Kinematic subtleties in Einstein’s first derivation of the Lorentz transformations Alberto A. Martı´neza) Dibner Institute for the History of Science and Technology, MIT, Cambridge, Massachusetts 02139 ͑Received 30 June 2003; accepted 14 November 2003͒ We analyze Albert Einstein’s derivation of the Lorentz transformations in his paper, ‘‘Zur Elektrodynamik bewegter Ko¨rper,’’ originally published in 1905. The analysis clarifies various misunderstandings in the secondary literature and reveals reasons why Einstein’s work entailed interpretive difficulties. © 2004 American Association of Physics Teachers. ͓DOI: 10.1119/1.1639011͔
I. INTRODUCTION namics. In the words of Russell McCormmach, Einstein ‘‘was able to carry through a profound critique of the foun- As the centenary of the special theory of relativity ap- dations of physics using elementary algebra, differential proaches, readers may want to understand how Albert Ein- 1 equations ... and with that light mathematical equipment he stein originally derived the Lorentz transformations. On the was able to formulate the kinematics of special relativity.’’3 one hand, we learn these equations easily as the simple alge- But even though Einstein used only algebra and calculus in braic core of Einstein’s novel kinematics. On the other hand, formulating his theory, his arguments were complex. The Einstein’s kinematics early on became known as notoriously mathematical subtleties are best brought to light by recon- difficult to understand, both to his peers and to nonspecial- structing his derivations in detail. He summarized in only a ists. To help students and general readers today, the present few equations the results of hundreds of operations. His paper clarifies kinematic subtleties that have deterred even omission of intermediate steps may have obscured the intel- specialists from understanding Einstein’s derivation. ligibility of his kinematics, as has happened with other sci- Einstein’s first work on the relativity principle was diffi- entific and mathematical texts throughout history.4 cult to understand for many readers. Consider the following 2 Because of its complexity, Einstein’s first derivation has examples. In the spring of 1905, the physicist Josef Sauter not been used in physics textbooks. In the Annalen der was one of the first persons to hear about Einstein’s work in Physik, it occupies five pages, but once unraveled, it occu- depth. Sauter was one of Einstein’s co-workers in the Swiss pies approximately thirty pages of text. Einstein presented patent office, and because he had studied and published on his derivation in about fifty-five equations, but worked out Maxwell’s theory of electromagnetism, Einstein ‘‘gave him explicitly the derivation involves more than three hundred his notes, which Sauter criticized severely: ‘I pestered him equations, consisting of roughly five hundred algebraic and for a whole month with every possible objection’.’’ Despite differential operations. Einstein’s succinct presentation thus his criticisms, Sauter facilitated a meeting between Einstein and Paul Gruner, professor of theoretical physics at the Uni- allowed his readers to skip many details and to skim the versity of Bern. Consequently, in 1907 Einstein gave his pa- substance of his argument, but at the expense of understand- per ‘‘Zur Elektrodynamik bewegter Ko¨rper’’ to Gruner, in ing his derivation clearly. For brevity, we will not give every support of his candidacy for a faculty appointment at Bern. step of Einstein’s derivation. The present paper selects key In the words of Gruner, ‘‘I received his essay although the points in need of clarification, and is thus meant to accom- whole theory at the time seemed to me to be highly problem- pany a careful reading of Einstein’s paper. Once the kine- atical.’’ The reaction of the faculty was even less favorable, matic meaning of each term is understood, readers can then as they ‘‘declared the work as inadequate: it was more or less carry out the mathematical steps without doing so blindly. clearly rejected by most of the contemporary physicists.’’ What follows is an analysis of Section 3, ‘‘Theory of The professor of experimental physics, Aime´ Forster, re- Transformation of Coordinates and Times from a Resting turned Einstein’s paper with the remark, ‘‘I can’t understand System to a System in Uniform Translational Motion Rela- a word of what you’ve written here.’’ To be sure, a few tive to It,’’ of Einstein’s 1905 paper. The virtues of Einstein’s 5 physicists seem to have understood the gist of Einstein’s ar- first derivation have already been highlighted. For example, guments promptly, including Max Planck. But even Planck Einstein derived the transformation equations without mak- first wrote to Einstein asking that he clarify certain points in ing any hypotheses about the constitution of matter, nor of his paper. intermolecular forces. The transformations were best suited What aspects of Einstein’s work were problematic to early for the solution of problems in electrodynamics, yet Einstein readers? Some difficulties were conceptual, because Ein- did not base them on the presumed validity of a contempo- stein’s ideas diverged radically from ordinary notions. rary electromagnetic theory, nor on Maxwell’s equations. His Doubtless, the major difficulty was Einstein’s novel concept derivation also was independent of whether light is presumed of the relativity of time. But there also were mathematical to consist of waves or particles, as Einstein used mainly the subtleties. Difficulties in understanding Einstein’s concepts concept of light-ray, appropriate to both conceptions. More- have been discussed by many writers; in contrast, we analyze over, Einstein’s derivation involved aspects that show both here his use of algebra. the logical economy of his thought as well as its conceptual Historians have remarked that the mathematics employed roots. In particular, rather than postulating the invariance of by Einstein was rather simple compared to the analytical the speed of light in nonaccelerated frames of reference, Ein- methods then employed by leading theorists in electrody- stein postulated the constancy of the speed of light in a single
1 Am. J. Phys. 72 ͑5͒, May 2004 http://aapt.org/ajp © 2004 American Association of Physics Teachers 1 frame and then derived its invariance. Such expository de- ϭt, ͑2a͒ vices paved the transition from a physics based on the privi- leged reference frame of the ether to one based on observa- ϭxϪt, ͑2b͒ tional and formal symmetries. Because such key virtues of ϭy, ͑2c͒ Einstein’s paper are well known, our goal is only to identify and clarify formal ambiguities. ϭz. ͑2d͒ The best known analysis of Einstein’s paper is Arthur I. Miller’s 1981 study,6 but it suffers, like others, from some These equations were seldom stated explicitly; in particular, errors that need correction. The present elucidation also will there was no need to express the equation relating the time expose difficulties that Einstein’s contemporaries may have coordinates, because time was assumed to be the same in all experienced in attempting to understand his work. reference systems regardless of relative motions. Before the 1880s, physicists used a single variable t for all such sys- tems. The other three equations, by contrast, were stated ex- plicitly at least occasionally, as done by Lorentz, for ex- II. KEY ASPECTS OF EINSTEIN’S DERIVATION ample, in his 1886 paper ‘‘De l’influence du mouvement de la terre sur les phe´nome`nes lumineux.’’7 Due to his research Compared to the conceptual analysis of measurement pro- on relative motion in optics and electromagnetics, he ad- cedures in the first two sections of Einstein’s paper, his for- vanced a series of modifications to the traditional transfor- mal derivation of the transformation equations was far more mations that eventually led to the equations advocated by abstract. It followed the mathematical tradition of J.-L. Larmor, Poincare´, Einstein, and others.8 Hence, Poincare´ Lagrange, S.-F. Lacroix, and others who dispensed with geo- gave the name ‘‘Lorentz transformations’’ to these new equa- metrical diagrams. It followed the descriptive approach es- tions, although Woldemar Voigt had published equivalent poused by Gustav Kirchhoff rather than explanatory ap- equations in 1887.9 In 1909 the simpler and older transfor- proaches involving models of causes or mechanisms. It mation equations were named the ‘‘Galilean transforma- involved a profound reliance on formal requirements such as tions’’ by Philipp Frank.10 What distinguished the new trans- linearity, symmetry, and the theory of functions. It did not formations in Einstein’s work in comparison to the involve the methods or concepts of vector theory, although equivalent equations in the earlier work of other physicists they had been advocated by influential physicists, such as was that Einstein introduced such transformations by means Peter Guthrie Tait, Oliver Heaviside, and August Fo¨ppl, to of general kinematic arguments, rather than introducing them replace the cumbersome methods of ‘‘Cartesian’’ coordi- exclusively for the solution of problems in optics and elec- nates, especially in electromagnetic theory. trodynamics. The end result of Section 3 of Einstein’s paper was a For simplicity, Einstein derived the four transformation group of four equations. To obtain them, Einstein began by equations given only the relative motion of the two systems positing two Cartesian coordinate systems, K and k, with along the X axis and ⌶ axis. Thus, only the relation between rectangular axes X, Y, Z, and ⌶, H, Z, respectively. He