The Theory of Relativity was formulated by Albert Einstein in two Parts at the beginning of the 20th century. The first Part, published in 1905, was described later as the Theory of Special Relativity - to distinguish it from the second Part, published about 10 years later - the Theory of General Relativity. This present brief account, which follows longer versions above, is concerned primarily (as were they) with the earlier Special Theory which first appeared in a German periodical under the title 'On the Electrodynamics of Moving Bodies'. It will also serve however as an introduction to the more general theory - to be considered subsequently.
Being a theory, the earlier Part was designed to provide a suggested answer to a particular problem - one which, around 1900, was being recognized as such only gradually by the world's major physicists. It's described here firstly in a form that was even less appreciated as a problem initially (nor even today) but which might serve to clarify the more fundamental difficulty that the special theory was intended to resolve.
To describe such problems, we require a set of two reference systems in relation to which we may measure the motion of various moving bodies. For such motion can not be measured against a featureless vacuum; some agreed reference criterion is necessary. Thus, a moving train may be imagined passing through a railway station. The train and the station can represent the two reference systems needed - one moving and the other relativley stationary. An aeroplane passing near a high hill, a fast ship passing a dockside or the moon circling the Earth could represent other mutually relevant reference systems. In any such case, it is convenient to describe the systems concerned as system M (for the one that more obviously moves in relation to the other) and system S (for the relatively stationary one). We shall use mostly the train and station example here. Within the moving system itself, various smaller bodies are also able to move and such motions can then be measured in relation to that, their more immediate local system, or in relation to the external 'stationary' system to which the train's motion itself is typically referred.
[We may mention here that when most people are asked what they think Einstein's theory of relativity is about typically reply in brief that "...it shows that everthing is relative". Well, many things are indeed relative - such as size, length, weight, noise, temperature, colour, etc, etc. Such things aren't very accurately described by simply saying something is long, or heavy, or loud, or hot or whatever, as all such descriptions are 'relative' to some agreed standard. One person's long or heavy may be someone else's short and light, etc. Einstein's relativity is however not particularly of this kind. Rather, the special theory concerns specifically the motion of smoothly moving objects (of any sort) relative to some (any) agreed reference system whose own velocity, if also smooth and uniform, does not affect the forces and laws that determine the motion of such moving bodies. Galileo was the first to address this matter and come to that important conclusion. But Einstein would greatly improve this idea with his more considered theories.]
We begin by imagining, in turn, three such moving objects (or 'missiles') that can themselves move within the moving system M. They could for example travel along the aisle of a moving train's carriage, as a ball thrown at 30 mph between two men standing at each end, a bullet fired from a gun at 500 mph from one end to a target at the other, or a pulse of light released from a light source to another such target (which light, as we know, travels at an extremely fast speed). In each case, we imagine the missile beginning its short journey at the left (rear) end of the carriage as observed from the station (system S) which travels down the aisle towards the right or forward end, being the direction in which the train is travelling. We may assume in each case that the missiles travel at a steady, uniform speed from start to finish of a measured distance - without a brief acceleration phase (conveniently ignored for present purposes). The moving system M itself (here the train) also moves in this smooth, uniform manner. [It was this qualification - of the systems' motions (and those of the bodies moving within or on them) being uniform that accounted for the earlier theory being referred to as the 'special' theory - in contrast to a later, more 'general' theory - where the motions concerned needn't be so restricted but could be variable, accelerated or even uniform (again, as a 'special' case of that more general conception.) But the special case of 'just uniform motion' happened to be addressed first because of a specific problem concerning motion which was probably best approached in the first instance by means of this more restricted perspective.]
In the first instance, we have an observer with quite accurate timing and/or velocity-measuring devices positioned within the carriage before the train begins its journey. He records the speeds of each missile in turn and so confirms that they do travel in relation to the inside of the stationary carriage at the speeds described. As a part of these measurements, he will utilize an accurate clock and measuring rod to obtain the relevant timings and distances underlying such speeds. Another such observer is in the station and he finds (by measuring what he can easily see through the stationary carriage windows) that the missiles' speeds noted on his identical timing devices, etc agree exactly with those found by the observer in the temporarily stationary train - whether measured in relation to his own reference system (the station) or in relation to the equally unmoving carriage immediately in front of him.
