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Light-speed constancy versus light-speed invariance in the derivation of relativistic kinematics.

248; see Torretti |1983~, note 8, pp. 292-3, and Zahar |1989~, p. 101). For a discussion of Einstein's earliest doubts about Maxwell's equations, see Miller |1981~, Section 2.4. By 1905, of course, Einstein was largely committed to the granular micro-structure of radiation--his revolutionary light quantum hypothesis of that year was an unambiguous demonstration that he regarded Maxwell's equations as possessing only statistical ('phenomenological') validity.

2 The independence of the light-speed with respect to the state of motion of the source 'proves to be the true essence of the old aether point of view', as was aptly put by Pauli in his I EINSTEIN'S 1905 LIGHT POSTULATE

The main purpose of Einstein's celebrated 1905 paper on relativity was to provide a 'simple and consistent theory of the electrodynamics of moving bodies taking as a basis Maxwell's theory of bodies at rest' (Einstein |1905~; for an English translation, see the Appendix in Miller |1981~). In order to derive, using the principle of relativity, the transformation formulae for the field components in the Maxwell-Hertz equations, Einstein wanted first to establish the form of the associated spatio-temporal coordinate transformations. The special theory of relativity was born in this first ('kinematical') part of the paper.

The spirit of Einstein's kinematical discussion is quite contrary to that of the subsequent electrodynamical discussion. He was concerned to derive the coordinate transformations by appeal to basic principles which, in so far as they appealed to electromagnetic phenomena, relied as little as possible on the details of Maxwell's theory. Einstein knew what he was doing was momentous. He was questioning the basic prevailing assumptions about the nature of space and time, and he could not afford to do so on the basis of a theory of electrodynamics which he (better than anyone else in 1905) knew was inconsistent with the micro-behaviour of black-body radiation.(1) But there was one gross feature of light that was common to all ether-based theories from Maxwell to Lorentz, and which in Einstein's view would survive any subsequent modification to the theory of radiation. This was the principle of the 'constancy of the velocity of light', defined in the following way:

Any ray of light moves in the 'resting' coordinate system with a definite velocity c, which is independent of whether the ray was emitted by a resting or by a moving body. (Einstein |1905~, Section 1.2)

That a theory whose effect was to render the electromagnetic ether redundant should rest on a postulate which happened to express the principal feature of ether theories(2) is not the only irony here; earlier in 1905 Einstein had introduced the revolutionary notion of the light quantum, one whose prima facie compatibility with the light postulate above seems doubtful.(3)

A word is in order regarding the nature of Einstein's 'resting' frame.(4) Einstein made it clear at the beginning of his kinematical discussion that this frame can be chosen arbitrarily from the infinity of inertial frames ('in which the equations of Newtonian mechanics hold'),(5) the term 'resting' merely distinguishing it 'verbally' from all the others. However, this implicit appeal to the relativity principle at the outset does little to clarify the origins of the light postulate, or indeed Einstein's subsequent use of it.

Throughout the paper, the physics being assumed to hold in the resting frame corresponds to what informed 1905 readers would expect to hold in the inertial frame relative to which the hypothetical, isotropic Maxwell-Lorentz ether is at rest. In 1907, Einstein was to write of the constancy of the lightspeed: 'It is by no means natural to expect that |it~ . . . should be actually satisfied in nature, yet--at least for a coordinate system of a certain state of motion--it is made likely by the confirmations which Lorentz's theory, that is based on the assumption of an absolutely resting ether, has obtained by experiment' (our emphasis).(6) Much later, Einstein was to reiterate the point: 'Scientists owe their confidence in this proposition |light-speed constancy~ to the Maxwell-Lorentz theory of electrodynamics' (Einstein |1950~, p. 56).

Lorentz's 1895 theory of the electron shared all the succeses of Maxwell's electrodynamics, as Lorentz's field equations reproduced Maxwell's for macroscopic phenomena. (A further irony to be noted here is that in abandoning the Maxwellian view that the luminiferous ether is comprised of ponderable matter, in which the light-speed is determined ultimately only by the elastic properties of this rarified material, Lorentz himself could provide no analogous mechanical explanation for light-speed constancy in the ether rest frame; see Saunders and Brown |1991~, footnote 18.) In assessing Lorentz's theory, Einstein was aware of some of the theoretical difficulties involved in constructing a 'reasonable' theory of electromagnetism on the basis of the alternative emission theory of light, (Einstein's pre-1905 analysis of the emission theory, in which the light-speed does depend on the speed of the source, is mentioned in Stachel |1982~, pp. 51, 52.) Furthermore, an important feature of Lorentz's theory, and one cited by Einstein in his 1907 paper, was that it provided the first satisfactory ether-based dynamical derivation of the Fresnel drag coefficient (later shown to be direct consequence of Einstein's relativistic velocity-transformation law by von Laue in 1907). The existence of this effect had been directly verified in the famous Fizeau experiment of 1851, and to greater accuracy in 1886 by Michelson and Morley; more indirect evidence was furnished by the null-result first-order ether wind experiments. All of this, then, gives some indication as to why Einstein felt relatively secure in postulating in 1905 the existence of a specific inertial frame--linked to Lorentz's theory--in which the light-speed is independent of the speed of the source and isotropic.

Moreover, that the constancy of the light-speed is not explicitly postulated ab initio to hold in all inertial frames (at least in Section I.2 of the 1905 paper which contains the definition of the light postulate cited above) ensures that the postulate remains logically independent of the relativity principle. This is an obvious desideratum in a semi-axiomatic derivation of the new kinematics of the type Einstein was constructing. As we shall see below, the corresponding behaviour of light in any ('moving') inertial frame distinct from the 'resting' frame was explicitly taken by Einstein in 1905 to follow from the conjunction of the light postulate and the relativity principle. This view would be reiterated in his later work. In 1921 Einstein wrote: 'The consequence of the Maxwell-Lorentz equations that in a vacuum light is propagated with the velocity c, at least with respect to a definite inertial system K, must therefore be regarded as proved. According to the special theory of relativity, we must assume the truth of this principle for every other inertial system' (Einstein |1921~). And in his Autobiographical Notes, he defines the principle of relativity as 'the independence of the laws (thus specially also of the law of the constancy of the light velocity) of the choice of the inertial system . . .' (Einstein |1949~, p. 57. A detailed analysis of the logic of the kinematical part of the 1905 paper is found in Williamson |1977~. In his more recent authoritative study of the foundations of relativity theory, Torretti also stresses Einstein's 'careful reasoning' in deriving the light-speed in the moving frame from the light postulate and the relativity principle; see Torretti |1983~, note 7, p. 295.)