iden- the coordinates x and , and between t and , would be ex- tified these systems with rigid bodies, each consisting of pected to vary. To visualize the systems in relative motion, three mutually perpendicular rods, as he argued that the we may suppose that at the initial time t their coordinate axes meaning of coordinates, lengths, and times should be given coincide. Einstein began: ‘‘First of all it is clear, that the by specifications pertaining to rigid bodies and clocks. To equations must be linear on account of the properties of distinguish the systems, he identified K as ‘‘resting’’ and k as homogeneity that we attribute to space and time.’’11 This ‘‘moving’’ ͑in quotation marks͒. He then derived the four requirement can be expressed by the following: transformations relating the position and time coordinates, x, ϭa xϩa yϩa zϩa t, ͑3a͒ y, z, t, of any physical event in K to the coordinates , , , 11 12 13 14 ϭ ϩ ϩ ϩ ͑ ͒ , of the same event in k. The transformations expressed the a21x a22y a23z a24t, 3b simplest relation between systems in relative motion: the ϭ ϩ ϩ ϩ ͑ ͒ case in which the axes of K are parallel to the axes of k, and a31x a32y a33z a34t, 3c the two systems move relative to one another in a straight ϭa xϩa yϩa zϩa t. ͑3d͒ line with a uniform speed . By letting K and k be displaced 41 42 43 44 only along the X and ⌶ axes, the equations he found were That is, each coordinate of k is a function of the coordinates of K and four undetermined constants. At any one time any 2 ϭ͑tϪx/V ͒, ͑1a͒ particular value of a coordinate in K corresponds to only one ͑ ϭ͑ Ϫ ͒ ͑ ͒ value of a coordinate in k otherwise, had Einstein allowed x t , 1b the equations to be quadratic, then to each coordinate value ϭy, ͑1c͒ of K there could correspond two coordinate values of k, as, for example, ϭy 2 yields two possible values for ͒. ϭz, ͑1d͒ By the homogeneity of space and time, Einstein presum- ably meant that no locations or directions in physical space where ϭ1/ͱ1Ϫ2/V2, and V is the speed of light in empty are distinct or privileged, and that time likewise has a certain space, which we will henceforth designate by c, as already uniformity. For instance, if one were to place two identical done in 1905 by some physicists. ͑For ease of reference to measuring rods end to end, anywhere in space and at any the 1905 paper, all other symbols are in Einstein’s original time, all observers would agree that the result would be twice notation.͒ the length of one such rod. All points fixed in one nonaccel- These four equations replaced the equations previously erated system move with the same velocity relative to an- used by physicists to transform coordinates between systems other nonaccelerated system, and the transformations be- in uniform rectilinear relative motion: tween systems must be indifferent to the choice of origin for
2 Am. J. Phys., Vol. 72, No. 5, May 2004 Alberto A. Martı´nez 2 either system.12 In any case, Einstein did not offer such spe- discussed the subject of transformation equations.18 In this cific arguments to justify the linearity of the transformation context, Einstein’s use of Eq. ͑4͒ may be reinterpreted. He equations. wrote: ‘‘We set xЈϭxϪt, so it is clear that to a point rest- Einstein proceeded: ‘‘We set ing in the system k belongs a definite system of values xЈ, y, z, independent of time.’’ This ‘‘point resting in the system k’’ xЈϭxϪt, ͑4͒ moves at a velocity in K; it would depart from the position xЈ, y, z, and reach the position x, y, z, after a time t, thus the so it is clear that to a point resting in the system k belongs a Ј Ј 13 value x would vary with time, whereas x would not. Thus definite system of values x , y, z, independent of time.’’ Eq. ͑4͒ can be understood as describing the uniform rectilin- Equation ͑4͒ is identical to the Galilean transformation 14 ear motion of a point along the X axis of K. In this sense it ϭxϪt. The primed notation, as in xЈ, was used occasion- does not imply a transformation between reference frames. ally by some writers to designate coordinates in an additional Yet this interpretation, too, is defective, as we will see. 15 coordinate system. So it might seem that Einstein had in- In any case, the point ‘‘resting’’ in k is described in terms troduced a third coordinate system, because he used xЈ in- of the values xЈ, y, z of the system K. But why did Einstein stead of for the coordinate corresponding to x. Accordingly, not describe the position of the moving point with the values in his analysis of Einstein’s paper, Miller argued that Ein- x, y, z instead? By employing the value xЈ instead of x,he stein had introduced a third set of coordinates, an ‘‘interme- could hope to simplify his derivation of the function diate Galilean system,’’ in addition to K(x,y,z,t) and (xЈ,y,z,t), because then all