the influence of the gravitational field on the course of all those processes which take place according to the known laws of motion when such a field is absent - ie which have already been shown to accord with the special theory of relativity. We can then proceed in any analysis of such motion according to the method explained previously when measuring a body's motion with measuring rods and clocks); (b) only the inertial mass of the moving body subjected to the non-uniform motion concerned (here due to being acted upon by a gravitational field) and in particular 'only the energy' (of that mass) is relevant in 'exciting' that field (to so act upon it); and (c) both the field and the body together must satisfy the law of the conservation of energy (and of impulse). With these stipulations, he claims, we can derive a law or theory of gravitation which has many advantages in understanding and predicting the behaviour and processes of moving bodies no matter by what form of velocity they are moved.
To this point, Einstein's explanations of those processes in this domain of general relativity have relied on a very abstract analysis with a seeming avoidance of any concrete examples of solid moving bodies referred to actual agreed reference bodies. Moreover, the latter category has been described in another make-believe manner (as a 'mollusc') which rather than being superimposed onto some actual reference body (as the rigid coordination frame with clocks was so superimposed onto such as the railway station or the train for the special theory) seems now to be considered as the reference body itself (and possibly the source of the 'force' which impels the body). At least with Alice, we had a given moving body (Alice) and an understood reference frame (the rabbit hole), although with the same source for the necessary force. And we may recall that it is 'how' the 'mollusc' reference is chosen, not 'what' is chosen to represent it, that apparently does not matter (despite us not knowing what are the range of possible 'hows', never mind the range of possible 'molluscs'). [We must recall, however, that in Section 22 (Inferences from the General Principle of Relativity), Einstein notes that he will be focusing on the properties of the gravitational field (in particular) as they (and thus the general theory per se possibly?) may be derived from the general principle of relativity in a purely theoretical manner - ie he has actually warned us! The wording of that principle may be usefully reviewed; see section 28.]
In Part C of his Foundation paper of 1915, Einstein addresses the 'theory of the Gravitaional Field' and we consider this at this point in the event that it may give a different slant on this area compared to that of his 1920 book. [Or should this fit in earlier ??] [Conts.]
We also interject here three paragraphs which seek to consider actual material bodies to which the foregoing abstractions concerning gravitation might conceivably apply (although they may be quite inappropriate). We imagine a scenario in which we have two stationary trains (A and B) side by side at the top of an extremely high mountain (say 10 times the height of Everest but much steeper) each on their own set of tracks and ready to have a race to the bottom. Normally, their forward (actually essentially 'downward' here) velocity is, as with our horizontally moving train in our special relativity cases, controlled such that they would be expected to travel (due ro the force engendered by their engines) at a constant speed which exactly overcomes any opposing (or, here, enhancing) factors or force - as friction, air pressure, gravity, etc. To effect this, the mountain-descending trains would have a system of extendable metal 'flaps' to 'catch' the wind and other braking systems which would result in their velocities again being uniform - at say 100 mph - and so not accelerated. If all was as it should be, they should thus finish in a virtual tie.
However, when they start the competition it is apparent that the braking systems of one train (say B) have failed from the very start and it is soon travelling at an enormous and ever accelerating speed; literally as though 'falling off a cliff'. Train A's motion would be appropriately analysed in the same manner as in the case of our horizontal trains with the stationary side 'embankments' (virtually vertical) now serving as the reference body K for the train and for any 'missile' experiments pursued thereon (their rsepective 'downward' speeds also being controlled to be uniform not accelerated; ie the effect of gravity on them is also temporarily suspended). As before, we would expect (and find) that the limitation in light's speed would, after all measurements were performed with the measuring rods and clocks required, result in the same discrepancies as noted before - depending on the v to c ratios - and that the correct transformation equations would then give results in accord with the special principle of relativity.
The non-uniform velocity of the uncontrolled train, subject to the unfettered force engenderd by the gravitational field of the Earth (which would be very largely below or beneath it), would also be analysed in relation to the stationarry vertical embankment that it would be travelling past - as its reference body. In this situation, we apparently must now apply our measuring devices onto this passing 'cliff' as though it were a mollusc-like, multi-armed creature (K') to arrive at certain conclusions regarding the train's accelerating velocity and how time and space and the missiles on board are affected therein - which conclusions should, it seems, accord with a general theory of relativity. Possibly, we could (?should) therefore analyse train A's motion by this same method - by treating its albeit uniform motion as a special case of otherwise non-uniform motion and obtain the same results as by the former, more usual treatment; this may well follow if the theory of relativity is indeed truly general; in this latter case, the quadropus would presumably become suddenly rather stiff and orthogonal in form. Or would it ?