We shall denote Einstein's principle of light-speed constancy in the resting frame by PLC. It is worth recalling that in 1905 there was no direct empirical evidence supporting PLC (distinct from the mentioned empirical success of the Lorentz theory), although today the principle is known to hold to an accuracy of better than one part in |10.sup.11~.(7)

We are interested in this paper in the generalization of PCL to all inertial frames, i.e., the claim that the properties of isotropy and independence of the speed of the source are universal (flame-independent)(8) properties of the lightspeed. We call this the principle of universal light-speed constancy, denoted by PULC. (As will be shown below, the non-trivial aspect of the generalization of PLC to PULC concerns only the question of the isotropy of light propagation.) We stress that this principle is weaker than the principle of light-speed invariance, which demands further that the numerical value of the light-speed is flame-independent.(9)

We shall examine in the following section two reasons for adopting PULC (given PLC), the first based on (a 'weak' version of) the relativity principle, and the second on the outcome of the 1887 Michelson-Morley experiment. Later, in Section 4, we derive the kinematics consistent with this principle (the 'PULC-transformations'), and show that light-speed invariance is now equivalent to the kinematic principle of 'reciprocity'--a result which also follows from the classic 1949 paper by Robertson on experimental kinematics. The familiar Lorentz transformations are finally recovered in Section 5 essentially by further appeal to spatial isotropy, and without the assumption of reciprocity.


(i) Theoretical considerations. Einstein's 1905 derivation of the Lorentz transformations was based on the joint application of PLC and the relativity principle(10) (as well as the standard spatio-temporal symmetries). It is important to see how these principles worked together. The principle of relativity (PR) Einstein expressed thus:

The laws by which the states of physical systems undergo changes are independent of whether these changes of states are referred to one or the other of two coordinate systems moving relatively to each other in uniform translational motion. (Einstein |1905~, Section I.2)

He argued first that 'light (as required by the principle of the constancy of the velocity of light, in combination with the principle of relativity) is also propagated with velocity c when measured in the moving system' (Einstein |1905~, Section I.3). Einstein was here referring to a ray of light propagating in the direction of relative motion of the two frames in question; he went on to argue that PR and PCL further imply that a spherical light pulse in the resting frame must be seen as a spherical pulse in the moving frame. That is, both speed and isotropy of propagation are universal, or invariant (frame-independent) properties of light.

Let us consider isotropy-invariance first. This, and nothing more, was the real result of the 1887 Michelson-Morley experiment involving stationary sources (an experiment of which Einstein was almost certainly aware, as shown in Miller |1981~, p. 91), as we shall see later. But how does isotropy-invariance follow from PR and PLC? The example of sound waves propagating in an isotropic mechanical medium is instructive. The speed of sound waves is independent of the speed of the source, but PR certainly cannot be taken to imply that sound propagates isotropically in a frame in uniform motion with respect to the medium's rest frame. (Of course, if the medium itself is also boosted, then PR predicts isotropic propagation in the moving frame. But in Lorentz's theory, the luminiferous ether is assumed to be static or undraggable, largely for reasons associated with the phenomenon of stellar aberration.) Now the difference between light and sound is effectively spelt out in the first page of Einstein's paper: the laws of electrodynamics and optics (whatever they are) are simply assumed to be on a par with the fundamental laws of mechanics, the latter taken to have the same form in every inertial frame. The assumption, in short, is that a static (immobile) luminiferous ether does not exist: there is for light no mechanism which (like the sound medium for the laws of sound propagation) breaks rotational symmetry under boosts.

The case for the numerical invariance of the light-speed is arguably more subtle than is commonly thought. As Einstein defined it, PR is a condition on the laws of physics. In the kinematical part of the paper, Einstein is careful to avoid stating what the true laws of electrodynamics and optics are, as we have seen. But whatever they are, questions may be raised as to what precisely the sameness of these laws under boosts means. In Maxwell's theory, the value of the light-speed c in vacuo in the ether frame is a function of the values of the electric and magnetic constants ||Epsilon~.sub.o~, ||Mu~.sub.o~ of the 'vacuum' in that frame. The numerical values of these constants, and hence of the light-speed are fixed by a mixture of convention and experiment. (See, e.g., the discussion in Rosser |1971~, pp. 82ff.) Suppose identical procedures are set up in the moving (primed) frame. It is not inconceivable that the value of at least one of these parameters could be found to differ from that in the resting frame, even though the wave equation for the electric field, say, is found to take the same form as in the resting frame (with c|prime~ replacing c therein):(11)

|Mathematical Expression Omitted~.

That this possibility is consistent with the relativity principle PR was claimed in a 1952 paper by Stiegler (Stiegler |1952~). More recently, the notion that PR and PLC jointly imply only PULC (and not light-speed invariance) found support in the 1984 work of Tzanakis and Kyritsis, who claimed that PR 'does not a priori mean that the speed of light (or any other parameter) characterising each of these |inertial~ systems has the same value in all of them' (Tzanakis and Kyritsis |1984~).(12) In analysing the validity of this claim, one is faced with a putative ambiguity inherent in the slippery concent of form invariance of physical laws. Recall that in 1905 Einstein had no access yet to the relatively precise language of geometric 4-tensors, a development awaiting the work of Minkowski. But it is important to realize that such developments would have been of no avail to him in 1905 anyway. Recall that Einstein was using PR to derive the Lorentz transformations: a formulation of the principle involving reference to such geometric objects--for instance, 'the fundamental laws of physics must be expressible as equations in the 4-tensor calculus'--would of course render such a procedure circular.

None the less, the ambiguity in PR referred to above is arguably avoidable in the pre-geometric formulation, as long as we are careful enough in enunciating the significance of the principle. (It is akin to the task of specifying what is meant by spatial isotropy without being able to appeal to the notion of 3-tensors). What is at stakes is a choice between two fairly unambiguous, if not wholly precise versions of PR--the 'strong' and the 'weak'.

