234. We may consider next what it was that Einstein was doing whilst Michelson, Hertz, Lorentz and Poincare were still grappling with the foregoing problems - as part of their ‘traditional lines of enquiry’ - seeking answers to the anomalies that science had recently thrown up - in the usual time-honoured ways of mechanics - albeit taken to the limit. Meanwhile, in 1894, Einstein was 15 and attending his Munich High school (Gymnasium) which he found rather rigid, and stifling of natural curiosity. He excelled at mathematics and while interested in physics, the subject there was based on rote learning of rather dated texts. Maxwell’s ideas, for example, were not yet taught, never mind Hertz or Lorentz - whose 1892 theory (still developing) had not long been published. His father’s business failed that year and the family decided to move to Milan but leave young Albert with friends in Munich so he could complete his high school diploma there. However, after 6 months of disaffection at school, he was asked to leave - albeit with an excellent report on his mathematical abilities. This suited him as he had himself actually sought to leave on ‘psycho-social’ grounds (missing his family, etc). In Milan, he continued self study in mathematics and probably some physics - although what texts he may have had seems unknown. But they must have started him ‘thinking’. On the basis of this and his good report from Munich, he applied in 1895 to the Zurich Polytechnic but failed the entrance exam, again doing very well in mathematics. The Director advised that he first study for a Swiss high school diploma - which he did in a sympathetic school in Aarau where he was allowed to pursue further his own interests in physics and related lab experiments. He soon obtained his diploma and the Polytechnic then accepted him in 1896 without further examination. He was about 17.
235. By this point, he had decided that mathematics required too much specialisation and preferred the more fundamental and general aspect of physics, later claiming that he gradually realised he had a talent for manipulating the more general principles of science and ‘scenting’ out where any inconsistencies lay. He also maintained that his major capacity wasn’t his intelligence but having the stubbornness of a mule - to keep persisting on any such problem - for years if necessary. It was thus as a near 17 year old about to enter Zurich, having read certain accounts of the physics of the day, that he formulated in his mind his famous ‘thought experiment’ concerning what one would observe if one could travel with a beam of light - at its speed. In terms of the accepted physical principles of the time - namely Newtonian classical mechanics and the later Maxwellian electrodynamics, (discussed by such as Helmholtz and Mach, whose ideas he must have read), the possible answers to this question threw up logical difficulties - from either point of view, besides being incompatible with one another. That he formulated such a question at all clearly implies that he was reading about current matters in physics by then (ca 1896-98) - especially concerning the movement of light through the assumed ether - even though Maxwell’s equations and Lorentz's ideas were still not taught in the Zurich Polytechnic at the time. He (more than others) seems eventually to have taken on board Maxwell’s conclusion that light’s speed was indeed a (true) constant (but just when he became convinced of this seems uncertain) and it was this that led to certain inconsistencies and anomalies. Would that wave of light of his thought experiment appear as a still, frozen, dark wave not able to enter his retina or, despite his own immense speed, would it nevertheless still 'zoom' both away from him and into his eyes as always? But what speed would it have to go to do that? He struggled with such thoughts for several years before he found the only possible resolution.
236. One must conclude therefore that there was at the time (ca 1897-1902) some concern abroad about certain inconsistencies in the physics reported over the previous decade - following growing acceptance of Maxwell’s ideas - as a stimulus to him periodically to re-visit his early thought experiment. Similar inconsistencies in earlier actual experiments and explanations about light - concerning its velocity, frequency and direction (as by Fizeau, Doppler and Bradley) of which he would become familiar - must have implied similar results to those of Michelson (of whom, on the other hand, he would later say he was not sure he was then aware) - in showing an unexpected constancy in light’s speed and/or some lack of expected evidence for the role of a still ether medium relevant to these aspects. With no support found for a still ether, light's speed would possibly not be thus hindered and so account for (or at least help explain) its odd constancy. One wonders if such findings, if not by Michelson then comparable findings by those others, may have led Einstein to an early acceptance or at least suspicion of that constancy. But light’s speed was apparently still expected to vary by all other workers whenever its source or receptor moved (despite Maxwell’s equations) and various interpretations were advanced (eg by Lorentz), consistent with the current principle of relativity and the mechanical model, to accounted for the awkward absence of such variation under such conditions and/or support for a still ether; presumably therefore, the realisation and implications of the actual constancy of light was thereby frustrated and delayed. Even Maxwell himself, it seems, hadn't really taked on board this reality and its full implications.
237. While completing his studies, continuing to keep abreast of contemporary ideas as best he could - as well as learning about earlier findings pertaining to this general area of concern in German publications, Einstein eventually gained non-academic employment just after the turn of the century. He would continue to wrestle in his mind with essentially this same problem for about 10 years (1895-1905) - before finally resolving it satisfactorily before anyone else. We may enquire how other researchers of the time were conceiving such problems? What were they? Clearly, they had something to do with the propagation of light in its accepted medium or carrier - the ether. But was it the motion or not of the latter or the velocity of the former that concerned them most? This major problem or area of concern by the early 1880s seems to have arisen mainly out of Maxwell’s conclusions regarding the character of light as it pertained to the contrary views of Fresnel and Stokes regarding the motion of that ether. This soon found expression in Michelson’s experiments of the later 1880s - motivated by the challenge implied in Maxwell’s suggestion as to the likely difficulty of measuring the relative motion of the Earth through the assumed still ether - even if this bore also (if less explicitly) on the matter of the velocity of light per se. Maxwell was clearly interested in confirming that his new electromagnetic form of light was propagated within a similar, if now non-elastic, ether as that in (or by) which light had always been assumed so to move - and presumably a still one. Neither he nor Michelson appeared to be seeking information that pertained directly to the question of the variability or otherwise of light's speed; its assumed variability (despite his own equations(?) was, it seems, generally assumed and was so simply as a convenient means to establish in particular the motion or not of the ether per se (rather than the converse). And there is no reference in the relevant papers by either author about any role such a still ether may have as a system of absolute rest - that might pertain to concerns about...anything. Just when and to what end any such role or concerns later(?) arose, I'm uncertain. Possibly it was (?first) implied in some of Lorentz's ideas? [See also below re its possible relevance to the 'oversight' Einstein points out at the beginning of his paper.]
238. If light's speed had been accepted (after Michelson) as a true constant, it would have provided the one and only known means by which it would (in theory) be possible to determine whether one was at rest or moving - something which the principle of relativity dictated should never be possible; all laws of mechanics should operate identically in all uniformly moving frames (and in any absolutely still frame if such was available) such that there would be no way to distinguish such motion (or stillness) from any other. Thus, all observers (or their measuring instruments) should find all laws of nature identical whatever their differing respective speeds relative to each other. As this went against the principle of relativity, it would very likely never be accepted that light had in fact remained constant. At least, this would have been a conclusion had this implication ever been considered or voiced. But, as far as I can see, it wasn't so considered - in these terms at the time - but only much later by. [Who in fact did first consider this aspect?] In any case, for seemingly other reasons, Fitzgerald and Lorentz then took up the challenge of Michelson's negative result in this regard - in order to maintain the consistency of the mechanical model vis a vis Fresnel's view of the mainly still ether. Such an orientation may have served also to rekindle interest in the idea of Newton's fixed reference system of an immobile, absolute space with a principle of relativity in which (without ever being mentioned seemingly) time and space were quite reasonably assumed to be independent, absolute and invariable - as they had always (and only) been. All this was riding on the ultimate validity of Lorentz's efforts - which some (eg Poincare) felt was near to being attained if he could just formulate a 'new mechanics' to resolve one or two remaining difficulties - for the old one had been stretched to (or even beyond?) its very limits - ie without quite succeeding. It may have been such generalists who realised that it shouldn't have been possible to determine one's motion by using a feature of mechanics (in which camp optics (as everything) was still assumed to fall) - ie the speed of light.
239. However, Poincare had suggested (seemingly before Einstein) that there was no such thing as absolute rest or motion and hence only a principle of relativity could apply - ie in all situations, whether mechanical or electrodynamic. {The line of logic here must be a touch more complex than this!] As such, the latter must somehow 'fit in' to existing principles of mechanics and thus do so within its own constraints (and not of those of 'absolutivity'). That is, that the aforementioned distinction of who is or is not moving should not be possible where there is no absolute fixed point or system of reference. But, paradoxically, that only available principle (in Poincare's eyes) still retained its 'absolutive' (and/or constant) conceptions of time and space. However, any conceptions of these that may have been considered instead as relative (and/or varying) would not reasonably emerge simply by noting this 'inconsistency' by virtue only of this odd 'balance' of converses. A much more indirect and analytic evolution of this idea would presumably be required - arising out of the insistent demands of the implications of the unyielding constancy of the velocity of light vis a vis a necessary, inevitable principle of relativity. Thus Poincare felt that the latter needed a more complete 'explanation' by which its application specifically to the matter of light's speed (still not accepted by him as 'always constant' seemingly) could somehow be better reconciled - ie in that 'new mechanics' he felt was needed. In effect, Einstein did this such that the central implication of the principle of relativity - of never being able to distinguish one's absolute motion vis a vis others by means of any law of nature - mechanical or otherwise (ie by utilising light despite its constant speed) - was somehow maintained.
240. The central 'problem' in physics by the 1890s was thus essentially the fact that while the speed of light remained constant, this was not appreciated or 'taken on board' as a truly universal principle/reality - as it didn't accord with the principle of relativity (as it then stood) with its quite reasonable implication that everything's speed relative to any defined frame of reference was the sum of its own speed plus (or minus) that of the frame on which it and its source may be moving. [Presumably, this is an example of the consequences of the aforementioned problem concerning being able to distinguish thereby one's motion, etc.] Thus, awkward results of experiments with light due to this unrealised constancy were explained instead by advancing various hypotheses concerning the motion or otherwise of light's assumed medium. But if light's constancy had been recognised earlier, there may have been more focus directed to certain unstated assumptions underlying all of science (including the principle of relativity) and thereby resolve many of the confusing results obtained in the past. In 1904, Poincare delivered a most prescient lecture in America in which he appears to have appreciated earlier than most that the principle of relativity must surely apply to all laws of nature and that because of the extreme accuracy of Michelson's experiment, its implications about the role of both the ether and the non-variability of light's speed indicated that (as mentioned above) the (existing) principle must be somehow more fully 'explained' in a way that was more consonant with those findings (and obviate the troublesome 'distinguishing' problem). He felt that Lorentz was close but that his contraction and local time hypotheses (designed in effect to overcome that 'problem') were too arbitrary (and also, contrary to Newton, restricted to speeds slower than light). Poincare apparently did recognise that a better interpretation of the role of time and 'simultaneity' of events was necessary. {I wish I understood how he came to this conslusion and what he meant by it.] How close he seems to have been in his publications on either side of 1905 - the year of Einstein's independent 'breakthrough'! But did Poincare relate such recognition to the effects, if any, of observations from differently-moving frames of reference ? And did he (then) appreciate that light's speed was indeed a (true) constant?
241. While Poincare hadn't by this stage formulated his ideas into a coherent final form, he did thus recognise the fundamental place in any such 'new mechanics' of a principle of relativity which required all laws of nature to function identically for 'stationary' and uniformly moving observers such that neither group could know if they were the ones moving or not. [But did he recognise that one such law was that concerning the constancy of the speed of light and if so, when did he so recognise it?? Possibly this was in his 1906 paper on dynamics of the electron.] If then one accepts that Michelson's findings were as precise as one could ever obtain, they must accord with that principle (ie provide support for it) and to do that (in view of the lack of an expected variability in light's speed or constancy in that of the ether) some improved 'explanation' of the principle was needed in which (therefore) adjustments in the components of such motion or non-motion - ie in our measures of distance (space) and/or time - must be somehow incorporated. Those proferred by Lorentz seemed not quite adequate and, to his credit, Poincare appears to have realised that (as mentioned above) a better conception of time - analysed in terms of 'simultaneity' may provide an answer. [But seemingly not in terms of the same logic as Einstein?] Newton's mechanics would predict that speeds beyond that of light should be possible whereas Lorentz saw that as an upper limit, after which problems of infinity apparently obtrude. Poincare saw in this the idea that inertia in a moving body could probably only increase up to the speed of light (as mass increases or ?). However, he still seems not to have considered that light' speed may have been a true constant or that there was no still ether. While he apparently believed [on what basis?] that all laws of nature had to conform to his dictum, he didn't appear ready to accept (as asked about above) that one such law - in the electrodynamic sphere - would allow one to know who was moving if that law was (wrongly) taken as allowing light's speed to vary and assumptions about time and space underlying science (and the principle of relativity) were not fully recognised and proper adjustments thereto made.
What was needed was someone who combined Lorentz's understanding of the physics of electrodynamics with Poincare's capacity to analyse and apply logically the most general principle of nature. Einstein alone seems to have possessed just this combination to which was added a clearer appreciation of the implications of Maxwell's equations - ie the real constancy of the speed of light and the lack of any 'real, absolute motion' (as Lorentz felt necessary for his electrodynamic contraction hypothesis). However, it should be mentioned that Henri Poincare (1854-1912) would in 1906 publish his important paper on the dynamics of the electron, based on the theory of electromagnetism (as advanced by Hertz and Lorentz presumably), and apparently deduced essentially the same conclusions about the theory of relativity as had Einstein - despite working quite independently of him and apparently having not yet read Einstein's German paper. His ideas were, however, restricted to a narrower compass than Einstein's - ie to electromagetic phenomena. Nevertheless, such independent approaches which arrive at basically the same general conclusions greatly strengthens the ultimate validity and acceptance of same. Poincare was an outstanding French mathematician whose enormous contributions to science in that field far out-shone his more peripheral and sporadic activities in physics - as he 'inched' his way nevertheless towards Einstein's slightly earlier-reported and more general conclusions.