We now have the train start its journey some distance before reaching the station which it then passes through, from left to right, at a steady 100 mph. The speeds of each missile are then again measured by both observers as the train passes through and beyond the station. The observer on the moving train finds that each speed, again relative to the inside of the now moving carriage, is exactly the same with regard to that reference system as it was when the train was not moving. The stationary observer however finds that the speeds of the two slower missiles, for example - the ball and the bullet - now appear to be travelling in relation to his station at speeds that are now 100 mph faster than he found previously; that is, he now finds they travel at 130 and 600 mph, respectively, as the train speeds through and away from the station. We may have to assume that the stationary observer can observe the moving bodies in the train more from behind the carriage than from its side, as it speeds away (which may require the use of trigonometry when calculating the velocities concerned). In any case, his attention is focused entirely on the speeds of the moving bodies, not on the train itself.
These faster speeds by the moving bodies, from the stationary observer's point of view, would be as generally expected, the extra speed shown by them being of course due to that of the train. The speed of our third missile - the light - is however so fast that it is unlikely that one could accurately measure any difference in its enormous speed that might be expected due to the comparatively small extra speed of the train. However, from the results of the first two missiles, any observer might well reasonably conclude that its speed too would most likely be 100 mph faster than found previously (had he been able to so measure it).
The stationary observer then confirms that the train did indeed travel through and beyond the station at exactly 100 mph and so by subtracting that speed from the readings he just obtained, he finds that the results now agree precisely with his earlier findings for the missiles in the previously stationary train. Both the observer on the moving train (system M) and the observer on the still station (system S) would jointly agree that the fast motion of the train had no real effect on the net speeds of the missiles - from either's point of view - once the train's speed was taken into account. And they would likely assume that the same would be the case for the net speed of the presently unmeasurable pulse of light - when the train's speed was similarly subtracted from that assumed combined speed.
The fact that a smoothly moving reference system (M) has no net effect on the measured outcomes of moving bodies within it (and on the laws of mechanics that underlie them) was something that Galileo had discovered in the early 17th century. Later, Newton would agree that such a principle of mechanics did indeed apply to such measures of the speed of all known moving bodies. It is effectively accounted for by the effect of 'inertia'. All movable bodies, including observers, on any smoothly moving reference system (as a train, a ship, an aeroplane, the Earth itself) share a common inertia so that any new forces applied to such smaller movable bodies cause the same outcomes - relative to that moving reference system - whatever the speed of the system on which they occur. One may well have experienced this reality if throwing a ball up into the air while travelling on a smooth train or pouring a cup of tea, etc. The ball, the person and the tea have all effectively 'accommodated' themselves (by means of their inertia) to the train's ongoing uniform speed and behave just as they would in any seemingly 'still' environment. One could similarly juggle balls in the air equally well on a smoothly moving train or on the still platform of the station. The laws and forces of mechanics work identically on both - due to the equating effects of inertia. This reality is sometimes referred to as 'the principle of relativity': Everything works (moves) as it should (according to the laws which apply to such motions) relative to an agreed reference, at whatever uniform speed that reference may be moving. The latter's uniform speed has no effect on the outcomes measured within that environment so that it is just as though the moving reference system concerned was absolutely still. If there were any absolutely still references that could be use instead (which there aren't) it wouldn't, in any case, provided results that were any more correct or valid.
Thus, when different uniform speeds are taken fully into account, the net results which remain are just as expected. The speeds of the moving bodies relative to the moving carriage where they occur are not affected by that movement when measured by an outside stationary observer, once the full speed of the train is thus accounted for. Subtracting that speed from the total speed measured, as viewed by the stationary observer, should leave the correct, unchanged net speed of the moving body - relative to its moving environment - just as the principle requires and predicts.
Or, so it was always assumed !
However, if and when the measurements were made with much greater precision it would be found that there were in fact, at most normal speeds, very slight discrepancies from those that were previously thought to be the case - as predicted by that original principle of relativity. Thus, for the thrown ball, for example, subtracting the full value of the speed of the train (100 mph) from the total speed that appeared to have been found originally for the ball on the moving train (M), when measured from the station (S) - ie 30 mph plus 100 mph = 130 mph - actually resulted in a net value not of 30 mph, as had always been expected, but one that was very slightly less - as 29.999.. mph, say. And for the bullet, subtracting that full 100 mph value for system M from its assumed total of 600 mph was found to give an even slower net value for the speeding bullet (compared with expectation) - of nearer 499.993.., say; that is, a bit slower than the 500 mph previously assumed. But the discrepancy, while still slight, was slightly greater for that faster moving body than for the slower ball.
If we had measured some even faster missile, one that travelled at, say, 10,000 mph - to which the train's speed of 100 mph is added - to make its apparent total 10,100 mph - its net value after subtracting the train's assumed 100 mph would again be found to give not that original 10,000 mph but more like 9,999.9450.. mph (say); that is, an even greater proportional reduction would actually be found. [Note that these examples are used to illustrate the point; the reductions are rather less.] And for light, at its much faster speed, the discrepancy may well have been even greater (as implied by these findings). Indeed; it would later be found to be just that.