Strong-PR states that both 'the form of the laws of physics and the numerical values of the physical constants that these laws contain are the same in every inertial reference frame' (Taylor and Wheeler |1966~, p. 13: see also Bowler |1986~, p. 10). This means in particular that the constants ||Epsilon~.sub.o~, ||Mu~.sub.o~ in the example above are numerically invariant by fiat. Weak-PR, on the other hand, is the restricted claim that only the form of the fundamental laws of physics is frame-independent; it allows for the numerical values of constants appearing in the laws--such as the light-speed in the laws of optics--to vary from frame to frame. (For a discussion of the covariance properties of Maxwell's equations which effectively distinguishes between strong-PR and weak-PR, see Rosser |1971~, pp. 82-4.)

Now there is no question at all that, however he might have expressed himself, Einstein throughout his 1905 paper was appealing to strong-PR. This is evident not only in relation to his treatment of light in the kinematical part of the paper, but also in relation to the electrodynamical discussion. And certainly the majority of physicists since 1905 have at least implicitly adopted this reading of his relativity principle, exceptions being Stiegler in 1952 and Tzanakis and Kyritsis in 1984, as we have seen.

Recently, the notion of 'form invariance' of laws used in the above definitions of both strong- and weak-PR has been criticized on the grounds of irrelevance. The possibility of a generally covariant formulation of theories which are designed to violate the physical indistinguishability of inertial frames is taken to show that the condition of form invariance is empirically empty, or at least unrelated to the relativity principle (Friedman |1983~, Section IV.5). This conclusion would have surprised both Einstein and Lorentz, who were able to derive the correct, and highly non-trivial transformation formulae for electromagnetic field components precisely by requiring the form invariance under boosts of the known field equations of electrodynamics. We shall say no more on this issue, except to comment briefly on another traditional formulation of strong-PR which appears at first sight to avoid reference to physical laws altogether.

This principle dictates that no experiment performed in a laboratory at rest in an inertial frame, and reproduced exactly in a similar laboratory in a distinct inertial frame with identical (controllable) initial conditions, can distinguish between the frames. The principle is squarely in the tradition established by Galileo, Newton, and Huygens in their earlier relativity principles.(13) But we must be wary in applying this principle to our optical problem: if we are to conclude from PLC that the light-speed is invariant, a substantial amount of theory about the nature of light must be adduced to distinguish it from, say, the nature of sound, which as we have noted above also satisfies the 'constancy' condition for signal speed in a 'resting' frame but not the invariance condition if we treat the medium as static. The fact of the matter is that PR is not a purely phenomenological principle. In any experiment, theory and hence laws are always required to say what it means to reproduce initial conditions in the moving frame, and what effects count as potential falsifiers of PR. If this point is overlooked, real effects such as the frame-dependent degree of anisotropy of the 2.7 |degrees~ K black-body cosmic background radiation can be interpreted as refuting strong-PR, an interpretation actually espoused by Dirac (Dirac |1980~).

If the precise formulation of Einstein's strong-PR is perhaps open to debate, what is not in doubt is the heuristic power of the principle in his hands in 1905. Like Poincare, Einstein asserted its applicability to optical and electromagnetic phenomena (a notion, incidentally, already implicit in the work of both Galileo and Newton).(14) But unlike Poincare, Einstein used it as a universal constraint, or 'restricting principle' on physical laws.(15) (It is noteworthy that such positive use of the relativity principle to constrain dynamics dates back to the writings of both Newton and Haygens on the problem of collisions. See Penrose |1987~, pp. 22, 49; a detailed discussion of Huygen's use of the relativity principle is found in Barbour |1989~, pp. 464-7.)

The question that will be taken up shortly is whether, when viewed as a constraint of kinematics, weak-PR is as consequential as strong-PR. The question, in other words, is whether Einstein could not have replaced strong-PR by weak-PR in the kinematical part of his paper.

(ii) Experimental considerations. In his classic 1949 study of the empirical foundations of the Minkowski space-time metric, i.e., relativistic kinematics, H. P. Robertson set out to establish the Lorentz transformations without recourse to the relativity principle, in either weak or strong form.(16) His starting point was Einstein's PLC postulate, and the standard spatio-temporal symmetries; the point of the study was to establish the constraints imposed on the linear coordinate transformations by the important experiments due to Michelson and Morley (1887). Kennedy and Thorndike (1932), and Ives and Stilwell (1938).(17) Now the significance of the result obtained in the first of these experiments (no significant fringe shifts resulting from rotation of the Michelson interferometer; Michelson and Morley |1887~) was correctly described by Robertson thus:

The total time required for light to traverse, in free space |in the moving frame~, a distance I and to return is independent of its direction. (Robertson |1949~, p. 380)

That is, the Michelson-Morley experiment establishes that the isotropy of the round-trip speed of light, which holds in the resting frame by hypothesis, also holds in the moving frame.(18)

Robertson proceeded to infer the consequences of this result for the coordinate transformations linking Einstein's resting frame and the moving frame; and we shall return to this calculation in the next section. What Robertson did not say explicitly was that PULC is also a consequence of the Michelson-Morley experiment (given the standard Einstein convention for synchronizing clocks, which Robertson explicitly adopted), despite the fact that variations in the speed of the light-source was not a feature of the experiment. The point is that so long as the coordinate transformations map equal velocities into equal velocities (as is the case with linear transformations), then ff independence of the speed of the source holds for the light-speed in the resting frame (PCL), it automatically holds in the moving frame.(19) (This holds for the Galilean transformations and, say, sound propagation.) Thus, the only non-trivial element in PULC (given PLC) is precisely the condition that the (round-trip) light-speed in the moving frame is isotropic.

(iii) We have seen, then, that given Einstein's light postulate PLC, the principle PULC can be justified by appeal either to the weak principle of relativity or to the Michelson-Morley experiment. Now in their 1984 work referred to above, Tzanakis and Kyritsis raised the question whether the Lorentz transformations follow from PULC, weak-PR and Einstein's remaining postulates. The question, put differently, is whether Einstein's original argument goes through without strong-PR, and therefore without light-speed invariance. Tzanakis and Kyritsis successfully showed that it does. (Recall that for these authors Einstein's PR just is weak-PR).