242. It appears then that ‘the problem’ as seen by most other contemporary physicists through the previous decade (1890s) (including Michelson’s and Lorentz's views of it) was being approached by a different, if more traditional, route than it would be either by Einstein ('from elementary considerations involving light signalling') or by Poincare (pertaining and 'limited to electromagnetic phenomena'). Einstein would (eventually) see it primarily as a matter of making compatible those two seemingly incompatible basic realities of physics (or nature) - viz: the constancy of light’s speed - as indicated by Maxwell’s equations if not yet fully appreciated by most (and certain related findings pertaining to measurements of light), and the long established principle of relativity - which at first appeared to be incompatible with the former - if and where anyone ever considered if these may or may not be compatible; before Einstein, it appears not to have been a consideration. For how could anything not vary its speed relative to differently moving frames of reference - if such was indeed ever considered? But the evidence before Einstein, that he at least perceived, pointed to the need to adjust, if possible, some aspect of one or other of these otherwise apparent truths in order that they became mutually compatible. They couldn't both be right as they stood. But to accept that light's speed alone may not be variable and that the principle of relativity must somehow adapt to this proved a hard nut to crack. It seemed to go against all common sense (and 'certain unspoken assumptions' underlying both that principle and all of science - as mentioned above). Would it mean that a principle of 'absolutivity' with its idea of absolute motion was the more valid after all (as discussed earlier) - one that also automatically assumed (as comprehensible, consistent concomitants) - the ultimate constancy and absoluteness of our measures of time and space? Paradoxically, it didn't; at least not completely.
243. Certain related problem(s) had been recognised by most other workers - ie as had arisen out of Maxwell’s findings (and certain earlier problems not fully addressed) - as needing attention in terms of their immediate mechanical difficulties - to be somehow ‘patched up’ and kept consistent with the constraints of the existing principle of relativity - within that mechanical model of science, whereas Einstein would approach them as above - in terms of more abstract, general principles (of the nature of motion - including that of light). Eventually, this forced him to recognise, confront and question those unspoken assumptions - be they in the absolute or relative camps. Those unresolved remnants of absolutivity which had 'stuck' to the old principle of relativity were thus finally removed - because of our eventual comprehension of the nature of light. And with a 'new' principle within this realm, relativity would finally shake off all other remnants of absolutivity. Thus, our earlier question (parag. 41) as to whether investigations into imponderable phenomena (like light) may help resolve the fundamental dichotomy of the absolute vs relative world view (ie better than did studies of the ponderable alone) would be clearly answered - in the affirmative - ultimately.
Einstein's Answer of 1905: 'A Simple and Consistent Theory'.
244. Science normally progresses by a series of small steps provided by a sequence of researchers each addressing in turn problems and difficulties thrown up by earlier workers. Hypotheses are tested, results reported and subsequent hypotheses advanced by later workers. Sometimes a major advance is made. But generally, it is possible to trace the gradual development of most scientific ideas. Thus, after a century of incremental contributions by many well known workers, Lorentz had provided, by about 1900, a fairly well formulated theory concerning the electron and electromagnetic phenomena including, amongst other things, the behaviour of light and its medium, the (still) ether. This background to his ideas would be traced with little difficulty through the introductory remarks in the scientific papers written by himself and all earlier workers in that progression. It likely continued in the 1906 paper by Poincare referred to above. We may contrast this with the style of reporting to be displayed in 1905 by Einstein in what his biographer, Ronald Clark, referred to as "...possibly the most important scientific paper written in the twentieth century...". It was, he said, a perfect example of a paper whose aim, as described by Hermann Bondi (a respected philosopher of science) was "to leave as disembodied and impersonal a piece of writing as anybody might be willing to read...(but one that was)...very likely to tell the reader almost nothing about how the result was found." He might have added '...nor indeed exactly what the problem was that he was addressing' (other than 'certain difficulties'). Moreover, we may note that nowhere in this paper of such renown is his theory (of the electrodynamics of moving bodies) ever referred to as the 'theory of relativity'. It would appear that some time after its publication, Einstein must have realised that his new theory was but a special case of a more general theory of nature - to which was given the name of the 'general theory of relativity' - and after its publication, the earlier work gradually became known as the 'special theory of relativity'. The motion of concern (of moving bodies) in the earlier paper is constant or 'uniform' while in the later, more general theory, it can be thus or accelerated. But this important differentiation is not stressed in the earlier paper; the motions concerned are properly described there (at least on one occasion) as uniform - as inertia and the principle of relativity rely on same - but this feature is not particularly emphasised.
245. We would eventually learn also that, as a basis to his earlier paper, Einstein had been wrestling with a particular problem (relating to light) for about 10 years and that after 'abandoning many fruitless attempts, being visited by much conflict and confusion, 'at last it came to me...' (that 'time' was the key, as it were) - just 5 or 6 weeks before he actually wrote up and submitted his paper in the early summer of 1905. But the paper itself was almost cryptic in its presentation. It would contain no specific references to earlier studies by others or a single footnote. There was just one brief acknowledgement - to help he had received from his friend M. Besso - 'on working on the problem here dealt with'. So, we can't easily analyse his conclusions in terms of their development from the ideas (as focused on some specified problem) of his contemporaries or of those who came just before him (although he does refer initially to Maxwell). The 'door' that he decided to unlock with his newly discovered 'key' was thus a side door - not the main entrance at the end of the more well trodden path by which others were approaching hoped-for answers inside - to fairly well described problems. He must have realised that what he had discovered was that profound and fundamental and his confidence in it so certain that he felt no need to persuade the reader or scientific community of its bone fides by setting out its detailed historical development; it would stand on its own succinct merits - forever.
246. We do have however three brief introductory paragraphs - before he pursues the tight mathematical development of his thesis at a highly technical level. We can at least seek to analyse these to see what they may reveal about the basis of his argument and just what was 'the problem here dealt with' that he was addressing (and, hopefully, the answer he suggested to resolve it). His paper was not the report of an experiment in physics the result of which is offered as evidence that some new hypothesis suggested as an answer to some prior problem is thereby supported. Rather, it is a mathematical analysis of theoretical arrangements of 'bodies' in motion, over distance and time, designed to show that unsuccessful previous results of certain actual experiments in that field (not overly specified) can be better explained thereby and to indicate what they should have been if thus interpreted, as well as predicting what future results concerning the same spheres of physics should be if also analysed in this same new way. But somewhere within the body of his paper there must be, even if only implicit and theoretical, a 'result' (as referred to by Bondi) which supports his general hypothesis relevant to some problem in the sphere of physics that he is addressing. Once the following analysis of his paper is complete, we may see if this problem and 'result' have indeed been thus revealed. Presumably it will also suggests future phenomena which his theory (and no others) can explain and be thereby further supported.
[Note: the following six paragraphs are by the way of an aside; the original paragraph numbering continues further below.]
Before we analyse those three opening paragraphs, it may be useful to consider the way his paper and its main thesis is typically described today, in retrospect, shorn of its more technical development. That is, just how are its main and most general conclusions briefly described? Professor Stephen Hawking provides one such useful overview in his recent book on 'The Universe in a Nutshell'. He points out that by about 1900, it was believed that an all-pervasive ether was the medium by which light waves travelled - at a fixed speed - but that if one travelled through the ether in the same direction as the light (as on the moving Earth with a measuring instrument), it would appear to travel at a slower speed than otherwise, and at a faster one if one travelled in the opposite direction. That is, its apparent constancy of speed was not independent of all other elements in its measurement (despite Maxwell's conclusions whose implications were clearly not yet fully appreciated). Hawking then refers to the Michelson-Morley experiment of 1887 (seemingly as a test of these assertions) which concluded that the speed of light did not in fact vary, whatever the direction the measuring apparatus moved through the assumed ether. To explain this within the traditional mechanical model of physics, Fitzgerald and Lorentz offered their similar hypotheses which would predict that the speed of light should indeed be found not to vary - due to exact counter-effects of an hypothesized contraction of the apparatus's measuring arm and an associated slowing of time measurements - as such experiments moved through the still assumed ether. This is all as described elsewhere although it doesn't indicate exactly what it was that such experiments were actually seeking to establish. In any case, accounting for that experiment's failure by this means meant that the ether's role was still a necessary concept. However, there were still some problems with this new interpretation - as pointed out by Poincare.
Hawking then moves on to Einstein's 1905 paper in which he apparently pointed out that "..if one could not detect whether or not one was moving through space (which seemingly one could not as he had already shown that there was no still system to serve as an absolute fixed criterion), the notion of an (still) ether became redundant. [This seems to imply that such detection would have established a reality for the ether and vice versa. However, it would seem more logical to say that it wasn't just unnecessary but rather was impossible; I must have missed the real point here somewhere). In any case, Einstein faced the need to explain findings such as the Michelson-Morley failure (although not that particular experiment, of which he was apparently then unaware) by some theory that did not require the concept of a (still) ether - eg to provide a means for the seeming contraction and time slowing effects postulated as so caused by Fitzgerald and Lorentz; this seems to suggest a 3rd function for the ether!] "Instead, he started from the postulate that (all) the laws of science should appear the same to all freely moving observers. [Ie a new version of the principle of relativity - being the first postulate of his (published) theory.] In particular, they should all measure the same speed for light, no matter how fast they were moving. The speed of light is independent of their motion and is the same in all directions." [This being the second postulate of Einstein's theory.] This latter aspect was in fact not compatible with the principle of relativity as it stood formerly and, without explanation, was still 'apparently' not compatible with its now amended form either, as implied in that first postulate. That explanation would in fact constitute Einstein's theory of (special) relativity.
[Of course, the laws of science before Maxwell did already operate identically for all observers whatever (uniform) speed they travelled at - due, for example, to the influence of inertia) but the second postulate, following from Maxwell's discoveries, to be now incorporated within the generality that 'all laws of science' should fulfil that first postulate (indeed 'particularly so'), required a major adjustment in the original version of that principle of relativity since this (the second postulate) was the one such (new) law that dealt with the speed of motion of anything or 'body' (ie light) that had somehow to not be affected by the speed of its source or its receiving observers; no other 'laws of science' had ever entailed this particular and unusual demand. [However it was represented in the scientific literature of the day (ca 1895-1903, say), this 'problem' - of the stubborn constancy of light's speed in all cicumstances and thus its incompatibility with the traditional principle of relativity - must have been the main focus of Einstein's concerns.] To make that new law fit, it was necessary to discover that such observers' measures of the components of speed - ie of time and distance - must actually vary from that of other observers not travelling at the same speed (although they appeared to be no different to those actually travelling at that different speed). While it took a phenomenon of such immense speed as that of light to help reveal this reality (where such differences in measures of time and distance become much more apparent), it still applies at whatever different speeds different observers or frames of reference may be moving, even if relatively slow. Time was thus not a universal, independent constant (nor was space) but depended on the speed of the observer relative to that of the situation in which that relative motion (over time and space) of some body was taking place. Hence, the pertinent theory was called the theory of relativity. [It would be useful to be able to analyse exactly what happens (in terms of Maxwell's electro-magnetic constants, ratios, etc) when such as a beam of light (or apparently anything?) travels from a moving source yet doesn't benefit fully (depending on what proportion its speed is of that of light) from that 'starting advantage' - as observed from whatever reference frame.]
However, it seems to me most improbable that Einstein began his reasoning with this first postulate as a kind of self-evident reality and then said, in effect, " 'Eureka! - if this is the case (as I have myself just this minute postulated a priori - rather as a gift from heaven), then it follows as night follows day that Maxwell's law of light's speed (being such a law of science) must also fulfill this now obvious criterion and in order to do so, I can see almost immediately that there must be a new understanding of the conceptions of our measurements of time and space - which are not actually constant as formerly believed but must vary - and do so for the motion of all bodies (not just for light)." Indeed, did he not say himself that he had struggled with this problem for 10 years before he realised that 'time' at least seemed to hold the key to its possible resolution. And while there may then have been a subsequent series of 'mini eurekas' soon after, the order of his long-laboured deductions and their subsequent re-ordering in a format of the more general principles he gradually realised underlay that struggle, would seem to be two quite different things.]
[I think I finally 'see' a certain priority (in the overall flow of reasoning that led Einstein to his conclusions) of the realization that without a fixed reference point in space (as a fixed ether) so that all motion was relative, the equivalence of all known natural phenomena (laws of nature) in whatever apparent (uniform) speed of their associated frame of reference they transpire, they operate identically (due to inertia) - since those various 'speeds' are really quite unknown and meaningless - only having ostensible, relative magnitudes vis a vis some (any) other differently moving frame. Its all so arbitrary, there must therefore be some over-riding reliability or absoluteness in the laws of nature wherever they occur [Why? Because our universe and us wouldn't have evolved in any reliable, survivable way to even consider all this. This may be what ....... was referring to when he mentioned the more fundamental principle of the unity of nature or whatever.] This being the case, then any new law - even one pertaining to some constancy of motion itself (as light's speed) - should apparently fulfil this same 'requirement'. At least, such an hypothesis would seem to be a most reasonable one to pursue in view of the otherwise appparent generality of this principle. One's speed is just not of any relevance - it not being absolute or real; all possible environments are moving relative to each other. But while laws perform identically within all these differently-moving environments, different measurements are found by those observing from other reference points - although account can apparently be made for these differences by means of appropriate transformation equations. However, this is not the case with light's speed (alone). This realisation was the key to resolving the difficulties. More general, universally relevant transformation equations would now be needed which took account of the underlying variability of time and space, a variability that was previously imperceptible (with bodies moving at usual speeds); indeed, its existence was quite unappreciated. It only became so (perceptible and increasingly obvious and measurable) with speeds at a significant proportion of that of light.]