As originally understood, the velocity (v) of the moving system M plus that of the smaller body moving on it - which we may symbolize here as (w) - ie v + w, would equal the total combined velocity (V). This would be appropriately 'transformed' back (it was always believed) to that of the net velocity value of the body concerned (w) by the simple transformation equation as used by Galileo or Newton - ie:
But, as explained, it was later discovered that this result, when based on very precise measurements, would actually prove to be in error. In our present examples, we have simply accepted the velocity values - at v = 100 mph and w = 30 mph, 500 mph or 186,000 mps (for the ball, bullet and light, respectively). But velocity is the speed of a body based on the time it takes to move a given distance. If the velocities concerned are not known, one would normally have first to determine them by recording the times and distances taken - if these are available. On this basis, the velocity (w) of any of the bodies would be a function of the time (t) it takes to travel the distance (say, x) concerned; the transformation equation needed to determine the velocity w of a body on a system M moving at velocity v, as viewed from a stationary system S, would be a variation of the above equation but with the necessary values for all relevant times and distances shown in place of the given velocities. That is, in the case of the thrown ball (with distance and time on the Moving system shown as x' and t', say, and those net values - as on the stationary system - as x and t):
ie (100 mi/1 hr + 30 mi/1 hr) - 100 mi/1 hr = 30 mi/1 hr
From this, one could calculate by algebra the values for x' and t' separately when knowing only those for x and t, or vice versa. This was the basic tranformation equation for classical mechanics. It shows the relations between the variables in the two systems. By knowing those of either, it should be possible to calculate those of the other. By so calculating x' and t', say, one would learn the net velocity value (w) of the moving body (be it the ball, the bullet or the pulse of light) on system M (with its velocity of v (or x'/t')). [Note: The value of v can however also be considered as the relative velocity of the 2 systems from either point of view. Thus, from system S, system M is seen as moving at velocity v while its own velocity may be taken as zero but from the point of view of those on system M, its own velocity may be considered effectively as zero and that of S as the one 'moving' (relatively) - at velocity v - but in the opposite direction. The arithmetic is exactly the same. This symmetric equivalence is very important in understanding the theory.]
Thus, if system S wasn't so stationary and 'fixed' to the ground (as the station in our example) but was the slower moving of two ships, say, with a faster moving ship seen as system M (gradually passing the much slower ship S), there would still be a value v to represent the relative velocity between them (and again from either point of view) even though neither has a relative velocity of zero mph (relative to the ocean). The difference in velocities between them could be identical to that where one was still and the moving one passing at the same rate - if at a slower relative velocity, or if moving in the other direction. The same could be described for two trains passing or being passed by the other or with regard to the relative motions of the Moon and Earth, or Earth and Sun, or whatever. While it is convenient to consider that it is the Earth that moves around an apparently stationary Sun, for example, it is arithmetically equivalent to view the Sun as the body moving around the Earth (as indeed it appears to us so to do).
If, as we have concluded, the above equations do not give quite the correct answers, the source of the errors (ie the small discrepancies noted) may well reside within one or both of those more basic measures on which velocities are determined - that is, within those of distance (x') and/or time (t'). It was just such discrepancies, especially that for light, that would prove to be the problem for which a new explanatory theory was required - namely, Einstein's theory of special relativity as published in 1905. As mentioned, the essential problem to be resolved by this theory did not present itself initially in the form of the very slight discrepancies described here (for such as moving balls or bullets), at least not directly, although they arose by virtue of the same basic problem. And Einstein's theory did indeed provide an explanation as to why such very precise results should actually have always been the case, but this would be even more clear-cut when the moving body was as fast as that of light - which we haven't attempted to measure in our discussion so far. But it was initially only at those kinds of immense speeds that the problem was, finally, much more likely to manifest itself...and so it did.
The problem was eventually then 'fixed' when it was realised that the speeds of system M to be subtracted from the totals (when measured from system S) shouldn't be the full speed of system M (as 100 mph in the case of the train) as had always been reasonably assumed, but rather, were found to be some value slightly less than this - the amount less depending on the total combined speed (V) of system M and that of the missile concerned (which latter, moving body was of course the main focus of the observations) - that is, on v + w). For the ball, the amount subtracted from the total V should have been about 0.00..(10 more 0s)..1 mph less than the full 100 mph used; for the bullet, it might be nearer 0.00..(9 more 0s)..7 less, say, and for light very much less (no 0.000s needed at all). Einstein had developed a formula that could calculate just how much of an adjustment was needed depending on the combined velocity V (ie of system M and the body moving within it), and he had a related explanation for just how and why such reductions of the perceived speed of system M (as part of that combined speed) actually comes about. For he did look at the assumptions underlying the magnitudes of the time and distance (space) involved - being the two variables underlying all velocity - as pointed to above. Those assumptions had previously suggested that the values of time and space for all possible given motions were always the same (for any given situation) whatever the circumstances of the observer and the moving reference systems. But they weren't.