Like Einstein, Tzanakis and Kyritsis assumed in their derivation not only the usual spatio-temporal symmetries, but made use also of a basic kinematic principle (commonly known as 'reciprocity') stating that the direct and reciprocal relative velocities between the resting and moving frame are equal and opposite. They were apparently unaware that their result is virtually a direct consequence of Robertson's 1949 work. For it follows directly from Robertson's analysis that PULC and the 'reciprocity' principle, together with the standard symmetries, jointly imply the Lorentz transformations up to a scale factor, as we shall see later. (This also follows from the above-mentioned work of Stiegler, who in 1952 also did not seem to be aware of Robertson's 1949 paper.) It will also become clear later that the scale factor can readily be reduced to unity on the grounds of weak-PR and isotropy.

Now although it is not unusual to regard the kinematical reciprocity principle as a consequence of (strong-)PR, this was conclusively shown not to be the case by Berzi and Gorini in 1969, in an important paper which established that the principle is essentially a consequence of strong-PR and spatial isotropy jointly.(20) (We discuss this result in detail in Section 6 below. A simple demonstration that reciprocity is not a consequence of strong-PR alone is provided in the next section.) However, in the Tzanakis-Kyritis approach (which appeals to weak-PR), reciprocity must be considered an independent and non-trivial assumption. Notice that in Robertson's 1949 analysis reciprocity was not assumed at the outset; rather it was shown to be a consequence of the Michelson-Morley and Kennedy-Thorndike experiments, given PLC.

The question arises whether reciprocity is strictly necessary in the derivation of relativistic kinematics based on PULC. To answer this question, we must first define our terms.


We start with the pair of inertial frames S, S|prime~ corresponding to Einstein's resting frame and moving frame, respectively. The coordinate systems adapted to the frames are assumed to be in the 'standard configuration', with S|prime~ being measured to move relatively to S along the positive x-direction with velocity v. (The space-time origins in both frames coincide, their spatial axes have the same orientation, and time in both frames flows in the same direction.) It is assumed that both the standard rods and clocks which were available to observers at rest in both frames are of the same construction and preparation (an assumption which looks more natural when weak-PR is upheld). Finally, it is assumed that the coordinate transformations from S to S|prime~ are linear.(21)

Notice that the temporal coordinates in S and S|prime~ have no physical significance until the convention is established which 'spreads time through space', i.e., which determines how distant clocks are synchronized. For the moment, however, we shall leave the synchrony procedure unspecified.

It can readily be shown from the assumption that the linear transformations do not depend on the choice of the y-axis (and hence z-axis) in S that the coordinates for an arbitrary event satisfy the following matrix equation:(22)

|Mathematical Expression Omitted~

where a, b, d, and e are functions of v, and perhaps also of parameters which are intrinsic to S and S|prime~. Now for some body or signal propagating along the positive x-direction in S with velocity u, the velocity transformation formula resulting from (1) is

u|prime~ = a |center dot~ (u - v)/du + e (2)

If we now put u = 0, then u|prime~ will be the 'reciprocal' velocity of frame S as measured to move along the positive x|prime~-direction in S|prime~, which we denote by |Mathematical Expression Omitted~. From (2), we have

|Mathematical Expression Omitted~.

Now the principle of reciprocity dictates that |Mathematical Expression Omitted~, i.e., that a = e. This principle is widely assumed in derivations of the Lorentz transformations in the literature, as it was by Einstein in 1905, and by Tzanakis and Kyritsis in their 1984 derivation based on weak-PR. (We mentioned earlier that reciprocity is in fact a consequence of strong-PR and isotropy; it is also a consequence of light-speed invariance.)

Notice that reciprocity depends on both the convention for synchronizing distant clocks adopted in S and that in S|prime~. An immediate consequence of this fact is that reciprocity cannot be a consequence of any version of the relativity principle alone. For even when it is decided to adopt the same convention in both frames, the choice of the convention is not determined by the relativity principle. If reciprocity holds for one convention, then in general it fails for another. (Relativistic kinematics is an instance of this.)

Reciprocity holds, of course, in classical kinematics, in the sense of being a consequence of the Galilean transformations. Clearly, however, this does not settle the issue when the relative velocity v between S and S|prime~ is large. Consider, for example, the following reciprocity-violating transformations (defined for Einstein's synchrony convention, which we shall discuss below), where |Gamma~ |is equivalent to~ |(1 - |v.sup.2~/|c.sup.2~).sup.-1/2~:

x|prime~ = |Gamma~ |center dot~ (x - vt)

y|prime~ = y

z|prime~ = z

t|prime~ = ||Gamma~.sup.2~ |center dot~ (t - vx/|c.sup.2~)

for |absolute value of~ v |is less than~ c. The situation described here contains standard relativistic space contraction and relativity of simultaneity, but no time dilation. The failure of reciprocity effectively disappears in the regime v |is much less than~c.(23)

Finally, the following remarks on length contraction and time dilation will be useful later in the paper. Suppose we have a rod stationary in S|prime~, and oriented along the x|prime~-axis. The length of the rod as measured in S is obtained by ascertaining the coordinates of the ends of the rod simultaneously. Then |(length).sup.s~ = C |center dot~ |(length).sup.S|prime~~ defines the contraction factor C. Similarly, suppose we have a clock stationary in S|prime~, measuring the time interval between two events occurring on its world-line. Then |(time interval).sup.S~ = D |center dot~ |(time interval).sup.S|prime~~ defines the dilation factor D. (Notice that both factors C and D depend on the synchrony convention adopted only in S.) It is easily shown from (1) that

C = |a.sup.-1~ (4a)

D = |(e + dv).sup.-1~. (4b)

Now it is a standard assumption in relativity that standard rods and clocks remain just that under boosts. That is to say, a standard rod or clock in S, when accelerated (not too violently!) to velocity v, may be considered to be standard also in S|prime~. Given this assumption, which is adopted throughout this paper, it follows that C and D indeed determine the extent to which rods and clocks, as measured in the resting frame S, contract and dilate, respectively, when boosted from rest to velocity v.


It is now time to introduce Einstein's light postulate. The first thing to note is that (the synchrony-independent part of) PLC warrants the adoption of Einstein's clock-synchrony convention. Recall that this is the convention whereby the one-way speed of light is the same in either direction along any spatial axis, although it may vary from axis to axis. Without source-speed independence holding for the round-trip light-speed in S (and hence in S|prime~), this convention is simply not well defined. (A convention is no less a convention for depending on an empirical postulate in order make sense.) It is worth examining the consequences of Einstein's convention.