The term 'apparently' in the foregoing needs explaining. It is the case that such traditional transformation equations did not in fact (quite) account for the different results as measured from the external vs internal frames of reference in any such case of 'normal' experimental velocities- even though they 'appear' to do so. If someone from 'Mars', say (or wherever) with super-human perceptual acuity had observed the same situation, he may have said: "No, not quite; for I notice that the actual difference values after your transformations should have been 10.000000000010004230012 (say) or 9.999999994162110011 (say), or whatever. You seem to have missed this (calling them 10.00) because either your measuring apparatus was too insensitive or you were actually utilising constant, absolute values for the elements of speed (ie time and distance) in your calculations. You seem not to have appreciated that such elements actually vary, as they are dependent on (relative to) the speed that the internal reference environment is travelling (in relation to the external one) - as a proportion of the one absolute criterion of speed that does exist - that of light. We might make this rather more obvious if we measured the speed of a super fast bullet I've brought along; it goes at 120,000 miles per second. In this case, we will find that applying the correct transformation equations - ie ones which take into account the effects this speed has on the values of time and distance relevant to the calculations - results in a much more apparent effect (of say 325 units vs 495) as the outcome of some independent factor affecting a dependent one involving our speeding bullet. The closer such speeds approach that absolute upper speed limit as a proportion, the greater and more obvious would be the effects of the affected magnitudes of the time and space elements of the speeds concerned. This absolute limit happens to be that of the speed of light, which also just happens to be an extremely fast constant - of 186,000 mps. But the same effects on time and space would occur (although at some other range of magnitudes) whatever was this ultimate criterion (of maximum, constant speed) - even with a much lower maximum". [But while all motion is thus said to be only relative, with no absolute speeds (nor absolute values of the associated time and space), as between all differently moving frames of reference (their actual speeds, if this was possible to acertain, being arbitrary and irrelevant), it would appear that speeds measured as a proportion of any such ultimate criterion do have a certain absolute value. This confuses me - as does how it is possible to find a reference system in terms of which the speed of light can be measured as a constant.]
247. We may return now to our previous topic. Because Einstein chose to write in such a concise, abbreviated style (as part of the apparent 're-ordering' of his reasoning as mentioned above), it may be helpful to first reproduce verbatim each of those short introductory paragraphs (in their English translation) and then try to 'interpret' them and their evolution as best as one can. He begins:
"It is known that Maxwell's electrodynamics - as usually understood at the present time - when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest, there arises in the neighbourhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighbourhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise - assuming equality of relative motion in the two cases discussed - to electric currents of the same path and intensity as those produced by the electric forces in the former case."
248. [Now, when Oersted discoverd that a current of electricity (ie electrons moving due to an electric force) in a conducting wire placed near a magnet at rest (ie a compass needle), the latter would move to point perpendicularly towards the wire - due (it was later concluded) to the creation around the wire by those moving charges of a magnetic field. Later, Faraday found a symmetrical converse of this phenomenon when, rather than electric charges moving near a magnet, the magnet itself was moved near a conducting wire. This time an electric field was engendered around the moving magnet to which the electrons in the conductor responded - by moving as an electric current thus induced. The latter phenomenon was formalised within Faraday's 'law of induction' and the theory associated with it. It led to the developments of useful electrical machinery in which it may have been more practical to have a fast moving magnet encircled by fixed copper wires, say, rather than the converse. [Although I'm not sure about this.] But, in any case, in both Oersted's and Faraday's cases, the same outcomes would have arisen had the motions applied to the other elements in their respective situations. It was their relative motions that produced their results. This was apparently long realised but tended to be forgotten in as much as Faraday's law was generally framed as though this symmetry wasn't the case. Einstein's first point in his paper is thus simply to point out this particular oversight in contemporary physics. Its relevance to anything in particular is not at that point discussed. [What did Hertz say on this aspect, if anything?] But he continues this topic in the next (his second) paragraph where its relevance becomes clearer.]:
249. "Examples of this sort, together with the unsuccessful attempts to discover any motion of the Earth relative to the 'light medium', suggests that the phenomena of electrodynamics as well as mechanics possesses no properties corresponding to the idea of absolute rest. They suggest rather that, as has already been shown (my italics) to the first order of small quantities, 'the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good'. We will raise this conjecture to the status of a postulate (hereafter to be called the 'Principle of Relativity') and will also introduce another postulate, which is only apparently irreconcilable with the former, namely that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body. These two postulates suffice for the attainment of a simple and consistent theory of the electrodynamics of moving bodies - based on Maxwell's theory for stationary bodies. [His theory thus states, in effect, that 'there is a Principle of Relativity which holds true for both mechanics and optics - even though we accept also (ie assert equally) a second Principle - that the velocity of light in this conception of optics is a constant, whatever the velocity of its emitting source - these two Principles not being mutually inconsistent even if appearing to be on first consideration.] The introduction of a 'luminiferous ether' will prove to be superfluous inasmuch as the view here to be developed will not require 'an absolutely stationary space' provided with special properties, nor assign a velocity-vector to a point of empty space in which electromagnetic processes take place."
250. [Firstly, we note that Einstein implies that other examples are available where such relative motion has been similarly overlooked and absolute motion wrongly assumed. However, he doesn't specify these. They, with the one he does describe, plus unsuccessful attempts at verifying any motion of the Earth relative to the ether suggests that both electrodynamics (which includes light) and mechanics (ie the motion of the Earth vis a vis an assumed ether) operate perfectly well without any involvement of a concept of absolute rest and do so (in his new theory) within the dictates of a (new) Principle of Realtivity. As Ronald Clark puts it in his biography of Einstein - such examples and attempts then suggested to him an "...inevitable consequence: the destruction of the idea of absolute rest...". (Poincare apparently arrived at the same conclusion about the same time.) And this briefly expressed 'suggestion' apparently leads logically and equally inevitably to his 'conjecture' which, nevertheless, he tells us 'has already been shown' (to a certain high degree of accuracy) although, rather naughtily, he again doesn't specify when or where it was thus 'shown'. This seems odd in that this conjecture, already thus mysteriously supported, turns out to be one of the most profound hypotheses in science: a 'Principle of Relativity' so defined as to apply to all of nature (including, therefore, to any constants of velocity which (perplexingly) may exist - ie in any sphere; although only one in fact does - in optics). It requires* all laws of nature, whether in the spheres of optics and electrodynamics, or of mechanics, to be equally valid, to apply fully, in all frames of reference (which, as it happens, are, in his view, all moving) and in so doing to not provide any means by which any frame of reference could be differentiable from any others - as the faster, or slower or more or less 'stationary or mobile, etc. through the operation of any such law. Such a principle will not tolerate different sets of reference criteria when measuring the motions of bodies from whatever sphere of nature - be it optics/electrodynamics or mechanics. The unsuccessful attempts (to establish absolute motion of the Earth) that formed part of this foundation didn't include that of Michelson apparently - of which he later said he wasn't then sure he was aware - despite that experiment being seen as the virtual sine quo non of this category of work; apparently it wasn't necessary for Einstein. [* Bondi says it is so 'required' because of the existence of a yet more fundamental principle - that of the unity of physics; this likely needs further analysis however.]
251. And while Michelson sought evidence supporting a still ether (and thus of implicitly the existence of absolute rest), there doesn't appear to have been a scintilla of any suggestion in either his papers or in Maxwell's prior comments about methods to so establish this that such a still entity if found would have particular relevance to...anything else whatsoever. Later, Lorentz would seek (or assume) evidence concerning such absolutely stationary ether - as an explantion of why charged atomic particles within Michelson's interferometer (traveling on the Earth through that assumed ether) would contract into a smaller space and thereby account for an unexpected constancy in the velocity of light (that was, in Lorentz's view, really variable) and its associated failure to verify the real motion of the Earth thereby. However, Einstein did not refer specifically to this failure. He would take exactly the opposite tack in his thesis - that there was no absolute rest (whether as a still ether or by any other means) and that light's speed was constant - regardless of any motion or otherwise of its assumed medium which therefore needn't be posited as a (desperate) means of accounting for light's stubborn constancy otherwise. It is thus intriguing that while Maxwell and Michelson were interested in the concept of a still ether - whose existence or not as based on such research would later be fundamental to the contrary theories advanced by Lorentz and Einstein - the former two themselves indicated at the time no concern or interest in the potential relevance of such concepts to anything beyond the concerns of Fresnel and Stokes regarding the assumed necessary medium for light, latterly as an electromagnetic phenomenon. They weren't part of that imagined group of absolutists or mechanists battling against those imbued with some contrary philosophy (as written about in such terms many years later); just contemporary scientists investigating current phenomena. Its existence might be supported or verified in terms of its motion being reflected in different velocities for light (ie as a convenient marker) without any concern that this might conceivably have regarding the fundamental velocity of light; even its assumed constancy otherwise.
[Note: It would appear that the word 'same' in the definition of the relativity principle presented by Einstein (which appears to me a touch ambiguous) seems to indicate that: If the laws of mechanics hold good (are valid) for any given frame of reference, the (particular) laws of electrodynamics and optics which are concerned with those same kinds of measurements' (ie with which the equations of mechanics are concerned) will also hold good (will be equally valid). This would seem to imply that there could be other 'laws of electrodynamic and optics' that are not necessarily concerned with such as the velocity of all 'bodies' and, if so, it is not those to which his definition refers. The word 'same' would thus refer to particular categories of measurements in nature - eg the speed of motion - to which the 'bodies' implied within both spheres (electrodynamic and optic (on the one hand) and mechanic (on the other)) will be equally subjected and influenced.]
252. [In any case, Einstein then introduces his second postulate - that the velocity of light is indeed a constant - which is independent of the speed of its source . He doesn't point out that this postulate was either implied or even explicitly stated within Maxwell's electromagnetic equations nor that certain of those 'examples' which indicate that the motion of the Earth was not measurable (relative to...?the still ether), could also have been cited or interpreted as indicating that the velocity of light did not, as expected, vary - but was constant. Whenever and however he arrived at it, he then points out that this light postulate is not really inconsistent with his new, all-encompassing principle of relativity but only apparently so. For clearly, it would have to be consistent with it in so far as he has already stated in the first postulate that all (relevant) laws of electrodynamic and optics 'must' be compatible with this principle, in that the 'law' of the constancy of light's speed is indeed a part of such laws of nature. There can be no exceptions to being so compatible if one accepts, as Einstein has, that there is no absolute resting place in space since that means that there is no other principle except that of relativity available to account for the consistency and validity of all laws of nature concerned with motion regardless of the (inevitable) motion of all possible frames of reference; there being no fixed frames of reference, everything must 'work' satisfactorily (whatever their uniform speed) in terms of the only relevant principle going - that of relativity - as it accounts for the total acceptibility of uniformly moving frames of reference for all activities and their completely valid measurments in nature for all observers. There must, therefore, be approriate transformation equations to satisfy the reality of that requirement. In one sense, he doesn't have to provide a rationale as to why all laws of nature must or should accord with the principle of relativity - since this is (?part of) his hypothesis or theory (the other part concerns his definitions of time and space differing from those of Newton) and if this allows predictions which are verified and nothing is found to disagree with these 'assertions' (definitions), its validity can be assumed until not supported - without a prior rationale (I think), although he appears to have given one in terms of the logic which follows from there being no absolute rest in the universe. Bondi appears to give another (related?) one by citing a principle of the unity of physics. I can myself see a certain inevitability of the all- encompassing principle of relativity in terms of the evolution of all laws of nature (that always work) within our universe in which there are only (uniformly) and differently-moving parts and their associated inertia. Otherwise, an infinity of different laws would have to have evolved - one set for every different moving environment. But this hasn't successfully 'evolved' (nor we with it) and therefore we have the one that proved more sustainable. If there is a universe somewhere whose constituent parts are all mutually 'still', some other single set of laws would presumably have evolved.]
253. [It was of course his resolution of the above 'apparentness' (cf 'obviousness') and its expression in such new transformation equations (rationally based) where the crux of his theory lay. That resolution in fact equates to his "..attainment of a simple and consistent 'theory - of the electrodynamics of moving bodies'" or, as it was only later termed [when exactly?] a 'theory of (special) relativity'. It was presented as a theory rather than as an 'irrefutable truism' simply because this is how science progresses; all such advances must be capable (in theory) of being falsified. It holds only until and if anything ever is shown to be inconsistent with it. So far, I believe, nothing has - now a century later. That he was convinced that there was a need for some kind of resolution between two apparently irreconcilable principles or truths in nature seems to have arisen after his lengthy anaysis of his thought experiment and comparable anomalies suggested by research - with only one certain way out.
[Note: It is, however, my own view that the Constancy of the velocity of light (ie c) - and the eventual realisation of its truth and universality by Einstein - was the prime 'force' which ultimately insisted upon an altered conception of nature, but that the consequent recognition that a more validly-based principle of relativity was (subsequently therefore) also required indicated that this latter principle - once discovered - would prove to be an equally necessary contribution before this new conception could be fully and finally realised. That is, once it was so appreciated, the Constancy of the velocity of light (c) had either to be a single exception in nature and thus an exception to the existing principle of Relativity or a way had to be found to 'modify' that principle so that it could incorporate (be compatible with) that Constancy (and all other laws of nature) - and so would not be an exception in that regard. But the former was not really an option, since without any absolute resting place in space, there was nowhere in which laws of nature could be (or had ever been) distinguished in their operation on that (moving vs 'still) basis (ie and so identify such a unique environment) which left as an alternative means some law which alone could allow such a differentiation. All our laws of nature have evolved in relation to our variously moving environments on which they prove consistent, reliable and valid (almost by definition; if they hadn't, we wouldn't be here; nor they!). But while the reality of the Constancy of the velocity of light had always been the case (ie one such evolved law), even if only recently recognised as such, there had never been any evidence pertaining to light's speed which could be cited as indicating that any particular frame of reference (environment) could be distinguished in terms of being 'faster, slower, more stationary or more mobile (or whatever) than any other - on that basis.