As implied above, the difficulties that had arisen by 1900 centred more on the much faster end of the spectrum of possible speeds - not thrown balls, nor fired bullets, nor even any missile that might travel at 10,000 mph, say, but those at or near the speed of light which, in a vacuum, travels at an amazing 186,000 mp second, and through air, water or glass just a little slower. If the odd discrepancies were slightly more apparent for the case of the bullet compared to the ball and even more so for any missile travelling at, say, 10,000 mph, what discrepancy might occur at the very much greater speed of light - ie at 186,000 mps !? It was fortunately of a value that was much more likely to become apparent and measurable than those for the much slower missiles described above. In fact, that was the focus of the difficulties reported by various physicists up to about 1900; for the behaviour of light did not appear to obey the usual expectations for the motion of bodies in general. Was this because it travelled at such an immense speed, or because it was such a very small particle or bundle of waves, or what ? Did it fall outside the usual category of more 'normal', tangible moving bodies - ones that we can see, feel and more easily measure accurately ? No, it wasn't, and it didn't.
If we mark along a horizontal line on a sheet of paper about a foot wide the speeds at which various bodies can move - for, say, 10 mph, 30 mph, 500 mph, 5,000 mph (the approximate speed of the moon around the Earth) and maybe even 65,000 mph (the approximate speed of the Earth around the Sun), that last speed would, on this chosen scale, be placed very near the right edge of the sheet and those slower speeds would likely be crowded into the first inch or so near the left edge. If the speed of light (at 670 million mph!) was then placed on that same horizontal line (on that same scale), the sheet of paper would now have to extend to the right for about 18 miles(!) before the mark for light could be properly placed. Between about 100,000 mph and that for light, there would be no other marks on the line over those 18 miles as nothing else moves at those still immense, but intermediate, speeds. If we drew an oval around the line on the foot wide page to encompass those speeds up to 65,000 mph, we might describe the category of speeds of moving bodies falling within it as those for most tangible bodies which fall within our general experience and obey the same laws of physics. We may well imagine that one single speed that falls so completely on its own - 18 miles out (!) - way beyond that encircled category (on our graph) - would be easily and generally accepted as not being a part of that same fairly close-knit category of more everyday motions of normal bodies. It was part of Einstein's genius that he extended the category of moving bodies which should all obey the same laws of physics over those 18 miles of our graph - to incorporate that lonely single velocity (for light) situated so far out on its own, making it also fall within that same one category - that is, a single category for all moving bodies, including light - all obeying the same single set of laws and principles - whatever its immense speed or small size !
If, as Galileo's principle of relativity requires, the speed of any reference system M should have no effect on the net speed of any body measured within that system (by an observer there or in a stationary system S), then this general rule - which reflects the logic of the transfomation equations and of the inertia that account for them - should, said Einstein, apply to (the laws that determine the behaviour of) light as much as to (the laws that determine) the motion of any other moving body. But there was a problem with light. Not only was it extremely fast with an extremely small 'body' (or even a discrete very small bundle of waves), its speed - unlike that of all other moving bodies - was always the same; it was a universal Constant. It wasn't its extremely great speed or very small size that mattered, but the invariable constancy of that particular speed. So that, unlike the ball or the bullet, say, measuring its speed on any moving system M - from some position S - shouldn't provide any increase in its combined velocity - due to the velocity (v) of system M. Thus, when we come to subtract that velocity of system M from the combined speed of the light (w) plus that (v) of M, as measured by the observer in S - that perceived speed (v) of M to be subtracted must somehow be effectively reduced to zero! That is, not just reduced a little bit, as in the cases of the ball or the bullet, or even quite a lot more (as in the case of anything that could travel at 10,000 mph), but reduced totally ! An explanation of that 'somehow' would be provided by the theory of special relativity.
In the case of the ball or the bullet, as we've shown above, the amount of M's perceived speed (v) that is subtracted is certainly not reduced to zero - but it is reduced - if very little. In the case of the ball moving at only 30 mph, it was an exceedingly small reduction. In the case of the bullet, it was just a little more. For the speed of the Earth (M) around the Sun (S), it would be a more measurable proportion than these, but still rather small. So, for all of these moving bodies, their potentially variable gross speeds could be shown to be increased by most, if not quite all, of the speed (boost) provided by any such moving systems M on which they occurred - be it a train, a plane, a rocket or a planet - in each case relative to any agreed reference.