terms, xЈ, y, z, are constant, k(,,,). Miller claimed that Einstein related the coordi- that is, ‘‘independent of time.’’13 Could Einstein have ob- nates in K and k ‘‘through an auxiliary set of space and time tained the same transformation equations had he used x in- coordinates in k(xЈ,yЈ(ϭy),zЈ(ϭz),tЈ(ϭt)) whose spatial stead of xЈ throughout? This question will be answered portion xЈ, y, z transforms according to the Galilean trans- shortly. formations; but every time coordinate is relativistic.’’16 Like- To ascertain the form of the transformation equations, Ein- wise, Roberto Torretti claimed that Einstein introduced ‘‘an stein began by seeking the relation between t and .He auxiliary coordinate system,’’ although Torretti argued that it wrote: ‘‘We first determine as a function of xЈ, y, z, and t. would be wrong to describe it ‘‘as a Galilei coordinate To this end, we have to express in equations, that is nothing 17 system.’’ Despite such claims, there is hardly anything in other than the embodiment of the data of clocks resting in Einstein’s paper to indicate that he introduced such an aux- system k, which have been synchronized according to the iliary system. Einstein did not refer to such a system, nor did rule given.’’19 Thus he based his derivation of the transfor- he introduce the terms yЈ, zЈ, tЈ. Moreover, he used the term mation equations on his procedure for synchronizing clocks: xЈ alongside the terms x, y, c, , and t, which he repeatedly ‘‘From the origin of the system k a light ray was sent out at identified as values of system K, as he emphasized, for ex- the time 0 along the X axis toward xЈ, and from there at the ample, that t ‘‘always denotes a time of the resting system.’’ time 1 is reflected back to the coordinates-origin, where it Einstein did introduce, explicitly, a third coordinate system arrives at the time ; thus it must then be: 1( ϩ ) ͑ ͒ 2 2 0 2 later in the paper. But only Eq. 4 can be interpreted as ϭ .’’ Note that because Miller interpreted xЈ as a coordi- suggesting that a third system was involved from the outset. 1 Ј nate of k, he stated that Einstein’s X in the passage just The interpretation of x as indicative of a third system quoted ‘‘was an oversight,’’ and that it should have been ⌶ creates difficulties for the validity of Einstein’s arguments. instead, ‘‘for consistency.’’20 But if we interpret xЈ other- Above all, how could Einstein expect to derive new transfor- wise, it is not necessary to change Einstein’s X; the light ray mation equations on the basis of a traditional transformation? emitted from the origin of k is described with respect to K. This problematic question is of basic importance for under- To relate the values of the function to the coordinate standing the transition from classical kinematics to Einstein’s values of system K, Einstein expressed , , and in kinematics. Also, why would Einstein assume the validity of 0 1 2 terms of the corresponding values of x, y, z, and t of K: his principle of the constancy of the speed of light only mid- way through his derivation? Moreover, why would he tacitly ϭ ϭ assume outright that yЈ y and zЈ z, without explanation, 1 xЈ xЈ whereas he then spent pages demonstrating that ϭy and ͫ͑0,0,0,t͒ϩͩ 0,0,0,tϩ ϩ ͪͬ 2 cϪ cϩ ϭz, rather than likewise assuming these relationships? Fi- nally, toward the end of his derivation, when Einstein explic- xЈ ϭͩ xЈ,0,0,tϩ ͪ . ͑5͒ itly did introduce the terms xЈ, yЈ, zЈ, tЈ as coordinates of a cϪ system, he used them as the coordinates of a system identical with K. So why would he use this very notation if he had first used it to designate coordinates of system k? To explain Eq. ͑5͒ we must clarify an ambiguity: the expres- Such problems disappear once we realize that Eq. ͑4͒ need sions cϪ and cϩ seem to indicate variations in the speed not be interpreted as a transformation equation, and that it of light. Miller, for example, interpreted them in this way.21 may have an entirely different meaning. Specifically, it can Yet Einstein’s analysis led to the conclusion that the speed of describe the uniform motion of an object departing from a light relative to k should be c. Thus we should rightly doubt coordinate xЈ, and moving along the X axis of K with a that he could derive this conclusion by letting light in k velocity , such that it is located at a coordinate x at time t. propagate at speeds of cϪ and cϩ. Likewise, these ex- This relation is a basic equation in kinematics: xϭxЈϩt, pressions are not speeds of light relative to the ‘‘resting’’ ϭ ϩ sometimes stated as x x0 ut, as for example in Kirch- system, because Einstein specified from the beginning that hoff’s Mechanik18 ͑a work studied by Einstein͒, where it des- the speed of light in K was c. So what do these expressions ignated the rectilinear motion of a point just before Kirchhoff mean?