It was concluded therefore that, as far as one could tell, the principle of Relativity must in reality be so based (ie in 'relative' or variable magnitudes of time and space) as to be compatible with all laws of nature, including that concerned with the velocity of light (and must of course have been so based all along). This strikes me as providing (a little) more in the way of explanation than simply asserting, without explanation, that 'all laws of nature 'must' accord with the principle of relativity' per se (even if they do) - as a basis for justifying...anything. However, another justification of similar form is provided by Hermann Bondi when he says, in effect, light's speed can't be an exception to the principle of relativity because of something he calls 'the principle of the unity of physics'. That is, he says: 'The Principle of the unity of physics requires that systems that cannot be distinguished by internal dynamic experiments (ie in mechanics) should be indistiguishable by any internal experiments'. We are thus driven, says Bondi, with virtually no means of escape, to Einstein's Principle of Relativity.' The 'force' of that drive was (?prior) realisation of the Constancy of light's velocity. Well, if ever asked, I can quote this but must admit, I'd prefer rather more concrete detail or an actual example of what he means by that generalised abstraction concerning the apparent primacy of something called 'the principle of unity'. We may also ask ourselves whether Einstein felt that it would be better to present his theory as though it unfolded in his mind in some particular order that he felt was the more logical development even if he 'happened upon' certain insights in some different order, worked it all out from that perspective and only later 're-arranged' it as presented. Will we ever know?
But see further discussion on this point below in which the demands of there being only relative motion and only moving parts to our universe (no still platforms) resulted in the evolution of laws of nature (ie those that survived and work consistently here) adapted to this reality. They all must work equally in all moving environments such that there is a consistency (a unity) of their operation from whatever perspective/platform/frame of reference they are perceived. [But, the idea of 'perception' brings observers (like us) into the equation and surely the laws must work equally whether we happened to have evolved or not? True, but within each differently-moving environment all laws of motion operate identically in any case; it is just that they must appear to do so to any (who have evolved) viewing same from other platforms. Why!? Because they're 'our' laws...or? Yes, partly; we construct concepts which prove consistent with our sensory and perceptive apparatus as our only way of knowing our ever-moving universe and its apparent 'laws'. Thus, we have concepts of space and time (now combined into one) by which means we can make sense of our perceptions of differing motions of bodies of either differing masses or acted upon by differing magnitudes of force, or both from our constantly moving perspectives. There are only moving perspectives and smooth, uniform motion - whatever its relative speeds - does not affect our laws - even that one entailing motion itself - and it apparently does so by virtue of adjustments in our perceptions of time, distance and mass.
Note: The theory of special relativity can be defined in a fairly general and succinct manner as a theory based on the idea that all laws of nature should be the same for all observers whatever their differing speeds. But this tends to subsume that most unique law of nature - that pertaining to the constancy of the speed of light - within some anonomous totality of such laws and so masks almost entirely the crucial role of having to find the only way by which the one law of nature pertaining to speed of motion (of anything) could be made compatible with the principle of relativity. It would seem too simple to suggest that Einstein merely reasoned that his assumption that the laws of nature do not depend on one's motion meant that the speed of light too must therefore be found to be the same by different observers whatever their differing speeds - ie simply because it too was just another 'law of nature'. Such conclusions arrived at solely on the basis of such grand general principles and reasoning seems almost too easy. Rather, he would seem to have concluded that all laws of nature, including that relating to the constancy of the speed of light, may well not depend on one's speed - but only after he figured out how that latter constancy could somehow be reconciled with an appropriate principle of relativity. Once he had resolved that, he could see that observers moving at different speeds could report that light's speed did remain the same for both of them - but only by disagreeing on the time and distance each believed the light travelled. This important latter aspect would seem to have been concluded only after much analysis and confusion. Measurements of distance and time thus depended on one's speed relative to some relevant reference frame. Seemingly, measures of mass would also so depend (see below).
And, having worked this out, Einstein (or others?) could then state the more general truth that all laws remain the same whatever one's speed - and call that 'the (special) theory of relativity': all motion is relative to a given frame of reference and any such motion (of bodies) implies time, distance and mass - which are thus all relative themselves to the relevant reference system. Everything remains internally consistent therein but may be perceived as having different values when observed from a system moving at a (relatively) different speed. The lengths of feet and seconds, say, to measure some standard motion of any body, including those of light, will appear quite normal to those in one moving environment but will appear different (ie longer or shorter) to those observing same from their own, differently moving (ie faster or slower) environment, while their own measures for these same motions will appear to themselves just the same as those used in the other environment appear to them and vice versa. Their perceived magnitudes are thus relative to the relative speeds of each other's frames of reference. At slow speeds (little different from each others), these magnitudes will appear almost identical but if their respective speeds are somehow vastly different, they (lengths of time and distance) may appear quite significantly different in the others' locality despite measuring the same things. Is it all similar to the fact that a house or tree you're standing beside looks quite tall but ordinary, yet those someway down the street look much smaller - yet we quite accept this without question or surprise?]
254. Finally, in his 2rd paragraph, Einstein adds, almost as an afterthought, that "...his theory (?unlike Maxwell's) will not require 'a concept of a luminiferous ether..." (my underlining) - ie an assumption of an ether medium for light - the term 'luminiferous' presumably implying its role as such a medium. And yet, he says its doesn't require such a concept, not because he has anything here to say about any other means or medium by which light might be propagated, but because 'it (his theory) does not require 'an absolutely stationary space...'. This clearly implies that another of such an ether's possible properties was its (generally accepted but unproven) 'absolute stillness' and it was that aspect that his theory would not require. Seemingly, it would not, inter alia, require its assumed property as a luminiferous medium either; It must simply propagates itself somehow (neither aided nor hindered by any such 'ethereal substance') despite its assumed wave-like character. But primarily it is not required in Einstein's perspective on it, as a repository for a concept of absolute rest - which would follow from the primacy he gave to his denial that such an entity existed. There was thus, in his conception, only relative motion. [We may contrast Einstein's brief reference here (and not earlier) to this concept of the ether and the relevance its stated non-existence (or non-necessity) had on his theory with that seemingly implied by Hawking - as a more definite raison d'etre for elaborating the theory in the particular form and order he did.] And while this 'stillness' aspect was unrequired by him, something like an ether might still exist as 'a wave medium' - although Einstein's reading of Maxwell's theory may have indicated to him that there was no need for this other (medium) role either - something that Maxwell himself appears not to have accepted before his early death. However, Einstein seems to confound these two roles of ether - without making any clear differentiation in terms their relevance to his own theory. My later conclusion that the main development of his reasoning might best place this assertion and premise at or near the beginning of his logic (even if it may have been placed there only in retrospect - by any trying to set out a more logical format of general principles) is certainly not supported in any obvious way when he inserts this aspect where he did. He also denies the need for any 'velocity-vector to be assigned to a point in this otherwise imagined still space for the proper functioning there of electromagnetic processes. [I'm not yet sure to what this latter aspect refers.] He then continues with his third paragraph]:
"The theory (ie of the electrodynamics of moving bodies) to be developed here is based - like all electrodynamics - on the kinematics of the rigid body, since the assertions of any such theory have to do with relationships between rigid bodies (systems of coordinates), clocks, and electromagnetic processes. Insufficient consideration of this circumstance lies at the root of the difficulties which the electrodynamics of moving bodies at present encounters." (my italics) That is, his theory will focus on (have to do with) those relationships - so analysed via kinematics. [One would like to have a list of the exact papers of ca 1900-1905, in which this topic encounters such difficulties (and descriptions of the latter) at the root of which may thus be revealed that this circumstance was given insufficient consideration (as he is presumably now going to give it).]
255. [Note: Kinematics deals with the motion of bodies (even materila 'points') in time and space without reference to the forces (as electromagnetic ones) or masses involved; that is, simply with their dynamic geometry, temporal and spatial.] His analysis will thus deal with the relationships between the positions of such bodies (or points) within a system of 3-dimensional spatial coordinates - as established typically with the use of perpendicular measuring rods - and of time coordinates - using timing devices or clocks and electromagnetic processes (light). Exactly which of many kinds of potential relationships that are possible within such a melange of variables that Einstein will be analysing is not made clear at this point. However, he appears to be developing his theory in a similar way to that of Newton - who based his laws of mechanics/motion on a set of definitions of the basic concepts involved in the mechanics/motion/velocity of bodies - over space and time - firstly as abstract geometric kinematics but later applied to the actual physics of same - entailing forces and masses. Einstein follows this model also but does so with respect to the laws of motion of all 'bodies' (including light) and this requires that he add the additional category of electromagnetic processes into his final analysis. His model, his theory, is thus expanded from the limited sphere of mechanics to the broader one of electrodynamics - of moving bodies.
In both cases, it should be possible and relevant to show the application of the principle of relativity to the motion of all such bodies. But in order that 'everything behaves just the same' whether one's surroundings are moving uniformly or are stationary (as qualified earlier) - as required by that principle - for all laws of nature - will now entail a new consideration - not previously appreciated; for now a law of nature has been (belatedly) recognised that concerns motion itself - the 'bodies' pertaining to same of which must remain constant in both types of surroundings regardless of relative velocity differences. To accommodate this reality, the velocity of all other bodies (as viewed by those in the relatively stationary environment) must now be equally recognised (again belatedly) as actually composed of a variation in the perceived magnitudes of its two component parts - distance and time - and not of a constancy of same, as previously assumed. The greater the velocity of such other bodies (as perceived from the stationary or slower-moving perspective), the greater is the extent of this perceived variation. Describing the bodies concerned as 'rigid' apparently obviates any subsequent explanations in terms of bodies actually 'contracting' or time dilating (as per Lorentz's hypotheses). It is only as viewed from a differently moving environment that the effect of the additional velocity of that other environment on activities therein can be appreciated. It is thus only from such a perspective that some allowance or adjustment in what is so perceived must be made in regard to the components of velocity of any and all bodies so observed except those of light whose constancy of velocity (both actual and perceived as such) is responsible for and necessitates those adjusted perceptions of the distance and time elements of the velocity of all other bodies moving in and with the differently moving environment.]
256. Newton began in his model by defining space and time as independent absolutes (see discussion elsewhere). I believe he set out his stall, his theory - of 'absolutivity' - fairly unambiguously and without too much preamble. As such, these two fundamental aspects of motion were seen as not varying according to any other independent factor(s) but remain unaffected in all circumstances - as two independent background elements to all the events of the universe. As mentioned elsewhere also, this 'orientation' to scientific investgations soon became 'second nature' to all concerned for about 3 centuries; it wasn't a position to be defended against; there was no opposition to it. Einstein, on the other hand, sets out his stall, his theory - 'of 'relativity' - rather more indirectly and less explicitly. He begins with an analysis and definition of simultaneity although we (eg those initial readers of his article in 1905, say) are not really aware of what his actual goal may be in this regard. We may assume (based admittedly on our later knowledge of his ideas) that it will eventually pertain somehow to a definition also 'of time' itself. We might also look out (subsequently) for any comparable analysis and definition of space - again as part of his theory. Presumably, they will differ from those proferred by Newton - which seemed to work so well - until about 1900 at least. He could, for example, be laying the groundwork for an eventual definition of time (and space) not as independent absolutes but, on the contrary, as dependent 'relatives' (say) - as foreshadowed above. And as such, they could be applicable to both meanings of that term (and in the case of one of them, very close relations at that (ie as per Minkowski; see later re the new single concept of a four-dimensional spacetime). They would thus not be independent constants (which never vary according to some other independent factor, but rather dependent variables whose values do so depend; or rather now, a single dependent variable). And, as with Newton, the validity of the ensuing theory and of the definitions of time and space on which they depend, will only be determined by the later availability or otherwise of supporting evidence (and not just on logical derivation). Both of these scientists thus went out on a limb, as it were, with their suggested interpretations (definitions) of time and space (as integral parts of their theories) and not necessarily as isolated concepts which they would necessarily 'swear by' as some kind of God-given truths to which they alone were privy. They were but parts of their overall theories and thus open to questioning and testing.
257. It may be useful to clarify, if we can, just what Einstein meant in the concluding sentence of this last (3rd) paragraph of his Introduction - that is, by the terms 'this circumstance' and 'the difficulties'. By 'this circumstance' he seems to be referring to the above mentioned relationships as they had been inadequately investigated within a large body of previous research in which, inter alia, the constancy of the velocity of light was not recognised as the actual basis of various anomalies so found. The variables which inter-relate in electrodynamics will do so in ways which go beyond that of mechanics and insufficient consideration had thus been given to a kinematic analysis of same - specifically that the true constancy of the speed of light wasn't generally appreciated or its implications so analysed as it didn't accord with the mechanical interpretation of the principle of relativity - so was not the focus of sufficient attention of that kind. To so correct that insufficiency would thus (it seems implied) provide the answer to the problem - which, in turn, was implied in the phrase: 'the difficulties' - by which he seems to be referring to such as the negative results of (such as) Michelson's studies and the not quite adequate 'explanations' provided by Lorentz and/or Poincare and/or to those of Fizeau and others (Bradley and Doppler) which were equally ambiguous in their interpretations of matter germane to his present concerns. However, he fails to exemplify the dictum proferred by a scientist heard recently on the BBC, who said that "The most important role for the scientist was 'To define the Problem". I don't know to what extent he fulfilled that requirement in his two other important papers published that same year (in the same Journal), but in the one in which he presented what he described to his contemporay (M. Besso) as 'my great discovery' (or some such) - and the subject of the present account - Einstein appears for some reason to have been purposely vague in that regard. He certainly is very precise regarding the many 'definitions' on which his theory (ie 'the answer') is subsequently developed (each, in a sense, a mini-hypotheis) but was not so in respect of any definition of the precise 'problem' being addressed. However, that actual problem may well be implied and derivable from that very 'answer'! We shall see.