But for light, there was no such potential variation (increase or boost) of its speed by means of any faster system M. For light's speed couldn't be increased (or indeed decreased) at all because it was a Constant and was so, in any case, at the Maximum speed at which anything could move. It provided the upper (and fixed) limit of all motion. And yet, as explained, it still fell within the same one category as all other moving bodies - which had to obey the same one set of laws of motion. All other bodies moving at slower speeds - which didn't have light's constancy restriction - could be increased (or decreased) in speed (as viewed from outside) - for a given set of conditions (of force, mass, etc) by performing their journeys on some faster (or slower) reference system M - but now it was appreciated that such increases could only be up to that maximum possible speed. Morever, the extent (for any such increases) would generally have to be some proportion of that possible upper limit; that is, they were all placed somewhere (proportionally) on a single 'sliding scale' - by virtue of the same causative factor that prevented light's speed being increased at all - but with slower bodies at least benefitting by a considerable proportion of the speeds of their moving environments. Mathematically, this proportion was a function (f) of the ratio of the velocity (v) of the relevant moving system M to that maximum possible speed (c) - ie f (v/c). By 'function of' one means some particular multiple or fraction of that important ratio. Einstein worked out exactly what that function must be and did do on the basis of what he concluded (in May 1905) was the actual cause of this unsuspected proportional effect. [Note: we earlier referred to the velocity of light - when considered as the moving body of concern - as w, but its more general symbol is c (for its unique quality of having a constant velocity.]
That none (zero %) of system M's speed (v) could be utilized for boosting bodies already moving at the speed of light (c) meant that Einstein had, when deriving his equation, one value (ie 100% of M's speed was not available to add to light's speed - ie at the upper limit) and where v = 0 (as when the train was stationary as viewed from the station) he had the other extreme reference point (at the lower limit); between these, he had the speed of the Earth around the Sun - at ca 67,000 mph - where the effect on that velocity had been found by an interpretation by Lorentz of an experiment by Michelson, both contemporaries of Einstein. The equation so derived on the basis of such points (which provided the required new relations between the values of x and t of the stationary system and x' and t' of the moving one, as mentioned earlier) would allow one to predict the slight discrepancies found even in the relatively slow speeds of the ball and bullet examples given earlier. But it was the implications of the constant but immense speed of light through which all this was first investigated and concluded. How did he explain these 'sliding-scale' effects on the perceived speeds of system M - as viewed from a system S ? That is, as he concluded in about May 1905, just before he wrote up his important paper ?
The extremely slight discrepancies for the speed of the thrown ball, and the fired bullet described above, were clearly not those which early physicists noted, were troubled by or investigated; they were much too slight to have ever been noticed or measured at that time. But, (alluded to above), there were various measurements made then involving light which did appear to cause some confusion. Because of light's great speed, there was a greater chance that certain real discrepancies would (as mentioned) have eventually been noted, as they indeed gradually were. But until about 1900 or so, the basis of these was uncertain. By accepting, before anyone else, that Galileo's principle should nevertheless apply as much to the law of light's constant (and maximum) motion as to the laws of all other moving bodies' - with their potentially variable motions, Einstein sought a way by which any attempted boost to light's speed by a smoothly moving system M (as viewed and measured from a system S) would always be totally inaffective or neutralised so that its law (of constant net motion) was, as Galileo's principle required of all such laws of nature, not affected whateoever by same.
This principle also required the laws (of variable net motion) of all other, slower moving bodies to be not affected either - by such moving systems when so measured (although their gross motions (V) of the body plus the system M speed, would be). To neutralise the effect of any relatively moving system on light's speed, the total amount (proportion) of that system's speed had to be found (as perceived and measured) to be somehow validly reduced to zero. For slower moving bodies (as the ball or bullet), it would be much smaller proportions of system M's speed that would, by this same mechanism, be unavailable (as viewed from S), so reducing its measured boosting capacity only slightly. If there were other bodies that could move at, say, half or even a tenth of the speed of light - that were boosted by some moving source (M) so measured - then M's perceived speed would be found to be reduced less than that for light (which was to zero!) but still much more reduced than the very slight amounts we've indicated for the ball or bullet. That is, the sliding-scale would apply - based on Einstein's new transformation equations, which reflect the newly appreciated relations between the variables of velocity concerned - again accounted for by the very same mechanism.
This mechanism (and explanation) for these strange effects concerning the varying amount of extra speed (varying on the sliding-scale between 100% down to 0% (in the case of light) by which any body may in theory be increased - by virtue of the velocity of their moving source (M) - lies within an understanding of the two components of all velocity - distance and time. As mentioned, we earlier expressed the effects on moving bodies in terms of velocity (speed) alone (ie directly) - with only brief reference to these two fundamental determinants which underlie velocity. The theory of relativity is in fact generally defined in just such terms - that is, about an effect on time and space. The key to this explanation is found in the measurement of these variables on system M when viewed from system S.