Consider a ray of light moving along the positive x-axis in S; we have in (2) above that u = c, and u|prime~ = c|prime~. Einstein's synchrony convention implies that at least for |absolute value of~ v |is less than~ c (where it is assumed that the direction of the light ray is invariant under the transformation),

(-c)|prime~ = -c|prime~

which with (2) yields

d = -ev/|c.sup.2~. (5)

From (1) and (5), we obtain the transformations(24)

x|prime~ = a |center dot~ (x - vt) (6a)

y|prime~ = by (6b)

z|prime~ = bz (6c)

t|prime~ = |center dot~ (t - vx/|c.sup.2~) (6d)

So far we have not used the requirement that c in S and c|prime~ in S|prime~ are independent of the spatial orientation of the light ray, which is a consequence of PULC. Once PULC is adopted, it becomes fairly obvious that reciprocity is equivalent to light-speed invariance. Let us conceptually replace all the standard clocks in S|prime~ by contrived clocks, whose rates are so adjusted as to make the lightspeed in S|prime~ equal to c. The coordinate transformations appropriate to this contrived case are familiar: light-speed invariance implies:(25)

x|prime~ = |+ or -~k|Gamma~ |center dot~ (x - vt) (7a)

y|prime~ = |+ or -~ky (7b)

z|prime~ = |+ or -~kz (7c)

t|prime~ = |+ or -~k|Gamma~ |center dot~ (t - vx/|c.sup.2~) (7d)

for |absolute value of~ v |is less than~ c, where |Gamma~ is again the Lorentz factor |(1 - |v.sup.2~/|c.sup.2~).sup.-1/2~, and k is a positive dimensionless function of v/c.(26) The minus part in the |+ or -~ signs in (7) is removed by requiring that the transformations reduce to the identity transformation in the limit v |right arrow~ 0. It is now easily checked that reciprocity holds. Conversely, the switch back to standard clocks in S|prime~ will affect the value of v, but not of v, so reciprocity clearly fails when c |is not equal to~ c|prime~.

The use of contrived clocks also allows us to derive in a very simple way the coordinate transformations appropriate to the standard clocks when PULC holds. (A more systematic derivation without recourse to contrived clocks will be given below.) The transition from the contrived clocks back to the standard docks affects only the element in (7) which determines the time dilation factor, this being contained in the factor k|Gamma~ in (7d). (To see this, recall that the v/|c.sup.2~ coefficient of x in the bracket in (7d) is a result of source-speed independence and the Einstein synchrony convention--see (6d)--which do not depend on clock rates.) It follows that the time coordinate transformation appropriate to the standard clocks will take the form:

t|prime~ = K |center dot~ (t - vx/|c.sup.2~) (8)

where K remains to be determined. The velocity transformation formula resting from (7a,b,c) and (8) is:

u|prime~ = k|Gamma~ |center dot~ (u - v)/K |center dot~ (1 - uv/|c.sup.2~). (9)

If as before we put u = c and u|prime~ = c|prime~, we obtain

K = k|Gamma~ c/c|prime~ (10)

which fixes our PULC-transformations:

x|prime~ = k|Gamma~ |center dot~ (x - vt) (11a)

y|prime~ = ky (11b)

z|prime~ = kz (11c)

t|prime~ = ky c/c|prime~ (t - vx/|c.sup.2~) (11d)

for |absolute value of~ v |is less than~ c. We remark that k may be a function of v, c, and c|prime~, and that it is straightforward, though tedious, to verify that it is a sufficient condition for the transformations to form a group that k = 1. Furthermore from (9), (10) and putting u = 0, we have for |Mathematical Expression Omitted~ defined as above

|Mathematical Expression Omitted~

which can be checked directly from (11) and (3), showing again that reciprocity is equivalent to light-speed invariance given PULC.

This result was essentially anticipated by Robertson in 1949. The transformations (11) do not appear in his study, but they follow from the conditions Robertson derived as a result of PLC and the Michelson-Morley experiment,(27) which (as we saw in the last section) jointly imply PULC. Robertson's more lengthy derivation made no use of contrived clocks and the transformations known to hold for them, but was based on first principles. A new derivation of this kind, which follows more closely the standard derivation of the Lorentz transformations based on light-speed invariance, is as follows.

Consider a point source of light which emits a pulse at the space-time origin in S. PULC implies that the wave-front equation in S

|c.sup.2~|t.sup.2~ - |x.sup.2~ - |y.sup.2~ - |z.sup.2~ = 0 (12)

is 'weakly' preserved in S|prime~:

|c|prime~.sup.2~|t|prime~.sup.2~ - |x|prime~.sup.2~ - |y|prime~.sup.2~ - |z|prime~.sup.2~ = 0 (13)

for t, t|prime~ |is greater than~ 0.(28) In the standard case of light-speed invariance (c = c|prime~), it is well known that joint satisfaction of (12) and (13) implies that for an arbitrary pair of events, the 'square interval' they define is invariant up to a dimensionless conformal factor |+ or -~|k.sup.2~:

|c.sup.2~|Delta~|t|prime~.sup.2~ - Delta~|x|prime~.sup.2~ - |Delta~|y|prime~.sup.2~ - |Delta~|z|prime~.sup.2~ = |+ or -~|k.sup.2~ (|c.sup.2~ |Delta~|t.sup.2~ - |Delta~|x.sup.2~ - |Delta~|y.sup.2~ - |Delta~|z.sup.2~) (14)

Now it transpires that even when c |is not equal to~ c|prime~, the weak version of (14) for arbitrary pairs of events

|c|prime~.sup.2~ |Delta~|t|prime~.sup.2~ - |Delta~|x|prime~.sup.2~ - |Delta~|y|prime~.sup.2~ - |Delta~|z|prime~.sup.2~ = |+ or -~|k.sup.2~ (|c.sup.2~ |Delta~|t.sup.2~ - |Delta~|x.sup.2~ - |Delta~|y.sup.2~ - |Delta~|z.sup.2~)

follows from the joint satisfaction of (12) and (13) for events on the (now non-invariant) 'light-cone' centred on the space-time origin in S, S|prime~. The proof is given in Appendix I. In the special case where one of the events in question occurs at the space-time origin, we have for the coordinates of the other event:

|c|prime~.sup.2~|t|prime~.sup.2~ - |x|prime~.sup.2~ - |y|prime~.sup.2~ - |z|prime~.sup.2~ = |+ or -~|k.sup.2~ (|c.sup.2~|t.sup.2~ - |x.sup.2~ - |y.sup.2~ - |z.sup.2) (15)

One can now substitute (1) into (15) and compare coefficients in the resulting quadratic forms. Assuming (a) that in the limit v |right arrow~ 0, the transformations reduce to the identity transformation and (b) that a light ray does not invert direction under a coordinate transformation for |absolute value of~ v |is less than~ c, the PUCL-transformations (11) are obtained straightforwardly, as the reader can verify.