258. Thus, in the first three paragraphs of his paper, he at least does provide some clues about what was 'the problem' that he felt needed to be addressed and answered'. It 'had to do with' the need for a more accurate and logically consistent understanding of the principles which determine the motion of all bodies in nature - whether electromagnetic or otherwise. Part of the answer was that they could apparently now all be explained within the one framework of electrodynamics. The need for such a theory is revealed in Einstein's reference to the 'difficulties' which this topic was then encountering (presumably over previous decades). They were thus manifested within various inconsistent results reported within this general sphere of research over the previous 25 years or so, presumably including difficulties explaining certain results of Bradley, Fizeau and Doppler. In particular, the constancy of the velocity of light (as and when it became appreciated) appeared to conflict with the original principle of relativity but, before this was appreciated, there was equal confusion concerning the motion or otherwise of light's assumed medium, the ether, and of its role as a fixed reference criterion - seemingly of importance to many as an explantion of other anomalies as mentioned earlier by Michelson; see paragraph 182). The work of Hertz and especially Lorentz in the 1880s and '90s further revealed these 'difficulties' in electrodynamics. And the answer to this 'problem' was implied within his comment regarding the 'circumstance' (of the kinematics of certain relationships - as just mentioned) which required sufficient consideration (ie in order to resolve those difficulties/provide that answer). Examining any role of an ether had thus confounded recognition of the real problem - seemingly.
259. Evidence that the Earth moved through a still ether - based on an appropriately varied velocity of light - would have saved them a lot of trouble. But this wasn't found and rather than point the finger at a stubborn constancy of that light's speed (which would have been incompatible with the current principle of relativity), they sought to account for the failure to establish that fixed ether by assuming that it existed nevertheless (despite that lack of evidence) and that it served not only as a substantive medium for light (consistent with the prevailing mechanical view) but also (by the 'real' motion implied) 'to contract the materials of the interferometer travelling at 30,000 kph relative to it (if not, as initially claimed, physically 'against' it. [Seemingly, Lorentz felt that the electromagnetic forces involved in that contraction of a body's particles were only generated when the inherent charges moving with same did so in absolute terms (ie 'relative' to an absolutely still ether in which they, as everything, existed). Did this reflect the same 'misreading of Faraday's induction law as pointed out (as a most fundamental aspect of the basis of his theory) by Einstein? And if it did, were there other recent or contemporary ideas then 'abroad' to which Einstein's destruction of the idea of absolute rest also prove relevant? (As Hertz or...?) Or, was it only vis a vis Lorentz?? In any case, it is ironic that Einstein would say that the motion of the sub-atomic charges underlying Lorentz's contractions needn't actually be 'absolute'; relative motion of them itself would have been sufficient - if such contraction actually occurred thereby. But he would deny that it did so it was academic in any case.] And, by some alchemy, the 'local time' was also conversely (and conveniently) 'dilated' in Lorentz's equations while this was occurring. By appropriately adjusting the usual Galilean transformations (accordingly - in terms of both these hypothesised contractions and dilations) when applied to such measurements (later called 'Lorentzian transformations), everything appeared to come out alright in the end (when so re-interpreting Michelson). It was rather 'too neat' - even though, amazingly, the arithmetic was spot on.]
260. [While Einstein sets out his two postulates (based on his minimally supported assertion that (a) there is no absolute rest and (b) unstated findings as have 'already been shown', and (seemingly) on Maxwell's equations, respectively) and his equal assertion that there is no need for an ether - either as medium or as a location for such a stationary system (for which, whether as ether-based or otherwise, there was no evidence and even some against), there is no indication that these elements in his reasoning transpired in any particular logical order or sequence. Once arrived at - by his admitted years of mental application - he could set them out ('after the fact', as it were) in whatever form and order he felt would best (most logically) reveal the underlying truths (thereby revealed) about the electrodynamics of moving bodies. That is, after he had 'figured it all out' - by who knows what means and in what order. We do know, however, that he did say at one point later that, after his long 10 year struggle "...suddenly it came to me...that 'time' was the key.." - ie presumably to unlocking a door of the room which contained the answer to the problem. As he reveals to us the 'kinematics' of moving bodies, the place and role of time (and no doubt of space as well) should become apparent. The original principle of relativity assumed that everything known in nature should vary its velocity according to any variation in the velocity of relevant frames of reference (eg as their source). This was a major part of the more general requirement that all (known) laws of nature should operate identically in all differently (but uniformly) moving environments once the difference in their relative velocities was accounted for. But accepting that light alone did not obey this represented a huge problem for that principle.
261. [It may well have been that it was only when Einstein took on board the validity of that constancy - ie first - that he (only then?) realised that something assumed as part of the original principle of relativity must therefore have to be adjusted to incorporate this later realised reality (and so manifest itself within transformation equations which were compatible with these facts). While he may also have convinced himself early on that there was no absolute rest, and that therefore, the principle of relativity was the only means in terms of which the motion of all bodies could be explicable, he seems not to have concluded at about the same time that, with no absolute rest, another feature of nature must also (immediately) follow - namely that time and space can not be absolute themselves but must vary. Rather, the latter conclusion likely only came to him (ie "suddenly...") when he realised that it was the incorrect assumptions of their absolute natures (underlying the original principle of relativity - which had gone some way to denying the need for a principle of absolutivity) that had to give in order to allow that principle, so adjusted, to prove compatible with the (already accepted?) constancy of the velocity of light. For they were the only elements left in the kinematics of the motion of all bodies that could so adjust - given that there was only relative motion and its associated principle; without a system of absolute rest, there was no 'absolutivity' principle to turn to - with its own set of laws. All laws functioned equally in all differently moving environments - the only kind there were; there was no stationary absolute environment that would require laws unique to itself. All laws had 'evolved' to prove mutually consistent within our universe of (only) moving parts. Uniform movement has no effect on them. Its where they 'grew up', as it were. They know nothing else. Hence the principle of relativity.
262. [It must have been while mentally manipulating the various factors of motion that one day...'it suddenly came to him ...that 'time' was the key...'. This must have been followed shortly by considering that the speed of anything, entailing as it does both time and distance (space), indicated that as such speeds approach that of light, so the extent of the alteration in the variable magnitudes of both these elements of velocity would have to alter accordingly - ie as such speeds became a greater proportion of that of light. They would no longer be seen (and wrongly assumed) as being absolute and constant. The original Galilean transformations (which Einstein was probably mentally manipulating when 'it suddenly came to him...' wouldn't accommodate the new reality of a constancy of any phenomena. Lorentz's transformations adapted to the anomalies of Michelson's results in ways that proved accurate but based on false conceptions of why such adjustments were necessary. He hadn't accepted that Michelson's result was due to light's speed being a constant. Einstein realised that such adjustments (in the transformation equations) could be accounted for in a more realistic (if very surprising) way. There would be no actual physical contraction or time dilation. What there would be was superficially very similar to these variations but rather difficult to explain. We may now continue with an analysis of the remainder of Einstein's paper where the basis for this conclusion, and the implications of light's speed being an upper limit of the velocity of anything, will hopefully be revealed to us]:
263. Following his Introductory paragraphs, Einstein presents his theory in two Parts - focused on Kinematics and Electrodynamics. He begins his elaboration of the basis of his theory rather ambiguously in Part I - by defining in Section 1 - the concept of 'Simultaneity'. This is of course something not immediately at the forefront of most people's interests or concerns. However, its relevance will no doubt become apparent. (We may note here that Poincare at least also made reference to this aspect of analysing time, although in what context he felt it was relevant I'm unaware.) After he has deduced his laws of the Kinematics of moving bodies (as part of his overall theory of same), he develops in Part II the electrodynamic aspects (with more theory?) in which he can apply the laws so deduced.
1. Definition of Simultaneity.
Einstein begins by defining a stationary three dimensional system of coordinates (x, y, z) - as one in which 'Newton's mechanics holds good'. A point can be defined therein - in terms of x, y, z - which is 'at rest' in this system - its position there being measured by rigid standards of measurement (ie measuring rods or 'rulers'). If this point moves - to a new position - it does so over time. Now, we must be clear, says Einstein, just what we mean by 'time' in such a context. Here we see how he is probably leading us to an eventual definition of this latter basic concept - but via this prior analysis of simultaneity. We may note that, at this point at least, he doesn't also say we must be careful as to exactly what we mean by a 'position in space' - although he has possibly already ensured clarity in this regard by defining same carefully - with cartesian coordinates and standard measurements. Presumably these prove just as robust when measuring the point's movement over space. The use of a standard time measuring device isn't as objective seemingly as are rigid rulers for distance in that he points out that when using a clock, we are assuming (making a judgement) about a simultaneity between what it shows and the occurrence of the event (a movement of the point) being timed. (This of course entails two such judgements - at the time it begins to move and when it arrives at its new position.) Thus, we can now better appreciate why he approaches his analysis of time from the point of view of defining simultaneity. [Possibly some parallel argument could be developed concerning space entailing assumptions about a 'simulspaceity' (or some such) between what a ruler shows and the (?actual movement/distance) of the point over space being measured (?judged).]
264. Einstein then simply asserts that such judgements of time - while accurate for events which occur 'near to where the clock is' - 'are no longer satisfactory...when they occur remote from the clock'. No rationale is provided at this point as to the basis of this important assertion. {POssibly thismis why he focused initially in'time'; maybe this remoteness doesn't apply with respect to space (think out).] He then gives an example of a way in which this claimed inadequacy (where such remoteness of the clock may apply) may be overcome (after giving initially a similar one - ie by using light signals - without explaining why this might be seen as a reasonable way of overcoming the inadequacy which he asserted earlier). He gives this initial method after saying that 'of course, we might content ourselves by simply doing this...(such and such)...in using such light signals' (as though this was the more obvious method) but then informs us that this method would in fact be inadequate - 'because it has the disadvantage that it is not independent of the 'standpoint of the observer' with the clock - as we know from experience'. [Unfortunately, I know not to what the 'standpoint of the observer' refers exactly nor do I, at least, know of any consequent disadvantage - whether from experience or otherwise. One wonders why Einstein feels he must be so opaque sometimes. Seemingly, the place of the observer is relevant to the validity of his ?judgement of time.] In any case, he then describes a much more practical (and presumably adequate) method of establishing the accuracy of our timings and thus better guaranteeing the valid simultaneity (or synchrony) by which our timings can apparently only be validly 'judged' - that is between an actual event and a time reading for it - even where the clock (and observer?) is at a distance remote from the event (seemingly). This method also uses light signals to which methodology we have in a sense been introduced in the prior inadequate method - without its rationale being explained; it is as though we have been thus 'softened up' to its suitability for this purpose without its relevance, necessity or suitability being really explained.
265. This adequate method is then described thus: Two observers are at separated points A and B in space (eg A could be at x1, y1, z1 and B at x2, y2, z2) and each has an identical clock with which they can both accurately time events at their respective points. But neither can accurately time events at the other's location (without certain assumptions, which are not revealed) nor therefore make mutual comparisons. The two respective times are defined as A time (tA) and B time (tB); no single 'common Time' (for A and B) is defined however - as Newton would assume with his definition that absolute time was the same everywhere. Of course, thus far, we have no reason to believe that the two times are not identical nor, therefore, the common time for both. We may accept that they may be different however. The only way that we can define the common time for A and B, says Einstein, is if we establish, by definition, that the time for light to travel from A to B equals that for it to travel from B to A. This assumes (ie by definition) that both the speed of light and the distance between the two point remains constant. If now we let a beam of light leave point A at its A time and travel to B, it will be reflected from B at the latter's now B time back to A, where it will arrive at a new A time - t'A. In accordance with our definition, the two clocks will be synchronised if:
As such, if either clock synchronises with a 3rd clock at C, then the other will do so as well. All such clocks are assumed to be stationary - relative to one another and are thereby synchronous according to our definition. TIME (ie of an event) itself is so defined in these terms - as: that which is given simultaneously with an event by a stationary clock located at the place of the event when that clock is synchronous with (?another) specified stationary clock (located...?where). {For some reason, Einstein doesn't cover the two aspects here questioned.) He then continues with a further assumption (which he says is in agreement 'with experience' - again not revealed here) that the quantity
That is , that the round trip distance 2AB divided by the time light takes to travel from A to B and back to A is equal to the speed of light in empty space and accepted here as being the universal constant c (as shown in Maxwell's equations). Such Time as shown by stationary clocks so synchronised in the stationary system is called 'the Time in the stationary system'. Such a definition objectifies what would otherwise be judgements of the simultaneity of clock readings and actual events anywhere in a stationary system. [One wonders if it might have been possible to define SPACE by like means - only by using what would be assumed to be an already accurately defined Time by which to do so - just as Time has been so defined (albeit later) - by using what one assumes was already acceptably accurate measures of distance (space); a bit of a chicken and egg situation. His frequent qualification as to the stationary status of his described systems (we may have simply assumed that they were so) would seem to imply that he will be contrasting these with 'moving systems' to be described subsequently.]
266. In any case, in Section 2, Einstein addresses the matter of the relativity of both Time and Space - and presumably in situations where the systems concerned are not (just) stationary, but move.] ie:
2. On the Relativity of Lengths and Times. This section seems to contain the kernel of his theory and is, he states, based upon the two key principles - of relativity and of the constancy of the speed of light. He thus starts by defining each of these carefully. The Principle of Relativity has a number of slightly different definitions in the scientific literature (including more than one by Einstein. On this (early) occasion he defines it thus: 'The laws by which the states of physical systems undergo change are not affected, whether these changes be referred to one or the other of two systems of coordinates in uniform translatory motion'. In other words, different environments moving at different uniform motions have no discernible effect upon the operation of all laws of nature; They are oblivious to such motion and work the same everywhere regardless. The Principle of the Constancy of the Speed of Light (one of those laws) is defined as: 'Any ray of light moves within a stationary system of coordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving source (therein)'. As such, the speed of light (c) must always = the distance of light's path (as defined...where?) divided by the duration of the time interval (as defined above) taken by such light.