It may be suggested that we could have envisaged a comparable train race on horizontal tracks in which the control mechanisms of train B's engined-powered force is (again unexpectedly) not opposed by the usual factors (of air pressure, gravity and friction) so that its initial speed of a steady 100 mph, being unopposed, soon fulfills Galileo's prediction and keeps increasing in velocity (especially if gravity is replaced by a coalman furiously shovelling more and more coal into the pre-war engine) while train A again continues at its uniform 100 mph. Our analyses should proceed in exactly the same manner as in the vertically falling situation - with the same results. In this case, the train station or nearby embankment would be treated by means of the measuring devices applied either to the rigid coordinate frame superimposed thereon (as reference body K for the uniformly-moving train) or to our malleable, octopus-like reference body K' - for our accelerating train - with essentially the same results as described above (especially if we assume the acceleration matches that due to gravity, say). However, whether either of these attempts to exemplify the situation for non-uniform motion of actual bodies can be deemed legitimate or even useful, one is still in the dark as to how Einstein's 'encounters' of the lines (succeeding points) of the curved space or time (and thus velocity) of non-uniform motion arise (at some mutual 'cross points') with respect to any moving body and its chosen reference(s). Are they not solid objects ? How do the lines cross ? Maybe they are utilized not to calculate values for specific cases but simply as a means of determining theoretical relationships by which we may arrive at the most general laws of motion - to eventually apply and confirm in real situations elswhere ?
We now resume our analysis of Einstein's Section 29 on deriving a law for gravitation based on the general principle and theory of relativity (eg possibly in order to fulfill exactly that latter suggestion). [We might usefully review Section 22 (A Few Inferences...) re deriving properties of the gravitational field in a purely theoretical manner. In any case, the influence of the gravitational field on all the processes which accord with the known laws of motion as initially determined for the special theory of relativity (with no such field present) can, he points out, be determined by the same methods that were used to analyse the use of measuring rods and clocks when measuring the motion of freely-moving material bodies (or their representative 'points') within that special domain. [Note: Einstein doesn't appear to detail this again here so that we may also have to review such methods (as in Section 23).] Rather, he simply notes that the theory of gravitation so derived from the general principle of relativity (or vice versa?) - which he is effectively keeping from us short of our reviewing that material - has many advantages in aiding our understanding of the nature of motion generally (and thereby answering the question as to why we were apparently seeking such a theory).
We may list these qualities here: 1. The theory is 'beautiful'. Possibly this means that it is what is often termed 'elegant' in the sense that its clarity, balance and simplicity should prove more effective in predicting and understanding nature than any more complicated theory that did not possess this quality; 2. It removes the defect of classical mechanics as described in Section 21 (which, again, we should review); 3. It aids in the interpretation of the empirical law of the equality of inertial and gravitational mass; 4. It explains an observation from astronomy for which classical mechanics has no answer. This concerns the matter of the elliptical orbit of the planet Mercury which rotates in the plane of that orbit with respect to the fixed stars by a very small but observable amount (of 43 seconds of arc) per century. This is very significant with regard to the validity of the Theory (see below).
We are thus shown (here) neither how this theory was derived (even theoretically) nor exactly how it may be verbally described - ie how to define it. He simply proceeds by describing an application of this new theory (its actual character and detail, obscured in the mystical(*) realms of tensor analysis apparently) taken thus on trust) to the case underlying point 4. above where the gravitational fields concerned may be regarded as weak and in which any body (mass) moves essentially in terms of its (acceleration) influence relative to "the coordinating system" - with velocities (v) which are small compared with that of light (c). When thus restricted, we obtain Newton's theory of same, as a first approximation. [How odd that the reference system here (K') for non-uniform motion - on which so much complex theoretical explanation and description has been provided above (viz: the mollusc) - is virtually dispensed with here by the above meagre and unqualified phrase in quotes; it suddenly seems almost irrelevant!] The theory of gravitation as derived within the theory of general relativity (or vice versa?) does not rely on any particular assumptions in contrast to Newton's theory (of gravitation) which had to assume that the force of attraction (eg due to gravity) between mutually attracting bodies (and/or their material points) was inversely proportional to the square of the distance between them. If the accuracy of the calculations based on the new theory are increased sufficiently, however, deviations from the results as calculated by Newton's methodology (eg in regard to planetary orbits) begin to become apparent although they are so slight as to have escaped actual observation or measurement up to the 'present' time (1920s?).
(*) We may note here that even Einstein refers (in his original paper) to a tensor as some kind of 'a thing' which is somehow defined by its own 'components' ('boot straps'?) which apparently allows one to transform a body's non-uniform motion from one set of coordinates to another. One is somewhat reminded of the recent prime mortgage/credit crunch scandal in which many unsafe mortgages were bundled together with better ones then 'sliced up' and sold on as 'derivatives', which were then re-packaged and sliced further as derivatives of derivatives, etc until finally no one knew the value or significance of what was being bought or sold or paid for by whomever.