Finally, we see how easily the Tzanakis and Kyritsis result of Section 2 follows from (11). If we consider the joint transformation S |right arrow~ S|prime~ |right arrow~ S, we have from (11) and c = c|prime~ (assuming reciprocity) that

k (-v) |center dot~ k(v) = 1

since weak-PR now implies that k is the same function of velocity in all frames. Furthermore, given that the 'relative space' in S is isotropic, k must be an even function of v, since otherwise the length contraction factor determined by (11a) is anisotropic. It follows, then, from the requirement that k is positive that k = 1,(29) yielding the Lorentz transformations.

We turn now to the question, posed earlier, whether reciprocity is indeed necessary in the derivation.


Given that space in S is isotropic, inspection of the transformations (11) quickly leads to the conclusion that for any pair of frames S|prime~, S|double prime~ with velocities v and - v relative to S respectively, the light-speed in each must be the same, i.e., c|prime~= c|double prime~. For if this identity is not satisfied, equation (11d) implies that the time dilation effect associated with the coordinate transformations will be anisotropic. However, considerations of spatial isotropy in S alone will not yield c|prime~ = c and therefore reciprocity.

Recall that the role of weak-PR in deriving PULC from PLC was to ensure that the isotropy of light propagation in S is maintained also in S|prime~. Presumably, it is a consequence of weak-PR that the form-invariance of all fundamental physical laws under spatial rotations in S will likewise be reproduced in an arbitrary inertial frame, i.e., that spatial isotropy in a quite general sense holds also in that frame. Now in 1976, Levy-Leblond provided a semi-rigorous proof that if isotropy holds in all frames, the linear transformations between them must satisfy reciprocity.(30) The proof requires however that the coefficients in the transformations vary continuously with v. We prefer to allow for the (admittedly implausible) possibility that c|prime~ (as in (11d)) may not vary continuously with v.

We not introduce the 'midway' frame |Mathematical Expression Omitted~ between S and S|prime~, such that the velocity of S relative to |Mathematical Expression Omitted~ is - w, and the velocity of S|prime~ relative to |Mathematical Expression Omitted~ is w, where w |is less than~ v. (Such a frame always exists: the formula for its velocity relative to S as a function of v is the same as in standard relativity theory.) According to an argument due to Rindler, whatever the linear transformations are between S and S|prime~, isotropy in |Mathematical Expression Omitted~ ensures that reciprocity holds for those transformations (Rindler |1985~, p. 14). Rindler's argument is somewhat informal, an d in our opinion is inconclusive. Indeed, the transformations (11) furnish a counter-example to it. However, a modified version of the argument does go through when consideration is given to the length contraction and time dilation factors in |Mathematical Expression Omitted~ determined by (11). (Consideration of such factors in the general case is not part of Rindler's original argument, but it is a feature of the Berzi-Gorini proof, mentioned earlier, of reciprocity given strong-PR; thus the following argument can be said to be a mixture of elements from these two derivations.)

We wish to compare the transformations |Mathematical Expression Omitted~ with |Mathematical Expression Omitted~. If we denote the transformations S |right arrow~ S|prime~ as A, and the transformations |Mathematical Expression Omitted~ as B, then we are comparing A |center dot~ |B.sup.-1~ and |B.sup.-1~, which from (11) are found (recall that these transformations form a group when k = 1) to be respectively:

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~


|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

where |Mathematical Expression Omitted~, |Mathematical Expression Omitted~, and |Mathematical Expression Omitted~ is the light-speed in |Mathematical Expression Omitted~. (Notice that we have not used the consequence of weak-PR that for an arbitrary pair of frames the coordinate transformations take the form of (11). The reason for this will become clear in the next section.) We take it that isotropy in |Mathematical Expression Omitted~ implies that the length contraction factors |Mathematical Expression Omitted~ for |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~ for |Mathematical Expression Omitted~ are identical. Given (4a), (16a), and (17a) this leads to k = 1. We assume that the similarly defined time dilation factors |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~ are likewise identical, which from (4b), (16d), and (17d) now yields c = c|prime~. This completes the (reciprocity-free) derivation of the proper orthochronous simple Lorentz transformations based on Einstein's light postulate PCL and weak-PR and isotropy:

x|prime~ = |Gamma~ |center dot~ (x - vt)

y|prime~ = y

z|prime~ = z

t|prime~ = |Gamma~ |center dot~ (t - vx/|c.sup.2~).(31)


(a) A diagramatic summary of the contents of the previous sections is now given. The 'isotropy' entries signify the principle that 'kinematic' effects such as length contraction are isotropic in the resting frame S; the bold arrows correspond to inferences which are proven in this paper.

(b) In Section 4, we mentioned that our transformations (11) follow from Robertson's 1949 analysis of the implications of the Michelson-Morley experiment given Einstein's 1905 light postulate (PLC), as well as the standard spatio-temporal symmetries in the 'resting' frame S. Robertson went on in his study to show that reciprocity (and hence light-speed invariance) is a consequence of the 1932 Kennedy-Thorndike experiment, and finally that the 1938 Ives-Stillwell experiment establishes that the scale factor k is unity even for large velocities v. Our analysis in Section 5, if correct, shows that appeal to the last two experiments in Robertson's approach is doing the work of a single principle: that space in an arbitrary moving frame is isotropic--as Robertson assumed to be the case in the 'resting' frame. (Recall that the argument in Section 5 did not require the assumption that the coordinate transformations between arbitrary frames take the form of (11). Such form-invariance is a consequence of weak-PR, but as we have noted Robertson avoided any form of the relativity principle in his study.)