3 Am. J. Phys., Vol. 72, No. 5, May 2004 Alberto A. Martı´nez 3 ϭ The question is answered by clarifying the algebraic likewise, t1 tB . Thus Einstein found that the time of arrival analysis Einstein presented in Section 2 of his paper where of the light ray at B is he used xЈ ϭ ϩ ͑ ͒ rAB t1 t Ϫ , 10 t Ϫt ϭ , ͑6a͒ c B A cϪ and hence the total time taken by the light to return to clock rAB A is ЈϪ ϭ ͑ ͒ tA tB ϩ . 6b c xЈ xЈ t ϭtϩ ϩ . ͑11͒ Here are the apparent variations in the speed of light. Ein- 2 cϪ cϩ ͑ ͒ stein obtained Eq. 6 by the following procedure. Consider a 23 rod of length rAB moving relative to K. One clock is at- Miller’s procedure to obtain these results for t0 , t1 , and t2 tached to end A of the rod and another to B. A light ray is somewhat different from the one here, because instead of ͑ ͒ ϭ Јϩ departs from clock A at its clock reading of t . It reaches using Einstein’s equation 4 , he used: x x t1 . Anyhow, A ͑ ͒ clock B at its reading of t , and then it is reflected back and given these results for t0 , t1 , and t2 , Einstein wrote Eq. 5 . B ͑ ͒ ͑ ͒ arrives at the first clock at its time tЈ . Because the rod is If we expect Eq. 5 to agree with Eq. 8 , then it should give A the values of the function in terms of the values of the moving, there is a difference between the time light takes to coordinates in K for the three events in question. However, travel from one end of the rod to the other and the return trip. Eq. ͑5͒ does not agree with Eq. ͑8͒. According to an observer in the stationary system, the ray of Ϫ Consider again the process of synchronization of moving light traveling toward B moves an extra distance (tB tA) clocks as observed from the ‘‘resting’’ system K. One clock in addition to rAB, because the end-point B of the rod moves is attached to the origin of system k, and the other clock is a away as the light ray approaches. Upon reflection at B the ray distance away on the ⌶ axis, such that both clocks move travels a distance less than rAB because A approaches it. uniformly with k, along the X axis of K. If the coordinate Hence the travel times of light in each direction are systems coincide at the instant when the light ray is emitted, the ray departs from the origin of both K and k. It then rABϩ͑t Ϫt ͒ Ϫ ϭ B A ͑ ͒ travels to the distant clock and is reflected back to the first. It tB tA , 7a c thus returns to the origin of k, because this process aims to synchronize the moving clocks. Because the origin of k has Ϫ ϩ͑ ЈϪ ͒ rAB tA tB moved away from the origin of K, then the x coordinate of tЈϪt ϭ . ͑7b͒ A B Ϫc the returning light ray cannot be 0 in K. Thus it is problem- ϭ atic to expect that x2 0 is the coordinate value in K for the Einstein didn’t give these equations, but notice that the mag- location of the ray on its return. Likewise, if we interpret nitude of the velocities of light in opposite directions is the Einstein’s xЈ as the location of the ray on its arrival at clock same: c. By solving for each time interval, we obtain Eq. ͑6͒. B, we also run into difficulties, because the light ray did not Thus we see that the expressions cϪ and cϩ did not travel only a distance xЈ to B, but an additional distance enter Einstein’s argument as speeds of light relative to K or k, because the clock moved away. but as the rates with which light approaches moving points In short, the problem is that whereas the values given for relative to K. That is, cϪ is the rate of light approaching the time coordinates t , t , t are values referred to K, the clock B, and cϩ is its rate of approach toward clock A. 0 1 2 values for the coordinates x , x , x seem to refer to k. Such expressions don’t imply any transformation, but in- 0 1 2 Normally, a statement of coordinates in parentheses, such as stead, the usual vectorial addition