The underlying times and distances involved in the velocity of the moving bodies concerned can only be made therefrom in terms of the information about them being conveyed from their sources on system M to the observer (or measuring equipment) on system S - by means of light waves which, while exceedingly fast, are not instantaneous. The transfer of such information thus takes time and does so relative to the distance and velocity that system M and the moving body concerned is moving away from or towards observers (or recording instruments) on system S (or vice versa). For slow moving systems and bodies nearby (as balls and bullets, etc) on such as trains, this information is, as mentioned, only very slightly delayed. But, at very high speeds and distances, the delay can be much more significant. Thus, the time shown on the local clock on M when measuring the time of events there would appear to those on S to be slowed by varying amounts (and distances shortened equally) depending on such factors and thereby influence the consequent velocities as perceived and measured at or from a distance, especially if travelling at a much different speed. This basis for the velocity measures so reported is quantified precisely in terms of the function of the ratio of v to c referred to above. These are shown below in the relevant new equations derived on this basis - for this very purpose.
Because observers on system M can see their own local system as the one that is effectively stationary (ie as a system S therefore) and the one we have more typically referred to here as system S as the one that is, to them, effectively moving - ie as a 'system M' - the local clock on that latter system would show to those on the other system (now called S) as equally slowed. And both perceptions are as correct and valid as the other! There is no absolutely valid time - somewhere 'out there', or even 'here'. There is only relative time. The same applies to space or distance. And thus perceived velocity, composed as it is of these two elements, must vary accordingly and not be of those unchanging values previously assumed.
For the law of the constancy of the velocity of light to be consistent with the principle of relativity, that unchangeable velocity of light must be what is found empirically when a light pulse is released from a source moving at whatever speed is measured from a relatively stationary system. Thus, if the system on which the light is released was, like the train, moving at 100 mph or even at some enormous speed - as, say, 50,000 mph - the light pulse so released couldn't be found to be moving at a new velocity of 186,000 mps plus 100 mph, or plus 50,000 mph, when measured from a stationary system, but rather at 'only' 186,000 mps. What happened to that attempted extra 100 mph or 50,000 mph 'boosts' ?? Its hard to credit but the information (by means of light waves) about those latter speeds couldn't 'keep up' with the velocity of the pulse of light being measured. The local clock time on M would (to those on S) appear unchanged at the time shown when the pulse showing the time started its journey. Effectively, system M hadn't moved ! [Or, had it shrunk as well ??] For slower moving bodies, the time and distance measures would be viewed from S as less affected by this 'information lag' - ie the time would be seen as only slightly slowed and the distance as only slightly shortened so that the speeds of such systems M would be only slightly slower than expected otherwise - and the net speeds of the moving bodies thereon would be as their laws required once the only slightly slowed (rather than totally stopped) speeds of their associated systems M were subtracted from the total combined speeds (V) perceived for the body and its moving system. This is the basis of the 'somehow' referred to above.
Einstein derived an equation which reflects the fact that when a body moves at a velocity w within a moving reference system M which is itself moving at velocity v, the total velocity V of the body - as seen from a stationary system S - ie w+v - must prove to be of a magnitude (as so viewed) equal not to that nominal sum per se, but to its value divided by an amount that is slightly greater than 1 - the amount greater than 1 being dependent on the extent by which the total nominal velocity of the body (w+v) is a proportional function of the velocity of light (c) - as that is the factor which determines the speed with which the relevant information can reach the observer or their recording instruments on S. That is:
It may be appreciated that where the total nominal velocity of the body (v+w) in the numerator is comparatively small, the amount by which the denominator is greater than 1 will be exceedingly small since the fraction added to 1 is the multiple of v and w over an enormous product of c squared. At slow nominal speeds therefore (of the body and system M together), the perceived velocity V will be barely less than the expected nominal sum of v+w. The amount to be subtracted from this value in order to determine that the net value (w) of the moving body has indeed not been affected by its occurrence on a moving system M (as required by the principle of relativity based on its inertial justifications), is represented entirely by the slightly reduced value of v (ie of the moving system M), not of the velocity of the body itself (w) or any small portion of it. It remains unchanged - as the principle requires. If however, the total nominal value of v+w is very large - as where the value of v and/or w are (is) very large, the amount by which the denominator is greater than 1 becomes much larger and thus the perceived value of V becomes much smaller than the otherwise large nominal sum of v+w. If the velocity of the moving system M could become as fast as the speed of light itself (ie v = c), then the perceived velocity of the body moving in that system at its speed w could not exceed that of light. This is revealed in the similar equation:
That is, nominal summation of v + w (where v = c) is perceived as a reduced value now equal only to that of c since the velocity (v) of M is only perceived to be of magnitude of c - w. Subtracting this amount from the total leaves a net value for the body concerned to be as required by the principle of relativity - ie unchanged - despite being propelled to the speed (w) on a theoretically extremely fast moving system M. Einstein also derived the equation necessary for calculating the exact amounts by which any such total velocity of a moving body on any system M must be seen as reduced (due to the information lag referred to above) in order to determine the correct (ie uninfluenced) net velocity values for the relevant moving bodies. These are based on the extent by which the distance over which the body moves appears (to those on S)
would appear slowed by: 1 - sq rt of (1 - v2/c2) secs per sec or very near to 1/2(v2/c2).