267. He then presents the logic by which he will demonstrate the relativity of Space and Time (which we will eventually see is an inevitable consequence of the only means whereby the two Principles concerned can be shown to be (as they apparently must be and have always been) compatible. Thus, whereas Newton began by defining Space and Time as absolutes - without too much justification or background - and built up his laws of motion and his model generally from that basis (associated with concepts of mass and force), Einstein (after providing some preliminary definitions about simultaneity and stationary systems of coordinates which will prove necessary in his subsequent developments) begins instead by defining or presenting as axioms the aforementioned two only apparently (but not actually) incompatible principles of nature - which, individually, do have historical bone fides. But applying these equally )as they had always stood) to the usual measures of motion - ie based on unvarying (absolute) measures of time and space - lead to difficulties (as reported by other researchers over the previous 20 years or more) which apparently pointed to a need to adjust those latter measures in ways that would overcome such difficulties but still allow the two principles to be seen as compatible. That adjustment entailed the proposition that Space and Time must vary (be relative) according to the speed of the immediate environment in which it was being measured when observed from a (relatively) slower environment. Their relativity turns out to be the only way that undeniable compatibility can be realized. His theory thus says in effect that these two principles are compatible and shows us the means by which they are so - ie by repealing Newton's absolutes and replacing them with their relative equivalents (which have been the case all along). By this means, one of the principles (that of relativity) was in fact itself adjusted in this respect so that it 'became' compatible with that concerning light. [Actually, its perception became compatible; it had itself always been so compatible but it had been wrongly perceived as implying in its operation absolute measures of time and space.] On this basis, Einstein could then proceed to develop his laws of (electrodynamic) motion with their apparently inevitable implications for mass and energy.
[Any analysis of the evolution of Einstein's resolution of the problem he is addressing in his famous paper will require an explanation of (a) exactly which problem it is he is so addressing (as mentioned above) and (b) the logic of his decision that the two principles as defined, not individually but in combination, could provide the resolution of that problem (and have other significant implications). My own view is that he probably first realised that the Constancy of light's velocity was a fact that had been generally overlooked by most researchers of the later 19th century and that when he analysed what the application of that constancy to various relevant research findings led to then (only secondly) realised that the Principle of relativity would appear to be thereby compromised. As this was apparently not acceptable in contemporary physics, he concluded that something within the set of factors involved in motion generally (as supervised, as it were, by the principle of relativity, would have to give - something that quite likely related to the factors which Fitzgerald and Lorentz had been manipulating within the relevant Transformation Equations - ie measures of Time and Space - when trying to fit awkward data into the usual mechanical model with its principle of relativity which assumed constant/absolute time and space. While both factors were seemingly involved, it seems it was Time through which Einstein first saw his eureka! light - that is, that its value/magnitude was not absolute and constant but that it varied according to the speed of the immediate environment in which it was being measured when observed from a (relatively) slower environment. The upper limit on the speed of anything as possessed by light, meant that everything below that speed had to to adjust proportionally. The same implications on the measured magnitude of Space (distance/length) must have followed soon after; That is, they must both be relative and so vary in new transformation equations.]
The ultimate acceptance of the validity of that means of (pre-existing but hidden) compatibility, and of the relativity of Space and Time as part of the newly perceived principle of relativity which allowed this, would depend upon the eventual verification of any predictions such a theory may present. That is, they don't necessarily have to be shown to be logically deduced at the outset although a prima facia rationale would no doubt help in their serious consideration and later testing. Apparently, Einstein's paper did not provoke any immediate interest and was only gradually appreciated - by about 1909 or so. Its predictions were of course fully verified later.But his purpose in establishing the relativity of time and space would appear to have been equally important as a means of explaining the reasons why many previous findings were so awkward and difficult to account for. It would seem to follow that if he can 'prove' that time and space do indeed vary when the two principles are fully applied, then he can equally assert that their compatibility itself has been shown. But what is egg and what is chicken here?
268. He begins his demonstration (ie proof) of the relativity of Space and Time (which his theory says must be the case given the existence, validity and actual compatibility of the two principles described - unless its the other way around!?) thus: 'Let there be a stationary Rod of length l which lies along the x axis of a stationary system of coordinates (the Rod's length measured by a ruler that is also stationary in the same system). This may be be called the 'length of the stationary Rod' (ie in the stationary system). Let the Rod move with velocity v along the x axis in the direction of increasing values of x and then have its length re-measured by two methods: (A) by the ruler held by an observer also traveling with the Rod. This may be called 'the length of the Rod in the moving system'; And (B) by the use of two stationary synchronised clocks. In this latter method, the observer ascertains at what spatial points (on axis x) in the stationary system the two ends of the (?moving) Rod are located at a definite time. The distance (space) between them, as measured by the ruler, is a length which may be designated as 'the length of the (moving) Rod in the stationary system'. According to the Principle of Relativity, the length of the stationary Rod (l) should equal the length as measured by method (A) in the moving system. Einstein then asks 'how do these (equal) lengths compare with the length of the moving Rod as ascertained in (B) by the stationary clocks? Does the latter still equal l ? [Note: for some reason he adds that in method (B) 'we shall determine the length on the basis of our two principles (or does this qualification apply to both methods?).] He replies that 'current (ie Newtonian) kinematics tacitly assumes that the lengths as determined by operations A and B should be precisely equal'. In other words, he adds, a moving rigid body at the time (epoch) t may in geometric respects be perfectly represented by the same body at rest in a definite position. But, on the basis of applying both Principles (which we have accepted/defined as being valid), we will find (ie to be explained later seemingly) that these two lengths will not be (?measured/perceived as being) equal. One must note that at this point, Einstein is again simply asserting that this will be so found. He doesn't yet proceed to show this to occur in a practical demonstration or how it must be the case by virtue of some logical argument. It has yet to be proven to us - in theory at least.]
270. Thus, he continues, we place clocks at the 2 ends of the moving Rod (at A and B) which have been synchronised with clocks of the stationary system (as observed by stationary observers) and so show the true time of the stationary system. But we have moving observers also - observing those two synchronised clocks located at the ends of the moving Rod. We then let a ray of light leave from end A of the moving Rod at time A (as noted in the stationary system) and travel to end B where it is reflected at time B to return to end A, at time A'. All this is as described before except that now the points A and B (the same distance apart) are Moving. Einstein continues: 'Taking into consideration the Constancy of light's velocity (oten ignored in the past), we 'will' find that:
where rAB denotes the length of the moving Rod measured as in the stationary system. That is, (tB - tA) will be greater than (tA' - tB) whereas, if light was assumed to vary according to the speed of its source (being truly additive and subtractive), these two times should (I believe) have been the same - as found in the earlier example. Observers moving with the moving Rod would thus report that the two clocks were not synchronous, while observers in the stationary system would report that they were! The conclusion is thus that we cannot attach any absolute significance to the concept of simultaneity (as Newton would assume) in that two events when viewed from a system of coordinates (ie from a given environment) appear to be simultaneous, appear not to be so when observed from a system which is in motion relative to the one where they do so appear. Time would thus seem to be dependent upon the motion of one's viewing platform and so not be invariably constant whatever may be such motion as always thought to be previously. But is the time as perceived from either platform more correct or valid than from the other? Apparently not. Both are equally valid. Both systems appear to be stationary to those in them (if closed off from other environments and the motion is uniform). And what about Space? Has Einstein thus 'asserted' (earlier) that two apparently equal lengths are not so and (here) that two apparently equal time intervals are also not so? If so, he would presumably next prove, explain or clarify such assertions.
271. In Section 3, he does appear to address these latter points, amongst others, and is entitled:
3. 'Theory of the Transformation of Coordinates (of Space) and of Times from a Stationary System to another System in Uniform Motion of Translation Relative to the Former.'.
['Translation' appears to be a technical term implying that everything concerned moves together in one direction to a new position along just one axis (here X).] That he describes this section in terms of a 'theory' seems to imply that, again, he is going to set out his thinking as though his conclusions follow inevitably from his premises but without, at this point, being able to completely justify or verify them; for it is but a theory. From the time of Galileo (ca 1600) upto that of Lorentz (ca 1900), it was accepted that one could describe the distance, time and speed pertaining to the motion of an object in its own immediate environment from the point of view either of that environment or, by the use of transformation equations, from that of any differently moving environment. Thus a ball thrown down the corridor of a train moving at 100 mph would appear to move at, say, 30 mph within the train itself but actually travel at 100+30 = 130 mph - at least as seen by those observing from the station through which the train runs. Such a simple sum is one such transformation equation. Others could be derived to show the difference in distances travelled as viewed from each perspective. Normally, however, no discernible differences in the times taken for the different views of the throw would be expected; the greater distance travelled by the ball as seen from the station would be accounted for in regard to the time taken by its extra speed. If the ball was thrown in the opposite direction, the equation would entail a subtraction - of 100-30 = 70 mph as the apparent resultant speed. If what was 'thrown' was a rather 'magic' ball (or a small bundle of 'slow' light) which, miraculously, always travelled at just 30 mph (ie its speed was a constant), the transformation equation would have to show that its speed as viewed from both within the train and from the station somehow always remained at just 30 mph. Clearly, some aspect of the transformation equation would have to be different to allow for this unexpected 'reality' that there would now be no additive or subtractive factor in the calculation. The same would apply to any comparisons made in respect of the (other) two components of a body's velocity - distance and time.
The above example (which we may call 'A') could be described equally from the point of view of the throwing 'event' occurring instead on the stationary platform and now being observed from the moving train. In this case ('B'), the platform can appear (to those on the train) to be travelling in the opposite direction to the train and, for the analogous situation, the ball would be thrown in the opposite direction. But the same transformation equations would apply. Also, 'the event' need not necessarily be conceived as motion of an object over space, but as occurring at a single 'point' (as simply turning over in the thrower's hand, say) in one or the other environment; the equations would simply calculate that the ball was travelling at 0 mph in respect of its surrounding carriage but at 100 mph (the train's velocity) in relation to observers in the station - in example 'A' - and vice versa in 'B'. The transformation equations would be to that extent simpler - whether for some specified distance travelled (by the train alone), the velocity of same or the time it so travelled. It may be pointed out that throughout the development of his theory, Einstein doesn't always make it clear just what 'the event' of concern is or what is the body that is moving. Is it a material point or a rod or anything or...?
272. In his next demonstration, Einstein sets out to derive just such equations (and in doing so, as dictated by his two principles, will presumably find that the values of the time and distance elements involved will no longer remain absolute and constant but will have to be relative - ie to the differing speeds of the two systems - and thus vary). [One says 'presumably' here as someone who has already 'read around' the subject; this would not necessarily be the case for those reading his paper for the first time - in 1905.] He begins with two systems of coordinates (reference systems) in stationary space which we may term system M (moving) and system S (stationary) whose X axes coincide (and seem to be considered as a single shared axis), while the other two, Y and Z of each system, are independent but parallel. Each system has identical measuring rulers and clocks associated with it. While Einstein doesn't specify which system is right or left of the other, we may take system M to be to the right of S initially (as observed before the reader). (If it was otherwise, the kinematics and arithmetic may simply entail more subtractions than additions but the final results would presumably be identical.) System M is thus seen as moving to the right - away from the stationary system S.
He then states that: 'for any event that occurs in S - at a point position (called here x(s), y(s), z(s) at time t(s) as measured therein by its own rulers and clocks - which could be at its origin where all 4 coordinates would = 0), there will 'belong' a comparable set of values which pertain to system M (to which Einstein gives rather obscure Greek symbols; for our purposes, they will be called x(m), y(m), z(m) and T(m) (or just T - for 'Tau'). [Note: We could equally assume that 'the event' occurs in M (as in our 'A' example of the moving train above) and the position of same considered similarly in terms of coordinates of M for which a comparable set of coordinates 'belong' in system S (eg 'the station'). However, his derivation is based on an arrangement that equates more to our example 'B' so we shall utilise that one.] He then says 'we wish to discover the system of equations which 'connect' these two sets of values. He thus uses the terms 'belong' and 'connect' without too much explanation as to meanings - in regard to the relationships implied. We may assume that both terms indicate that we are seeking the set of coordinate values in one system by or in terms of which an event in the other system may be validly described or measured. That is, as in the examples above, transformation equations are to be derived which allow one to describe the elements of motion from either point of view (S or M) - and, crucially, which are compatible with the two stated (mutually-compatible) principles.
To maintain some contact with our concrete examples, we may thus imagine the two systems M and S to be comparable to the moving train and the station, respectively, as described in example 'B' above. We may imagine in addition that the events of concern occur at the same height on the vertical axis (Y) - say at eye level - and at the same depth on the horizontal axis (Z) in the two systems - the Z axis extending into the background away from the station and railway line (ie perpendicular to them), the events occurring at a point mid-way between the rails, say. The other horizontal axis (X) could be a bank running parallel with and just beyond the tracks. In system M, the value of x(m) on its part of the X axis would be at its origin initially (ie = 0) and the position of a reflecting mirror (introduced below) some constant distance beyond that. In terms of system S, these values of x(m) and that of the mirror (x(m)+ ?) (in system M) will move from the origin of S at the rate of vt. While the two sets of coordinate values would appear to be directly measurable with appropriate measuring devices as eg in the case of the ball, so that the relevant transformation equation would, as always assumed in the past, be based on our usual values of distance, time and thus speed (v), the consideration of a 'body' (as light) - which travels at a constant speed (c) as measured from both systems (ie is observed so to do from either system) will require a modification of those values in the new and more valid transformation equations now to be derived. And that constant speed will have consequences for the perceived speeds (v) of any slower bodies and/or of the origins (and entire systems) of moving frames of reference as well.