of velocities within a (x ,y ,z ,t ), was understood to designate values of the single frame, which is valid exactly in both traditional kine- 1 1 1 1 matics as well as in Einstein’s. This important distinction coordinates in a single system. In contrast, Einstein seems to between the addition of velocities in one system and the have mixed values from two different systems. This ambigu- ity poses interpretive difficulties. We can eliminate them by transformation of velocities between two systems is ne- Ј glected often.22 disregarding x and determining (x,y,z,t) strictly in terms Einstein used Eqs. ͑6͒ and ͑4͒ to establish the value of of x, as well will demonstrate in the Appendix. At this point, each time coordinate in Eq. ͑5͒. As stated, he needed to as- however, there is still one way out that makes sense of Eq. ͑ ͒ cribe definite values to the coordinates in K of the three 4 , namely to give xЈ another interpretation, making it stand events: the incidence of a light ray on clock A, then on clock neither for a coordinate of the moving system nor for the B, and then back to A, given that the two clocks are moving fixed initial coordinate of a moving point. along the X axis. So we have: Suppose that at the initial time t0 , the origins of K and k do not coincide, but instead are separated by a distance t . 1͓͑ ͒ϩ͑ ͔͒ϭ͑ ͒ 0 2 x0 ,y 0 ,z0 ,t0 x2 ,y 2 ,z2 ,t2 x1 ,y 1 ,z1 ,t1 . In this case, if we let x represent the varying position of ͑ ͒ A 8 clock A, we have: Because Eq. ͑4͒ is valid for any time t, Einstein used the x ϪtϭxЈϭ0. ͑12͒ same symbol t to designate the arbitrary time of departure of A A the light ray from A, that is, he set Јϭ That is, for any time t we define a constant, xA 0, that ϭ ͑ ͒ describes the separation of clock A from the moving origin t0 t. 9 of k along the X axis of K. ͑Because clock A is attached to Јϭ ͑ ͒ ͑ ͒ ͒ By letting x rAB, Eqs. 6a and 6b serve to establish the the origin of k, that constant is 0. Likewise, if xB represents values of t1 and t2 . The time t0 corresponds to tA , and the varying positions of clock B:
4 Am. J. Phys., Vol. 72, No. 5, May 2004 Alberto A. Martı´nez 4 Ϫ ϭ Јϭ Ј ͑ ͒ xB t xB x . 13 xЈ ϭa , ͑16a͒ Ϫ2 2 Јϭ Ј 1 /c That is, for any time t we define a constant xB x that describes the constant separation of clock B from the moving y origin of k. Accordingly, for the times of emission of the ϭa , ͑16b͒ ͱ Ϫ2 2 light ray from A, its arrival at B, and its return back to A,we 1 /c have: z ϭa . ͑16c͒ Ϫ ϭ Јϭ ͑ ͒ ͱ Ϫ2 2 xA t0 xA 0, 14a 1 /c He simplified Eq. ͑16͒ by finally ‘‘Substituting for xЈ its Ϫ ϭ Јϭ Ј ͑ ͒ xB t1 xB x , 14b value,’’ namely, xЈϭxϪt, as stated at the outset, so that
Ϫ ϭ Јϭ ͑ ͒ xϪt xA t2 xA 0. 14c ϭa , ͑17a͒ 1Ϫ2/c2 The values 0, xЈ, 0, serve to explain the first terms in each parentheses in Eq. ͑5͒. This explanation implies that we may tϪx/c2 ϭa . ͑17b͒ ͑ ͒ Јϭ Ϫ Ј 2 2 understand Eq. 4 as meaning that xs xclock t, where xs 1Ϫ /c is any constant expressing the separation ͑in K͒ between any ͑ ͒ fixed clock on k and the origin of k at any given time. Thus, We include Eq. 17 only to highlight an ambiguity that fol- Einstein’s definition of xЈ as describing a point at rest in k lowed. Einstein provided no explanation or justification for with a value independent of time, should refer not just to one his next step; he did not even explicitly state it. Miller notes point but to any point fixed on k, in particular, to the two that ‘‘without prior warning,’’ Einstein assigned a value for points where the clocks are attached.24 Hence, to be more the undetermined term a, namely: precise about his notation, Einstein could have stated that he aϭ͑͒ͱ1Ϫ2/c2, ͑18͒ sought a function (xЈ ,y,z,t), rather than just (xЈ,y,z,t) s that is, aϭ()b as stated above. This tacit step eliminated because the latter corresponds