So, the greater the speed of system M, the greater the apparent shortening of the motion and slower the apparent time taken for it, and thus the slower the associated velocity (w) of the moving body concerned accordingly. By subtracting this amount of the total velocities so perceived (ie by means of equations for the distances and times involved), so the net values for the velocities of the bodies concerned (be they bodies of light or slower-moving ones) will be found to be uninfluenced by the speed of any system M - just as required by the principle of relativity. Thus, in place of the 'old transformation equations' (OTE) utilised by Galileo and Newton - which gave slightly erroneous answers (see (OTE) above), we now require ones which takes account of the contraction of distance and dilation of time according to the extent to which the body is travelling in M (v + w) as a proportion of the possible upper limit of velocity - c. That is, as a function of the ratio of v to c (ie f (v/c)). These have been calculated by Einstein in the form of revised, 'new transformation equations' (NTE) which replace those of Galileo and Newton which, we may note, did not in those equations take into account and include the value c for the constant and maximum speed of light. The new transformations allow one to calculate what the values of distance (x) and time (t) would be for a body moving along the x axis of either system M or S (coincident along axis x) moving at relative velocity v. That is:
[Note: These new transformations satisfy the equation: x'2 - c2t'2 = x2 - c2t2 . Thus, x' and t' refer to the distance and time of a body's motion on system M, and x and t refer to the same measures relative to system S. The relations between them are provided by the particular functions that each is of the other - in terms of the ratio v/c - as appear in the transformations. The key difference between these transformations and the original ones by Galileo and Newton is that (as mentioned) in these new ones the inclusion of the value c as a maximum possible velocity sets a proportional limitation or restriction on all the values of time and distance (of v) so determined. This is reflected in (and accounts also for) the informational delays underlying these proportional restrictions. The values of velocity calculated (necessarily) in terms of such varying values of time and distance mut differ therefore from what they would have been had one continued to assume that there two deteminants of velocity never varied whatever the circumstances of such potential informational delays.]
We may now better appreciate that the unusual characteristic (law) of light - its constant speed - can and does meet the over-arching requirements of the generalised principle of relativity (of its one and only net value being unaffected by the speed of any Moving system on which it may be measured (from a Stationary or slower one) - by virtue of the effect of the inevitably delayed information concerning the elements of velocity - distance per time - being received by the observer, these being not absolute and unchanging as previously believed but relative; that is, relative to the difference in the velocities of those two reference systems (being the source of those perceptual delays). That, in a nut shell, is the theory of special relativity.
It provided the answer to the problem first described in our introduction as to why the full velocity of any moving system M isn't seen from elsewhere to be added totally to the velocity of any body moving on it - when measured from outside that system. This effect was discovered by means of establishing first that, in the case of the moving body being a pulse of light, none of that full velocity of system M is seen as added to light's (constant) velocity - a finding that previously was not correctly explained. The extent and quantitative effects of the delayed information are represented by the revised transformation equations which were derived on that premise. They can be compared with the equation shown earlier. One wonders if the underlying logic in regard to why the laws governing the motion of all bodies (including light) must accord with the principle of relativity, with its basis in the principle of intertia, somehow entails a inherent, integral role for the perceptual delays referred to in this regard. Or do these pertain to slightly different aspects of this process ?