(c) It is well-known that given strong-PR and reciprocity, it is possible to parametrize the family of relevant coordinate transformations in terms of the invariant velocity. This result was first obtained in 1910 by Ignatowsky.(32) who erroneously considered reciprocity to be a consequence of strong-PR. In fact, as Berzi and Gorini argued in 1969, strong-PR and isotropy alone imply that the form of the transformations is given by (Berzi and Gorini |1969~):

x|prime~ = |(1 - K|v.sup.2~).sup.-1/2~ |center dot~ (x - vt)

y|prime~ = y

z|prime~ = z

t|prime~ = |(1 - K|v.sup.2~).sup.-1/2~ |center dot~ (t - Kvx) (18)

for |absolute value of~ v |is less than~ |K.sup.-1/2~, where K is a universal constant having the dimensions of an inverse square velocity: |K.sup.-1/2~ is the invariant speed. Now the prediction of an invariant speed is highly non-trivial. But it is not, of course, tantamount to obtaining relativistic kinematics, suggestive though the above transformations are. Further constraints are required to exclude the (Gallilean) possibility that K = 0, and to pick out the correct finite value of K.

It is important to bear in mind, when assessing the above result, that neither Ignatowsky nor Berzi and Gorini explicitly specified the synchrony convention adopted in their derivations. Let us consider the Berzi-Gorini argument in more detail. These authors sought to prove first that strong-PR and spatial isotropy jointly imply reciprocity, as we mentioned earlier. They appealed to isotropy of the synchrony-dependent length contraction and time dilation effects for general linear transformations of the type (1) above (as we did in Section 4 for the transformations (11) in the context of the Einstein convention). Now such an isotropy principle is neither well defined nor plausible until the synchrony convention is specified and show in some fashion to be natural. Alternatively, one might construe the kinematic isotropy condition in the Berzi-Gorini argument as a constraint on the still unspecified synchrony convention. Be that as it may, the argument actually contains a hidden synchrony convention. Berzi and Gorini assume in their derivation of reciprocity that relative to an arbitrary inertial frame, the maximal velocity of moving frames is isotropic. In other words, if |Gamma~ is the set of allowed inertical velocities relative to the original frame, then |Gamma~ = |-|Xi~, |Xi~~, where |Xi~ |is less than or equal to~ |infinity~. This requirement serves to fix the synchrony convention in that frame, and hence in all frames. Berzi and Gorini showed that given strong-PR (which makes |Xi~ the maximal speed in all frames) and the above kinematic isotropy condition, reciprocity holds in that convention. The group structure of the resulting transformations (taken to be a consequence of strong-PR) results finally in the transformations (18). It is thus shown that the original synchrony convention coincides with that in which the invariant one-way speed is isotropic.

It is a consequence of Berzi and Gorini's work that the reciprocity assumption is redundant in any derivation of the Lorentz transformations which, like Einstein's 1905 argument, starts with strong-PR, light-speed invariance and spatial isotropy. We encountered an independent proof of this claim in Section 4 above. which does not depend on the specific isotropy principle in question.(33)

It should be noted finally that the group structure of the coordinate transformations is a consequence as much of weak-PR as of strong-PR. Thus, the transformations (18) are derivable from weak-PR and reciprocity. This is the weak-PR version of Ignatowsky's 1910 result, and is proved in the 1984 work of Tzanakis and Kyritsis. However, the existence of an invariant speed (finite or otherwise) cannot, as far as we know, be deduced from weak-PR and isotropy considerations alone, unless the coefficients in the coordinate transformations are taken to vary continuously with v.(34) The main result of our paper is that the addition of PLC to these assumptions (omitting that of continuity) is sufficient to obtain the correct invariant speed.

(d) There is a difference between our PULC-based derivation of relativistic kinematics and that based on light-speed invariance that is worthy of mention. It is a consequence of work on the 'synchrony-independent' formulation of special relativity (in which an arbitrary synchrony convention is permitted in each frame) that given the invariance of the round-trip speed of light, the isotropy of length contraction and time dilation in the resting frame implies the isotropy of the one-way light-speed in that frame.(35) (The converse is clearly not true; the transformations (7) with k as an odd function of v constitute a counter-example.) Thus kinematic isotropy, taken as a constraint on clock synchrony, can be shown in this case to demand the standard Einstein synchrony convention based on light.

This state of affairs does not hold in our derivation of the Lorentz transformations based on PULC above. Here, it must be assumed ab initio that the synchrony convention in the rest frame that renders the one-way speed of light isotropic also renders length contraction and time dilation isotropic. Normally this coincidence is attributed to the general principle of spatial isotropy taken to hold in the rest frame (as Einstein did unnecessarily in 1905). The relevant, explicit version of the principle states that there exists a synchrony convention such that fundamental laws from all branches of physics, when expressed in coordinates consistent with that convention, retain their form under rotations of the spatial coordinates. We adopted this line of reasoning in Section 5 above. It follows that the success of our derivation depends inter alia on the plausibility of the general isotropy principle in the version just given.

(e) In Section 2 above, we briefly touched on some of the difficulties involved in defining the relativity principle within the conceptual framework of Einstein's 1905 paper. Now it would appear that in the subsequent geometric approach to special relativity, the derivation of the Lorentz transformations strictly requires no specifically light-related postulate, nor indeed the relativity principle itself; it has to do only with the symmetry group associated with the postulated geometric structure of the four-dimensional space-time manifold. (See Friedman |1983~, Ch. IV, Section 2). But it is frequently overlooked that physical kinematics (the behaviour of real rods and clocks in motion) cannot be derived directly from the symmetry group (the Lorentz group) of Minkowski space-time without the further assumption that the covariance group of all the physical laws governing the behaviour of all conceivable rods and clocks coincides with the said symmetry group. Thus, an appeal to more than the metrical structure of space-time is necessary in any derivation of relativistic kinematics, and first-principle derivations in the pre-geometrical style of Einstein's original 1905 paper should not be seen as rendered outmoded by the geometrical approach. At any rate, such non-geometric derivations continue to play a role in the teaching of relativity, and to arouse both historical and conceptual interest. For this reason we have wished to contribute here to the already large literature on them by showing that the assumptions required therein are weaker than perhaps is generally appreciated.


We wish to thank M. Bowler, R. Clifton, N. D. Mermin, and particularly O. Johns, S. Saunders, and G. Stedman for very useful comments; their agreement with the views expressed here must not be taken for granted. One of us (A.M.) wishes to acknowledge support from the Brazilian Federal Post-Graduate Education Agency (CAPES) during a post-doctoral fellowship at the University of Oxford.