In any case, the application of Newton's theory to the motion of the planets should conclude that they move around the Sun in elliptical orbits which maintain their positions with respect to the fixed stars - once account is taken of the influences of the stars and planets themselves. Tests of this deduction can be made to great accuracy and have been confirmed at the level of precision 'presently' (1920s) available (which turns out not to have been quite precise enough) for all the planets save one - that of Mercury. For that orbit alone, it has been observed and measured not to be stationary in this respect but to slowly rotate - as mentioned in point 4. above. The classical mechanics of Newton sought to explain this odd result (not consistent with the rest of that theory which predicted and found no such rotations for all other planets whereas in fact it was there) by applying certain improbable post hoc hypotheses with little basis in nature, devised solely for this one atypical planet. This contrasts with the prediction of the general theory of relativity (with its law of gravitation) which finds that the elliptical orbit of every planet around the Sun must necessarily rotate as described above - even if current measurements are not always accuate enough to confirm this (thereby maintaining for a time the false belief of the Newtonian theory). And by the same reasoning, the general theory predicts exactly Mercury's more clearly observed rotation of 43 seconds of arc per century and does so again without the need of any such one-off hypotheses.
Conveniently, there are two other deductions from the general theory that can also be tested by empirical observation. These are the curvature of light's rays by the gravitational field of the Sun (as observed by Eddington in 1919) and the displacement of the spectral lines of light reaching us from the stars as compared with those produced from the same types of atoms terrestrially (as reported by Adams in 1924). Both confirmed the validity of the theory. In all three tests we may note that the general theory of relativity which was initially intended to explain the laws of motion of all forms - both uniform and non-uniform - appears to have been focused increasingly on that particular form of non-uniform motion that arises due to gravity (and its field). The theory has thus become essentially a theory of the gravitaional field. This may be quite justifiable in that such a field is effectively everywhere - affecting all motion (and seemingly spatio-tempotal geometry). However, the theory also seeks to explain certain other observations for one of which, again, Newton's theory required questionable hypotheses. This concerns the matter of the structure of the universe where, again, gravitational factors likely apply. For a more understandable and detailed analysis of the general theory of relativity overall, we must, it seems, refer next to others' interpretations. I suspect however that they will require too much calculus and associated mathematical (tensor) reasoning for this particular 'analyst'. [An alternative option may be to examine the less technical approach to explaining general relativity now provided on Wikipedia (not available when this present analysis began) or from similar overviews posted recently on the Internet by which means one may be able to correct any mis-interpretations shown here and/or augment any deficiencies.]
Comment.
During the foregoing analysis of the general theory, one became gradually aware that this theory was increasingly more concerned with a particular theory of gravitatiion (as but one of many forms of non-uniform motion) rather than with a more broadly focused generalisation of relativity that might have grown out of the quite restricted special theory - dealing with uniform motion only. It was therefore of considerable interest when I noted a reference to a recent article of August 2008 by a division of the Springer scientific press (SpringerLink) which reproduced abstracts of a symposium of 60 invited scientists held in Berlin in 1965 on the 50th annniversary of the publication of Einstein's general theory - which followed a lecture by Einstein there on 4 Nov 1915 on 'General Relativity and Gravitation' (as an immediate precurson to publication of his Foundation paper on General Relativity in December). The symposium's papers were subsequently published (by...?...) as a 10 page article in the Bulletin of General Relativity and Gravitation in Jan 1966 (No. 10/1).
One of the contributors, Professor V.A. Fock of Leningrad University, delivered a lecture on 'The Fundamental Principles of Einstein's Theory of Gravitation' in which he expressed the view that (The theory of) "..General Relativity...is not a generalization of anything, but is a theory of gravitation..". (I don't doubt that others have since voiced that same view.) It rests, he said, on two assumptions (postulates?): (1) that space and time are unified (possibly as per Minkowski?) and (2) that the metric of the coordinate frame lacks any rigidity. Fock seems to replace the concept of Einstein's malleable coordinate 'tetrapod' (quadropus?) by the attractive idea of 'harmonic coordinate systems'. [I suddenly visualise my quadropus 'dancing to the music of space-time'!] Reference during the symposium was also made to the question of 'Mach's principle' (which assumes that 'Inertia has its origin in the distribution of mass over the whole universe" - a topic first written about by Einstein in 1913 and later the subject of some disagreement with relevance to his theory. I have not yet seen the full 10 page 'Bulletin', but hope to do so.
Fnally, we place here n Introductory coverage of Quantum Theory (and its Mechanics) as foreshadowed in our Introduction (to Relativity per se). It is contained within:
Introduction.