It may be noted that one could have proceeded towards the conclusion that time and distance must be relative from any one of three approaches: (1) as just shown - which accepts - as empirically found - that no further velocity can be added to (or subtracted from) a body already moving at the speed of light (eg as light itself (which is probably the only body which can move at that one constant, maximum speed)); (2) by calculating what relations could account for the also empirically existing discrepancies (if sought and found) for bodies moving at slower speeds than light, as referred to at the start of this account, and calculating in terms of the resultant rate of change that there would be a maximum and constant velocity (equal to that of light) beyond which no further velocity could be added (or subtracted) by any systen M; and (3) by calculating the effects directly of the perceptual delays inherent in actually measuring the required values of the distances and times involved when measuring the velocity of a body on M from the viewpoint of S. Thus, methods (1) and (2) follow logically from the general acceptance of the reasoning that the principle of relativity must be compatible with the laws underlying the motion of all bodies, including light, which in turn requires that the values for time and distance must adjust accordingly - these then proving consistent with empirical findings, while method (3) provides the explanatory mechanism by which direct confirming evidence can be obtained which agrees with the logically deduced conclusions of approaches (1) and (2), as also empirically confirmed. It is probable that without the reasoning behind method (1), method (2) would never have been considered or examined with the precision required.
A Condensed Version of the above Brief Account
On the assumption that the velocity (w) of a body moving on a reference frame (M) which is itself moving at velocity (v) can be increased without limit, by increasing either the velocity of the moving body itself directly or that of the reference frame M (or both), it was generally accepted that the body's velocity on M could be determined by virtue of the understood relationships between those velocities - whether measured from S or from M. Such relationships (and the particular function of which one set is of the other) were represented by 'simple' transformation equations (wherein, for example, v is subtracted from the sum v+w). This reflects the original principle of relativity, as formulated by Galileo and Newton. This interpretation also tacitly assumed that the magnitudes of time and space underlying those velocities (over any given intervals of same) were invariable/unalterable whatever the limitations of transmitting the time and distance data required. Thus, the velocity (w) of any such body and that of any reference frame relevant thereto (v) should always sum exactly to their arithmetic total (V), as viewed from S. From this total, the full value (v) was typically removed (and appropriately so, as formerly believed), in order to reveal that the net velocity w was indeed unaffected by the velocity of M - as the principle had originally always required. However, precise measurements of the net velocity w of moving bodies so determined (and of the time and distance measures underlying them) did not agree with the above expectations and eventually pointed to a need to re-examine the assumptions underlying them; that is, the assumptions that the basis of the magnitudes of time and space for any given situation never varied and that possible velocities so based were unlimited - for the inaccuracies increased as the velocities increased (relative to their reference frames) until, at the velocity of light, no further increases became possible. The assumed relationships between the velocities measured from systems M and/or S were thus incorrect and the original transformation equations derived on that basis had apparently given faulty results.
Einstein reasoned that because the principle of relativity appeared otherwise to apply so robustly in the above sphere of the mechanics of moving bodies, he felt that it should apply equally well (ie with equal validity) in the sphere of electrodynamics also and in particular to the movement of light as well as ro any other 'body'. But the gradual and greater understanding of light eventually made science aware of a factor which its analysis of the motion of other moving bodies hadn't properly considered. This was that there was an upper limit possible for the velocity of any moving body, including light, whose velocity at that upper limit was also a constant. And Einstein then reasoned that it was the underlying laws (which determined the motion of all moving bodies - whether variable (up to that limit) or constant (as in the case of light - at that limit) which must obey the dictates of the principle of relativity, for reasons of quite fundamantal physics pointed out by Einstein. In order for (the laws of) all moving bodies to conform to this newly realized reality, they had to accomodate themselves to the unrealized restrictions of the upper velocity limit and the constant velocity of light (which was now accepted as part of a single conception of the motion of all moving bodies). The restriction of this constancy effectively forced awareness of the actual and necessary variabilty of the time and space measures underlying the velocity of all bodies. Their relationship (that is, of the velocity of light and of the measures of time and space) had to be effectively reciprocal; they couldn't both be variable or both be constant; one set had to give - to accommodate that restriction.
To accommodate themselves to this restriction, the velocity of any moving system M to be subtracted from the sum value of w+v (V) must not be the full value of v but something less than this - with the amount less depending on the ratio of v to c (the constant and maximum velocity of light). This implies a need for a new predictive relationship between the velocities of bodies as observed from the two reference systems (M and S) - one on which new transformation equations must now be based. They should never in the past have been a function simply of v but of this more complex and ultimately determining ratio of v/c. As such, it may be appreciated that when v is very small in relation to the enormity of c, the effect on perceived velocity is minute and barely measurable whereas if v is a more significant proportion of c, the difference would be much more apparent - compared to the false conclusions that were formerly based on the original transformation equations. If bodies could travel at unlimited velocities (as originally assumed), so that light, for example, could transmit information virtually instantaneously, there would be no need to utilize the new equations, as the old ones would have proved accurate - by providing instantaneous time and distance data. But that isn't how nature works. There is a speed limitation for bodies and information can only be transmitted from a distance over time, not instantaneously.
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