Sub-Faculty of Philosophy University of Oxford Oxford, UK

Departamento de Matematica Aplicada UNICAMP, Campinas, Brazil

Department of Theoretical Physics University of Oxford Oxford, UK

APPENDIX I. The 'Weak' Invariance of the 'Square Interval'

Theorem: Consider an arbitrary pair of events P and Q where the coordinate differences between them are |Delta~x, |Delta~y, |Delta~z, |Delta~t in the coordinate system adapted to frame S, and |Delta~x|prime~, |Delta~y|prime~, |Delta~z|prime~, |Delta~t|prime~ in that adapted to S|prime~. Suppose that whenever P and Q are such that

|c.sup.2~|Delta~|t.sup.2~ - |Delta~|x.sup.2~ - |Delta~|y.sup.2~ - |Delta~|z.sup.2~ = 0


|c|prime~.sup.2~|Delta~|t|prime~.sup.2~ - |Delta~|x|prime~.sup.2~ - |Delta~|y|prime~.sup.2~ - |Delta~|z|prime~.sup.2~ = 0

and conversely, when |Delta~t, |Delta~t|prime~ |is greater than~ 0. Then it follows that for arbitrary pairs P, Q

|c|prime~.sup.2~|Delta~|t|prime~.sup.2~ - |Delta~|x|prime~.sup.2~ - |Delta~|y|prime~.sup.2~ - |Delta~|z|prime~.sup.2~ = |+ or -~ |k.sup.2~(|c.sup.2~|Delta~|t.sup.2~ - |Delta~|x.sup.2~ - |Delta~|y.sup.2~ - |Delta~|z.sup.2~)

for some real positive number |k.sup.2~.

Proof: Here we generalize Rindler's proof for c = c|prime~. (See Rindler |1985~, pp. 15-16.) First, introduce for arbitrary P, Q the quadratic forms

Q = |T.sup.2~ - |X.sup.2~ - |Y.sup.2~ - |Z.sup.2~


Q|prime~ = |T|prime~.sup.2~ - |X|prime~.sup.2~ - |Y|prime~.sup.2~ - |Z|prime~.sup.2~

where T = c|Delta~t, T|prime~ = c|prime~|Delta~t|prime~, X = |Delta~x, X|prime~ = |Delta~x|prime~, and so on. Given the linearity of the coordinate transformations between S and S|prime~, Q|prime~ is also a homogeneous quadratic form in the S-variables T, X, Y, Z. Now the most general form of such a quadratic form is:

Q|prime~ = a|T.sup.2~ + b|X.sup.2~ + c|Y.sup.2~ + d|Z.sup.2~ + iXT+ jYT+ kZT+ lYZ + mXZ + nXY.

The hypothesis of the theorem states that

Q=0 |if and only if equivalent to~ Q|prime~ = 0

when |Delta~t, |Delta~t|prime~ |is greater than~ 0. In particular, Q|prime~ must vanish for (T,X,Y,Z)=(1, |+ or -~ 1, 0, 0). This yields i = 0 and a + b = 0. Similarly, we find j = k = 0 and a + c = 0, a + d = 0. Q|prime~ must also vanish for (|square root of~2,0,1,1). This yields l=0, and similarly we find m=n=0. Hence

Q|prime~ = aQ

Writing a = |+ or -~ |k.sup.2~ we obtain the desired result.

APPENDIX II. Rindler on Reciprocity

Recall that in Section 5 we were considering a pair of inertial frames S, S|prime~, with a third frame |Mathematical Expression Omitted~ 'midway' between them, the velocities of S and S|prime~ relative to |Mathematical Expression Omitted~ being equal and opposite. Rindler has argued that spatial isotropy in |Mathematical Expression Omitted~ implies that the reciprocity principle holds between S and S|prime~, whatever the linear transformations between these frames are, the argument being this.

|any~ manipulation performed in S to determine the velocity of S|prime~ can be regarded as an experiment in ||Mathematical Expression Omitted~~. The 180 |degrees~ rotation about the ||Mathematical Expression Omitted~~-axis in ||Mathematical Expression Omitted~~ of that manipulation is a possible experiment in S|prime~ for determining the velocity or S. But by the assumed isotropy of ||Mathematical Expression Omitted~~ the two experiments must yield the same result, which establishes our assertion ('reciprocity'). (Rindler |1985~, p. 14)

One difficulty with this disarmingly simple argument is apparent when it is recalled that reciprocity is synchrony-dependent, and yet no details of the synchrony convention adopted in the three frames above are specified in the argument. To clarify this point, let us imagine Rindler's 'manipulations' being performed in a (standard) relativistic world for pre-assigned frames S and S|prime~. The Einstein synchrony convention is adopted, as usual, in all frames, and reciprocity reigns. Suppose we now alter the synchrony convention in S: the one-way speed of light is no longer isotropic in this frame. It follows immediately that reciprocity fails between S and S|prime~.(36) However, the symmetrical relationship between the 'manipulations' as described in |Mathematical Expression Omitted~ is of course in no way affected, nor the assumption of spatial isotropy in this frame.

Another counterexample to Rindler's claim, this time in which Einstein synchrony is retained, is provided by a world in which light-speed invariance does not hold, but in which our equations (11) and hence (16) and (17) above are satisfied. For we can imagine the Rindler experiment in the specific form of a pulsed light source at rest at the spatial origin in S (S|prime~) and a mirror situated at the origin in S|prime~ (S): the time taken by a pulse of light emitted at some known finite time to be reflected by the moving mirror and return to the origin will determine the speed of the moving frame. Although each procedure can be arranged to correspond to the outcome of a 180 |degrees~ rotation of the other in |Mathematical Expression Omitted~, it is not a consequence of the fact that the |Mathematical Expression Omitted~-observer sees everything related to the two experiments occurring symmetrically, together with the transformations |16~ and (17) when applied to the emission and absorption events in the experiment, that |Mathematical Expression Omitted~, even when k = 1.

1 Einstein expressed in 1955 the significance of his 1905 discovery that 'the Lorentz transformation transcended its connection with Maxwell's equations and had to do with the nature of space and time in general. A further new result was that the Lorentz-invariance is a general condition for any physical theory. This was for me of particular importance because I had already found that Maxwell's theory did not account for the micro-structure of radiation and could not therefore have general validity' (quoted in Born |1956~, p.
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Author:Brown, Harvey R.; Maia, Adolfo Jr.
Publication:The British Journal for the Philosophy of Science
Date:Sep 1, 1993
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