273. Einstein begins his analysis by saying that the equations to be found (by which distance, time and thus speed will be transformed) will be linear (as time and space are deemed to be homogeneous). But then (for no reason that is immediately apparent) he begins the actual derivation of the new transformation equations by defining a value on the X axis called x' which is equal to x(s) - vt. Now we have noted that the X axis can apply equally to either system; that is, it is common to both - extending in a sense from system S into and through system M as the latter moves increasingly further to the right along it. The value of v pertains to system M while that of x(s) and t(s)) pertain to system S. The value of the newly introduced term x' is thus a function of values in both systems. He described earlier an event as occurring at a point in system S - ie at x(s), y(s), and z(s). As it uses the X axis symbol of system S (ie x(s), and not the rather obscure one suggested by Einstein for system M (which we call here x(m)), the derived value x' would itself thus appear to represent essentially a value in the former system (S). Initially (when t(s) = 0), this position would be nearer the origin of S (than whatever value x(s) itself represented) - unless x(s) was taken as being at its origin (ie x(s)= 0) in which case x'(s) would become increasingly negative as t(s) increased - and system M consequently moved to the right at velocity v. If the origins of systems S and M initially coincided so that x(s) = x(m) when both t(s) and t(m), and both would then = x'(s) = x(s) - vt. As M moved to the right at vt, so the value of x(s) in terms of system M's coordinates would continue to = x(s) - vt = x'(s). Thus, the value of x(m) that represents in M an event at x(s) in S would equal x(s) - vt; that is, would equal x'(s). The position of the original value of x(s) in terms of M's coordinates would also be increasingly to the left of system M's origin - ie would have negative values. [The foregong provides some ideas about the derived value x'(s) which Einstein defined as above (as = x(s) - vt). But what its real meaning or relevance is may be in regard to the derivation of the transformation equations, I am as yet unaware.]
In any case, he then points out that any point at rest in system M - as, for example, its origin (ie 'at rest' relative to that moving system but actually 'moving' with it) would also have a set of positional values describable in terms of a set of coordinates of system S - ie as x(s), y(s) and z(s), say. While y(s) and z(s) would remain constant in value as M moves to the right, x(s) of this set would have to increase in value continuously at the rate of vt in order to represent the point in S where the point 'at rest in moving M' is located. But all components of the set of x'(s), y(s) and z(s), on the other hand, would remain constant and fixed in terms of system S, irrespective of passing time t(s); ie they would be independent of time - since x'(s) = x(s) - vt and so always exactly balance the distance that the origin of M moves to the right. Thus, whatever is the value of t(s) as system M moves increasingly to the right over time, the position of x'(s), y(s), z(s) would remain stationary within S. Of course, initially, when t = 0, x'(s) will = x(s) and the situation would not be differentiated from that which would be described for x(s), y(s) and z(s) (with t = 0). At that moment, the event in S could be described in terms of the coordinates of its own system S by either (ie both) of the foregoing sets, ie equally - as they would coincide. But over subsequent times of t(s), these two sets would differ. One may ask: 'What, respectively, do they each represent' - especially in regard to the set of coordinates of system M (that belong to one or other of those two of S - ie where 'the event' occurred) that we wish to determine? We should point out that the 'point at rest in system M' which Einstein says we can describe in terms of its position in S, is apparently not where 'the event' he mentioned earlier occurs (at least as far as one understands). However, as far as calculating the equivalent points (ie that mutually 'belong' to each other) in the two systems is concerned and the transformation equations of relevance thereto, it is probably not necessary to describe an actual event. The respective 'points' alone can represent the 'events' (which are either stationary or move in their respective systems). As such, it may be the case that 'the point at rest in system M' - as discussed by Einstein above - may be considered in that light. In any case, any conclusions reached regarding an event in system S (as represented in sytem M) would be identical to those reached in the converse situation. They're symmetrical. Thus, the station could appear to be moving to those on the train (and their own environment the stationary one) given sufficient blinds, one peep hole and a super smooth, quiet train.
[We must also presume that Einstein's purpose could not be achieved by calculating his equations in terms of x(s), y(s), z(s) and t(s) alone but that the role of the value x'(s) (kept 'stationary) is somehow necessary. The former set of coordinates may seem the more straight forward and understandable. By 'forcing' these measures to accord with the two principles, one could imagine (having previously 'read around' the subject) deriving a set of equations in which some factor or function incorporated the demands of the constancy of light's speed (c) in ratio with that of the moving system such that a variation and relativity of time and distance would have to emerge. However, for whatever reason, Einstein proceeds along his seemingly necessary if more complex path - ie by utilising this stationary/constant position on axis X - ie x'(s) - rather than the ever-increasing x(s) - at least in respect of the (non-event?) point in M that he has just described; (although to what end?).
In any case, he then defines the time (T - as 'Tau' in Greek)) in system M (also called T(m) here) as a function of the latter 3 coordinates of system S - ie x'(s), y(s) and z(s) - but now including that of time t(s) in that system as well. That is, the time T(m) in M will be a function of the time t(s) in S - somehow calculated in terms of x'(s), y(s) and z(s. Clearly this is a different position to the point x(s), y(s), z(s) and t(s). As the clocks are synchronised in the stationary system in which both x and x' exist, one wonders how the time t(s) in S (of which T is to be shown as a function) could be affected by this one way or the other and thus how it would affect time T in system M. One will shortly be introduced to the inclusion of a reflective mirror (see below) some constant distance from a light source at the origin of system M. The position on X of both the light source at M's origin and that of the mirror at some value of x(m) (in terms of system S) - would continually increase as M moves at velocity v over time t(s), whereas that of x'(s) would, I believe, remain stationary. We apparently must interpret all these positions and movements in terms of Einstein's initial description of an event occurring in system S - at x(s), y(s), z(s) - for which we seek to find the set of coordinates in system M that 'belong' to or 'connect' with same. That is, in terms of our train-station example ('B'), the 'event' at a point in the station occurs at this known position and time and we wish to calculate the position and time of that event (or point?) in terms of the coordinates of the train. This may (as suggested above) entail thinking of the station as (apparently) 'moving' (in the opposite direction) relative to what appears to those on the train as their own 'stationary' system. In either case, any point on the X axis of system S would be described as a point on that same axis with respect to system M as one that was increasingly left of that system over time - at the rate of vt.
The values of T (as To, T1, T2; see below) in its own moving system M would be as shown on its own moving synchronised clocks. But we wish to calculate their values as seen from system S which we were earlier led to believe would differ from t(s) of system S (or as measured as T within system M itself). [But our original example of those on the station observing the thrown ball on the moving train calculated its speed at 100 mph plus 30 mph = 130 mph. That is, 'the event' was in system M and we calculated its speed in terms of system S's parameters. Einstein focuses on the other direction - describing 'the event' as occuring at a point in system S and thus (one assumes) calculating its equivalent from the point of view of (ie as seen from) system M. But he now says we wish to calculate one of the parameters (time) 'as seen from system S !!?? Possibly this light signalling element in his derivation does not represent the moving body event per se (for which transformation equations and relative values of time and space are required or are entailed), but some necessary adjunct needed to calculate (derive) these?]
He thus continues: 'Let a ray of light be emitted from the (moving) origin of M at the time To (in M) and have it travel to (stationary) x'(s) along the X axis (presumably in system M) where it is reflected (at time T1) back towards the origin of M (travelling towards it) - where it arrives at time T2. The time at T1 (the middle point on its journey) is then shown by Einstein to be one half of the total time for the return trip. ie:
[Note: the distance from M's moving origin to x'(s) of system S (and effectively stationary re system M) and from it back to that origin (moving towards it) would thus differ. The distance from the origin of system S to x'(s) (and back) would however remain constant, I believe (as x' = vt) - but how that may or may not prove relevant, I'm uncertain.]
Despite the remarks made above, transformation calculations should still apply if the moving body concerned was such as a small 'packet' of light albeit moving at a very great speed - or at least would normally in the past have been expected to be the same - by anyone who assumed that the speed of light was not a constant (as the speed of everything else in the universe was similarly taken not to be). One could thus follow Einstein's reasoning and mathematics in which he focuses initially not on the speed of the projectile concerned (ie light) nor on the distance travelled, but on the times taken - as in the equation above re To, T1 and T2. But, by ignoring the constancy of the speed of light, such that the values of c+v and/or c-v would retain their full arithmetic values throughout, the value of T (in system M) would prove to be the same as that of t of system S - as seen from either system. If transformation equations were calculated on this basis, any factor or function applied to the values of time and distance (ie typically called Beta) would presumably equal 1 and thus have no effect. This can be contrasted with the case where the Constancy of light's speed is properly assumed. In this case, it appears that Einstein proceeds on the prior understanding that time in a moving system (as seen from a stationary one) is not equal to that in the stationary system. That is, he isn't going to discover this by doing the calculations; he somehow already 'knows' or suspects this. He also 'knows' that c is indeed a constant so that slower and shorter values of time and distance in M, respectively (as measured from S), will have to allow for this and be expected. That is, where in the false example it was assumed (wrongly) that a real value of c+v would apply and influence the outcome accordingly, in the proper calculation (or experiment) this sum (c+v), as well as (c-v) where applicable, must somehow always equal c - no more and no less. For the principle of relativity to continue to hold universally true, time and distance in system M as viewed from system S have to adjust to accommodate that actual constancy of light's speed. Whatever the complexities of his mathematics in showing the derivation of his equations, the 'bottom line' is that the 'sums' entailing any addition (or subtraction) of the velocity of either frame of reference vis a vis the other one to that of the velocity of light must in a sense be neutralised by virtue of an 'adjustment' in the relevant values of perceived time and space. This perceptual adjustment is required by the principle of relativity so that observers in either environment would be unable to differentiate their own state of motion by any difference in that law of nature (concerning the constancy of the velocity of light), nor indeed in any other law of nature affecting motion.
Thus he continues by saying that this ('half-way') time at T1 can also - "by inserting 'the 'arguments' of the function T' and 'applying the principle of the Constancy of the velocity of light' in the stationary system", be expressed (necessarily in terms now of both time and spatial coordinates (at time = 0 or ?) as:
[That is, the purely 'temporal' statement regarding time in M alone (with its origin's spatial coordinates understood) is converted thereby into one in which both temporal and spatial aspects of system S and the velocity (v) of system M and that of light (c), are now also incorporated. He is thus analysing Time in system M in terms of all relevant measurements involved in the motion of some body (ie here light) from both systems (one also 'in (relative) motion') - thus implying the basis of the transformation equations needed to calculate time T in terms of time t - of which the former is (partly) a function - ie because of the 'demands' of the constancy of light's speed. [Note: One would like to know how to verbalise tha above equation. That is, would it be such as: " Time in system M (of an event occurring at a point x(s)=x'(s), y(s)=0, z(s)=0 and at time = t(s)+x'(s)/c-v) equals one half of the sum of the Time at...etc)" - with the coordinates so described being of the relevant system(s).] Presumably, he could equally have analysed Space (distance) in system M in terms of similar measurements from both systems and so calculate distance (from origin to x', say) in that moving system in terms of the relevant Distance in system S - it again (or also) varying due to those same 'demands'. In any case, this 'resolution' of his equation would appear to be very fundamental in the overall derivation of his theory. By the phrase 'the arguments of the function T' he appears to mean that he will be calculating time T in system M as a function - of t in system S - and do so by 'inserting' into his simple equation for times (T1, etc) alone those necessary other parameters of the motion (of a body) - ie distance (origin to x'and back?) and velocity - of system M (ie v) and of the body concerned - light (c). These are seen to have been incorporated within the 4th (time) coordinates of both T1 and T2 - as t+(x'/c-v) and t+[(x'/c-v)+(x'/c+v)], respectively. That is, these represent the extent to which the perception of time (t) (alone?) is altered in system M (as viewed from S) to give its values as T. [Is Distance so perceived not also altered thus? Seemingly 'yes' - as the eventual transformation equations will apply equally to that parameter (along the X axis) as well.]
Had he sought the function by which Distance in M varied relative to that in S there should be equivalent parameters so incorporated, one imagines. In either case, one would be seeking to discover/calculate just what function T is of t and such other parameters (and/or what function Distance in M (as seen from S) is of some equivalent distance parameter in S and its other relevant parameters. That is, how much - in terms of t, v, c, and distance do the values of system M's time (T), space or velocity (as seen from S) have to be altered to 'allow for' the other element mentioned above by Einstein to be considered in expressing any such initial equation (re T1 or some Distance equivalent) in the more comprehensive form needed - viz: the Constancy of light's speed (c)? ('Needed' - to derive the transformation equations.) By seeking the answer to 'how much' - one is in effect seeking to calculate some constant (eg B (beta) by which time t and Distance are appropriately dilated, shortened, or whatever - ie to 'allow for' the constancy of light's speed at any given velocity of a moving system. When something is a function of something else, then, unless that function = 1 presumably, there would be more than one factor of which it is a function. Thus, if something (x) is a function of y (ie which equates to, say, .75 of y (vs 1)), then it must be .25 a function of something else (say z, or z and q or whatever). Thus, while T may well be a function of t, it is also partly determined by (is a function of) such as the velocity of system M, the Distances involved, and probably of c as well(?) - ie parameters that comprise that function beta. [Note: As suggested already, one could proceed as above but without imposing the constraint of that 'other element' to be considered - when expressing the equation in its more comprehensive form. This would imply that the magnitude of M's velocity would be added to that of the velocity of the body (ie light) whose motion is being so assessed. How would that affect the values of the 4th coordinates (ie of time) used in the more comprehensive equation - and its derivatives? The differences with those used above may help define just what those latter entail. This is touched on further below.]