only to the events at clock B. Likewise, his statement that the light ray travels ‘‘along the X the denominators in the equations for and and introduced ϭ ͱ Ϫ2 2 axis toward xЈ, and from there’’ is reflected back, should not the term 1/ 1 /c into the equations for and . be interpreted literally as meaning that xЈ is the coordinate of Miller asked: ‘‘But why did Einstein make this replacement? It seems as if he knew beforehand the correct form of the set the light ray when it arrives at clock B, because that coordi- of relativistic transformations... .’’26 Certainly his first pub- nate is actually x ϭxЈϩt . Rather, to make sense of his B 1 lished derivation cannot be construed to be his first investi- expressions, we have to understand him as meaning to say gation of the problem. Miller offered a few historical conjec- that the light ray is reflected from the constant point of k tures to explain how Einstein found the final form of the where clock B is fixed. transformations ͑in part from Einstein’s study of Lorentz’s Note that to interpret the text consistently, we have gone a paper of 1895, which, however, lacked the exact transforma- slight distance away from giving it a completely literal read- tions͒, but such arguments need not be reviewed here. Ein- ing. Only by doing so can we arrive at a consistent explana- stein wanted to isolate the term ͑͒ to then show that it is tion of Einstein’s analysis. Hence we may surmise that the irrelevant. But the important point is that Einstein did not subtle ambiguities involved may have caused some difficul- introduce the value of a in the explicit manner in which he ties in the understanding of his 1905 argument, especially for derived the values of the other terms. In view of this unsub- readers not sufficiently careful in kinematics, and perhaps for stantiated step, readers of Einstein’s derivation who did not anyone who didn’t have a good sense beforehand of the end have a predetermined notion of the final form of the trans- result of the derivation. formation equations might well have been puzzled by his By using differential analysis, Einstein next ascertained  Ј introduction of a and the factor. the dependence of on x , y, z, and t. We will not review his Finally, Einstein completed his derivation of the transfor- rather long procedure here, because the reader may consult mation equations by determining the value of the function Miller’s study or Einstein’s own paper. It is sufficient to say ͑͒. He ascertained that ͑͒ϭ1 by a long argument that that his use of differential analysis was an unnecessary de- ͑ involved the introduction of a third coordinate system iden- tour because he could have ascertained the function by tical with K͒, and the measurement of the length of a moving means of linear algebra alone. In any case, he arrived at: rod oriented perpendicularly to its direction of motion. Again, we skip those arguments for the sake of brevity. ϭaͩ tϪ xЈͪ , ͑15͒ 2Ϫ2 c III. COMMENTARY AND CONCLUSION explaining that ‘‘a is a function ͑͒ presently unknown, and Because we are focusing on the ambiguities in Einstein’s where for brevity it is assumed that at the origin of k, for expressions, consider again his use of the term xЈ. We raised ϭ0istϭ0.’’25 Although his words seem to suggest that a the question of whether Einstein used a third coordinate sys- ϭ(), he did not quite use this relation, but instead, a tem from the beginning of his analysis, and whether such a ϭ()b, as we will see ͑he should have just stated that ‘‘a system was described by the laws of traditional kinematics. is a function of ’’͒, but this was just a minor oversight. Presumably, Einstein introduced the auxiliary term xЈ to sim- Einstein’s analysis also leads to: plify his derivation, but did it really perform this function?