The above-titled book was published in 2010 as part of an irregular series in which Hawking describes, for the layman; the latest conclusions arrived at by himself and colleagues in their particular spheres of physics. Any later books by him on this subject (ca 2012-16, say) may advance these ideas further and, if so, they will be considered here on a later occasion. The title of the present book refers to the goal of much then current thinking in theoretical physics; that is, to achieve a comprehensive understanding of all the processes involved in the creation and subsequent evolution of our universe, including the emergence of life therein and our own existence; Rather a tall order ! A key feature in accomplishing this final goal is seen by Hawking as successfully applying in particular recent advances in quantum mechanics, in conjunction with aspects of general relativity theory. They have apparently contributed equally towards the best regarded theory (at this point) dedicated to this ultimate goal – called rather unrevealingly ‘M-theory’ (apparently developed from Superstring theory).
While earlier advances in physics provided some of the foundations for these general topics, they are mentioned only en passant by Hawking on this occasion. We shall begin therefore with a short review of our own on these important foundation aspects - seeking to fill certain gaps in his coverage of that pre-quantum era background. The present book thus gratefully provides a convenient structure on which to append such earlier material, as well as our associated comments.
[Note: Since writing the foregoing, one has become aware of a book similar to Hawking's - produced by the British scientific periodical 'New Scientist' in 2017 – called ‘Where Does the Universe Come From ?’ It too appears to address very similar issues (possibly motivated in part by Hawking’s earlier work) and, again, is based on a rather brief coverage in its first 2 (of 9) Chapters of the quantum and relativity foundations as a part of this much broader topic. We may similarly seek to scrutinise and comment on its various assertions therefore – if they seem at variance with one’s present understanding. Such quick ‘overview introductions’ to a subject often provide useful examples of some short-cut explanations which can either spur further study or encourage possible misinterpretations. In either case, one’s own analysis and understanding should hopefully be assisted by considering such contributions in like vein. Finally, Hawking’s ‘Grand Design’ may well have some parallels with Einstein’s long sought ‘Grand Unified Field Theory’ which seemed to be premised on there being no need to incorporate quantum mechanics in same, although Einstein continued, if separately, to give continuing thought to that then still developing sphere.]
General Background.,
1. Relativity Theory
In 1905, Albert Einstein published four important papers in physics, one of which was later referred to as the Theory of Special Relativity. It provided an improved explanation regarding the motion of bodies moving uniformly through space and time (on which many of the events of nature depend - which we then wished to better understand). This would include an awareness of the previously unsuspected effects of the constant speed of light on the perception and measurement of the light-conveyed information concerning measures of those many fundamental motions in nature. These procedures, which we may call ‘relativistic mechanics’, were an improvement on former methods of such calculation based mainly on Newton’s and Galileo’s earlier ‘classical mechanics’ - which hadn’t considered that latter role and effect (of light) – although otherwise was still perfectly satisfactory for most practical problems in engineering and everyday life - where motions transpire at more ordinary speeds. But this new Special theory and its mechanics did not incorporate methods to calculate motions that were not simply uniform (due essentially to the continuing inertia that results from such ‘one off’ impacts of forces onto various movable bodies) - but those, also, that were non-uniform (ie accelerated) in their typical expression - as arise especially through the on-going and virtually ever-present effects of gravity. Newton had considered this latter aspect also – in his early theories of motion - but did so in terms of a quite separate theory to that focused on uniform motions – designed to account in addition for such accelerated motions due to the presence of this continuously-acting, fundamental force; that is, as a kind of ‘add on’, separate theory, to his more basic conceptions of the general laws and mechanics of motion.
However, Einstein decided that he needn’t formulate a comparable separate theory for handling (analysing) such accelerated motions as well, but could incorporate understandable procedures to deal with them more directly - within his existing, if new, relativity theory (with its new mechanics) – providing that he simply generalised its relevant principles appropriately. He thus produced his broader General Theory of Relativity – in 1915 – to this end. ‘Classical mechanics’ was thereby further extended - to handle all forms and speeds of relatively moving bodies – in terms now of this further generalised ‘Relativistic mechanics’.
While the more general theory of relativity could (in theory) now handle all forms of motion, including those many accelerated forms due to gravity, the theory and law of gravity per se that arose and evolved out of this new general approach (by 1917) soon became the driving force in investigations of much astronomical research as well - where gravity’s newly conceived character - as a function of curved space-time (as well as of any attracting forces between large objects as planets and stars) - proved to be of more fundamental significance, with even wider implications. The General theory of relativity was thus a prime contributor to further advances made from that date - into that wider astronomical sphere of research in particular. Thus, the search began for evidence of gravity waves, black holes and theories concerning the creation of the universe itself - at the ‘big bang’. It is these latter phenomena with which Hawking’s presently reviewed book was chiefly concerned - seeking a ‘grand design’ and becoming thereby an even broader ‘theory of everything’, as it were. It is that post-1915 analysis of Relativity that more contemporary thinkers tend to focus rather than its earlier (pre-1915) fundamentals.
{Conts.]} To the Relativity Homepage