Einstein then continues his derivation of this crucial function (beta) by means of a calculus step - thus: 'If x'(s) be chosen infinitesimally small (does this mean the distance origin to x'(s) approaches zero or ?), the foregoing equation can then be expressed as:
[Which must say approximately the same thing - but as applied at the level of a ?point or whatever is implied via the apparent efficiency of calculus. [Sadly, I don't follow this particular step.] He continues by 'explaining that: 'As light is always propagated (as an expanding sphere proceeding from the origin of M) along all three axes equally - at the velocity (when viewed from system S) of the square root of (c2 - v2) (ie = c - v, with negatives removed?). [But does it not always travel at a constant c ?] In any case, we will then find that dT/dy and dT/dz both equal 0 - whereas dT/dx' would have a value which is less than 0 by an amount (v/c2-v2)(dT/dt). Referring back to the important assumption that T is a linear function, it then follows (says Einstein) that:
That is, the time T in the Moving system is some unknown function (a) of the time t in the Stationary system - less a (normally tiny) fraction of the speed of light; that is, time T in the Moving system is (typically) very slightly less (ie 'slower' or 'more dilated') than it is seen to be by those in the Stationary system - at least if we take the (previously unknown) function (a) to = 1 which in fact it is later shown conveniently to be. ('Typically' in the sense that at 'normal' velocities, the relevant fraction and thus the difference would be exceedingly small; but if and where v was very large (ie fast), the time T in the system perceived as Moving that fast (by those in the Stationary system) would apparently appear even more dilated (ie slowed) as compared to their own time t. In the above equation, it is assumed also that at the origin of system M, T = 0 when t = 0. This rather important latter feature is described by Einstein in the phrase "...and where for brevity it is assumed that..." ie that the foregoing values of t and T are as thus given. [As we know, measurements pertaining to the elements of motion (ie distance per time equals velocity) must be made with respect to a defined frame of reference, and do so in regard to both of the elements - of space and time. They can't be made in relation to a vacuum - for either. Thus, if there is no point in space that is fixed and absolutely at rest, then there is no point in time that is absolute 'zero' either - only relative space and time are available; all reference points (of both) are thus 'moving' and an appropriate/relevant comparative/reference set must be specified. [One wonders if the basis for the reference point specified for time in Einstein's explanation/derivation of relativity may be partially concealed in his use of the term 'briefly' - and the subsequent comment about time in system M at its origin being assumed to be equal to zero - ie 'when t = 0' - ie without further explanation.] Thus, at the origin of M, x(m), y(m), z(m) and T are all valued at zero (ie 0,0,0,0) as are x, y, z, and t in S - as these will apply to his important equation T1 = To + T2. [We may recall that Einstein did say that he struggled for years ...'until suddenly, it came to me...that 'Time' was the key...'. Was the contents of this 'eureka' moment concealed at all within this term 'briefly', or was this but one aspect of a more extensive insight and quite irrelevant to these matters?]
274. It is, in any case, with the foregoing equation - ie:
that it becomes possible to calculate the system of values in system M (equivalent to x, y, z, t of system S) and called here x(m), y(m), z(m) and T(m) (or just T) in terms of which an event* occurring at x, y, z - at time t - in system S, can be determined (measured?). [Note that one could equally derive the coordinate values in system M from those in S. To use Einstein's term - the two sets of values (in S and M) 'belong' to each other.) The equations so derived incorporate/express the fact that light is (also) propagated at its constant value c in system M (ie just as it is in S) - this being the requirement demanded by the two principles of concern. It gains nothing in speed from the speed of its source in M. We thus find that, for a ray of light emitted at the time T = 0 in the direction of increasing values of x(m) (in system M) that:
A system of transformation equations are utilised when one wishes to measures any aspect of the motion of a body (or of light) occuring in, say, a moving system for which there are already local measurments available but are also wanted from the point of view of another, slower or 'stationary' system. The available measurements are to be referred instead to the latter reference frame such that the speed of light is (as mentioned above) not shown to benefit in its speed whatsoever by virtue of the speed of its moving source (ie in the moving system) and that of anything else (any body or particle) only by some proportion of that system's speed - this proportion decreasing as the velocity of that system (as seen from a stationary system) increases until, at the speed of light, it would benefit not at all and so would travel at the same speed as it would in system S. The speed of that frame and anything else moving with it will be proportionally influenced such that the principle of relativity would continue to hold true. [This, I know, has been analysed elsewhere.] Thus, by using the appropriate transformation equations so derived (which recognise that the speed of light is the same in both systems - gaining no speed advantage from the Moving system - yet according with the principle of relativity), it becomes possible to describe the elements of motion for any event that may occur in one system as perceived from (ie referred to) a differently moving system - and do so in a way that allows the two principles of concern to prove mutually compatible. The crucial 'function' by which the normal (Newtonian or Galilean) transformation equations are revised turns out to be based on a ratio between the relative velocity (v) (of either system to the other) and the velocity of light (c). This was derived previously by Lorentz (for his electrodynamic studies) but on an incorrect basis. Einstein derived them independently - and in terms of the correct basis.
* Note: An 'event' would normally entail the motion of a body through space, over time, but one can imagine an event occurring at a 'point' (or particle) if it simply 'turned over' (on its own spot, as it were) or just suddenly 'appeared' or 'exploded' at one point. As such, an event could happen at a given point (x, y, z or wherever) and so avoid having to complicate its transformation by considering a body's motion - from a to b, etc. But an emmision of light rays, at least, does represent the motion from a to b or whatever of a kind of body and the moving point of an origin may represent another - as applies in the present analysis, I believe.
275. We continue by considering the equation which gives us the value of x(m) in the Moving system M - ie x(m) = ac(t - v/c2 - v2 times x'). The ray of light moves, relative to its origin in sytem M (ie at the latter's origin), with a velocity of c - v - when measured in the stationary system S. [Again this seeming inconsistency in light's otherwise Constant speed! Is he saying that 'the ray moves' - ie 'the velocity of light (c) is...' = (c - v)?? How can this be? Can 4 ever = 4 - 2 ?] However, he continues by showing (on what basis?) that the time t in system S may be shown as:
[I hope to see just what this equation, however derived, means and how it is the case.] If then we insert this value of t (the time in system S) into the equation for x(m), we obtain
In an analogous manner we can find that for rays moving along the other two axes:
Thus
By substituting 0 for x' (its value when t = y/sq rt (c2 - v2) [why?], we obtain a qualified set of the transformation equations we are seeking - ie:
where B (beta) = 1/sq rt(1 - v2/c2)
(in each case qualified by (being multiplied by) an unknown function of v, as calculated later). In addition, an additive constant would apparently be required on the right side of each equation - if no assumptions are made as to the initial position of system M and as to the zero point of T in that system. Thus, we have calculated the crucial value of the function as sought. [It is interesting to consider how the foregoing mathematical derivation would proceed if one assumed throughout that the speed of light (c) was additive with the velocity of system M (as was always assumed in the past) such that the eventual value of beta would equal 1 and thus that time T (as seen from S) would be found always to equal that in S itself (ie t) and that relevant distance values would also remain the same as viewed fron either system. By this means, it should be possible to see exactly where the effects of the proper value of c on the derivation ultimately occur and thus 'how it all works'!]
276. It is now necessary, says Einstein, to prove that any ray of light as measured in system M is only propagated at velocity c (as often indicated above) if this is its speed in system S. This would provide the proof that the two principles are indeed compatible - as has been asserted (and would require that time and space adapt accordingly - from absolutes to 'relatives', although this is not the immediate focus of this proof). Thus - at the time t = T = 0, when the origin of the coordinates of the two systems coincide, let a spherical wave of light be emitted therefrom and be propagated with velocity c in stationary system S. When a point (x, y, z) is just attained by this wave, then:
Transforming this equation with the aid of our recently derived equations of transformation, we attain after a simple calculation:
The wave (ie the light) under consideration is therefore no less a spherical wave (of light) with a velocity equal to c when viewed in or from either system. This shows that the two principles are indeed compatible. That is, by applying the adjustments in the values of time and space inherent in the transformation equations, it is possible to reveal the constancy of light's speed as observed from a stationary environment regardless of it travelling within and sourced from even a fast moving environment. The two principles are compatible because of those suggested 'adjustments'. Their validity, in turn, can only be established by the empirical realisation of relevant predictions in which such adjusted values have been incorporated.
277. The transformation equations derived above included the unknown function of velocity. This function turns out to equal unity and hence the final form of the equations is as shown above in blue. It apparently relates to the fact that the direction (sign) of the velocity of the moving system does not affect the dimensions of any Rod thereof set parallel to the y or z axes (ie perpendicular to x - in which direction is the movement concerned). But the kinematics of this proof is even more complex that that of the foregoing and we shall accept it in terms of its conclusions per se.
278. In his next Section:
4. Physical Meaning of the Equations Obtained in Respect of Moving Rigid Bodies and Moving Clocks,
Einstein addresses the matter of the apparent physical effects on the Space (length) of moving bodies and on Time during that movement as measured by moving clocks - in both cases as observed from a slower-moving or even relatively 'non-moving' perspective. Such effects would, presumably, be the manifestation of the relativity of these components of motion as concluded via the arguments presented in the first 3 sections discussed above. The body to be thus examined is a rigid sphere of radius R (ie seen as a sphere when examined at rest) which is now moving with system M (and at rest relative to it) with their common velocity = v (relative to system S), with its centre coinciding with the origin of system M. In terms of the coordinates - x(m), y(m), and z(m) (of the 3 axes of system ?M) - with time there (T) not mentioned), the equation of the surface of the sphere is given as:
In terms of the coordinates of system S - ie x, y, z - when time t in that system = 0 - that equation becomes:
A rigid body with the form of a sphere when measured in a state of rest (eg by those moving with it in system M), has when viewed/measured in a state of motion (by those in a stationary system) the latter form - of an ellipsoid (of revolution) with the apparent spatial magnitudes of its 3 dimensions X, Y and Z becoming:
Thus, whereas the Y and Z dimensions of the sphere (or of any rigid body of whatever shape) do not appear to be modified by the motion, the X dimension does appear shortened in the ratio of 1 : sq rt(1 - v2/c2). The greater the value of v, therefore, the greater the shortening. When v increases to the value of c, all moving objects, as viewed from a 'stationary' system, would be seen to have shrunk (with respect to the dimension of its motion) to zero. It would become a two dimensional plane. The speed of light at c is taken to be an infinite limit of speed for anything so that v can never be greater than c. The same results would hold for bodies at rest in the 'stationary' system when viewed from a uniformly Moving one of whatever velocity.
279. With respect to the effect of motion on Time, we may imagine one of the clocks qualified to mark the time t when at rest in system S, and the time T when at rest relative to system M, to be located at the origin of system M and adjusted to mark the time T there. When that moving clock is viewed from system S, what rate of time passage does it display ? Einstein continues: 'Between the quantities x, t, and T, which refer to the position of the clock (?), 'it is evident that': **
Therefore, T = t sq rt(1 - v2/c2) = t - (1 - sq rt(1 - v2/c2))t from which it follows that the time T marked by the clock (as viewed from the stationary system) is slowed by 1 - sq rt(1 - v2/c2) seconds per second - ie by 1/2 v2/c2 (where, he points out, magnitudes of fourth and higher order are neglected). The greater the value of v, the greater would be the apparent slowing (dilation) of time - as well (as shown above) the greater the degree of shortening of bodies seen as so moving.
**[This latter wording is a paraphrase of Einstein's (who actually says 'we have, evidently, x = vt...' which, in English, could suggest 'apparently' which is not, I believe, what he intends here); In any case, this equality is evident in that...while system S is stationary, the position of x, y, z therein is Not (necessarily) - at least with respect to its 'equivalence/connection/belonging to' the position in M of an event which occurs in M - itself moving at velocity v along the X axis. Thus, the position on that axis in S of the event in M is represented by x (of system S) = vt. The event thus has two sets of positional values: (i) within M - as x(m), y(m), z(m) - (at time T) and (ii) within S (as x, y, z and t. Hence, calculations entailing the extent of motion in terms of position x, can be substituted in the equations with vt.]
[A short time after Einstein's 1905 paper was published, his former physics professor at Zurich (Minkowski) suggested that the 3 dimensions of space and the one of time could be conceived as a single 4-dimensional concept of 'spacetime'; they always occur together and are interdependent. It occurred to me that this concept (and those of its component 'parts') is/are definable in terms of motion only. They have been so conceived (by man) in order to help us understand our universe and what 'happens' in it. One could thus imagine time to be: 'that which allows two identical events to occur at exactly the same place in space, and space to be that which allows two identical events to occur at exactly the same time'. In each case, such events, which manifestly do or can occur (as far as we can perceive), would be impossible without the construction of such concepts - or now, apparently, without that single concept. In this description, any event/happening must entail motion of a body. Without that, there would seem to be no need for time although a totally 'frozen' universe (or part of same; black hole?) would presumably always require space in which/where such bodies exist - except possibly at the instant of the big crunch or before the big bang - if all matter disappeared into (almost) infinite energy such that no space or time was needed?? But then, even an instant implies time and does energy not occupy space?] The foregoing effect of velocity on (perceived) Time leads, says Einstein, to a 'peculiar consequence': Time runs more slowly in (non-pendulum) clocks at the Equater than at the Poles! He explains this by showing that such a clock (A) synchronised with another of identical type (B) at the start of a journey which travels some distance, either in a straight or curved line at uniform velocity v, will no longer be synchronous with the stationary one but will lag behind it (as shown earlier) by an amount = 1/2 v2/c2 times t (ie the amount of time it so travels). If clock B is at a (relatively stationary) Pole and A on the Equater, after 24 hours the latter (having travelled about 24,000 miles will thus be very slightly slower than that of B which has remained relatively stationary (albeit both are travelling much faster through space with the Earth around the Sun also).
In the final Section (of Part I on Kinematics) - ie:
5. The Composition of Velocities ,
Einstein shows the impossibility of combining velocities to sum to more than the velocity of light (at c). The relevant proofs entail further algebraic manipulations of the variables of velocity (ie distance and time) - in terms of the restrictions on same inherent in his transformation equations. We may accept that adding together ratios of (shortened) distance to (increased) time (ie in eg high velocities) may well prove to be less than previously believed - when it was not appreciated that such variables were not apparently so altered.
Einstein next shows how the kinematic laws of his theory as deduced above apply not only to the mechanodynamics of moving bodies but to their electrodynamics in particular - in Sections 6 to 10 in Part II. Ie:
280. In the first section (6) of this Part, two related topics are covered - viz:
6. The Transformation of the Maxwell-Hertz Equations for Empty Space. and On the Nature of the Electromotive Forces Occurring in a Magnetic Field During Motion. [Is this the topic Poincare addressed in 1906?]
[Note: While one may not have understood the mathematics of every step in the foregoing development of his theory, we may ask ourselves now whether we can at least set out more clearly just what was the exact problem(s) Einstein was seeking so to answer and what essentially was that basic answer (ie his 'result') - even if every step is not fully comprehended. Possibly Part 2 will provide further clarification as to both the problem and its complete answer? ]