5 Am. J. Phys., Vol. 72, No. 5, May 2004 Alberto A. Martı´nez 5 To clarify this issue, we reconstruct Einstein’s derivation cisely to clarify the representation of physical quantities. without introducing such a term, in the Appendix. There we Thus, Einstein’s derivation focused on a clarification of the confirm that the term xЈ was not essential for Einstein’s concept of time, on the basis of measurement procedures and analysis. His choice to express the dependence of the func- algebraic analysis, but it admitted various ambiguities in the 28 tion on xЈ simplified the initial steps of his analysis, be- representation of other kinematic concepts. cause it is a bit more complicated to differentiate with Moreover, his derivation suffers from some deficiencies. respect to x because the position of the light ray varies with Consider again the factor  in the transformation equations. the changing values of t. Instead of performing every calcu- This factor is crucial, because it is a telltale feature that dis- tinguishes the Lorentz transformation from the Galilean lation with the coordinates x0 , x1 , x2 , Einstein simply car- ried the auxiliary term xЈ throughout the derivation to rein- transformation. But Einstein did not derive this factor ex- actly, but introduced it rather freely by setting the value of troduce its value at the end. Nonetheless, it also is correct to the variable a to obtain the form of the transformations that evaluate the function with respect to x on an equal footing he deemed to be correct. In addition to the  factor, the only with the terms y, z, t, and, after differentiation, the procedure other deviation from the Galilean transformations is the term even leads more directly to the desired results, especially Ϫ 2 because it yields the value of the  factor directly, rather than x /c in the time transformation. It seems awkward that ϭ most of the analysis in Einstein’s long derivation simply re- requiring the introduction of the function a ( )b. ͑ Ј sults in the introduction of this small algebraic term al- The role of the term x involves sufficient ambiguities to though physically the term is of crucial importance because suggest that Einstein’s derivation may have confused even it involves first-order effects in velocity , whereas the  careful readers of his paper, because even later physicists factor concerns second-order effects͒. Again, his use of dif- ͑ ͒ misinterpreted the role of xЈ. Because Eq. 4 is formally ferential analysis was an unnecessary detour. Thus the identical to a transformation equation, readers could easily amount of analysis does not seem commensurate with the misunderstand Einstein’s analysis. For example, even Tor- formal simplicity of its results. retti misconstrued Eq. ͑4͒ as a transformation. Likewise, Einstein’s derivation lacked mathematical elegance; its de- Miller misconstrued xЈ as a ‘‘Galilean coordinate’’ of system gree of abstraction along with its ambiguities serve to illus- k, and accordingly, he misinterpreted the expressions cϪ trate why many of his contemporaries had difficulties under- and cϩ as speeds of light relative to k, as though Einstein standing and accepting his kinematics. Back then, readers had relied on the Galilean transformation of velocities in his well-trained in theoretical physics constituted a small minor- analysis. He also construed Einstein as having employed xЈ ity of physicists. Many steps of his derivation could seem to avoid discussing the relativistic effect of length contrac- debatable or imprecise to readers who took his every expres- tion from the outset.27 It would then seem that Einstein de- sion literally, or to those who did not already know or accept rived new transformation equations by assuming as true the the final result, the transformation equations and their utility. traditional transformations and violating his own notion of The structure and details of the derivation engender the the invariance of the speed of light. On such grounds, anyone impression of Einstein groping tortuously to construct it, who approached Einstein’s analysis with a critical attitude having first discovered that the relativity of simultaneity could easily be confused or reject it as incoherent. But con- could provide a kinematic justification for the transforma- trary to such interpretations, the term xЈ does not correspond tions that he knew to work in optics and electromagnetism. to a Galilean or quasi-Galilean coordinate of system k. In- To be sure, the general approach and novel concepts made stead, it may designate the constant separation between the his derivation well worthwhile. He demonstrated that the two clocks as judged not from k but from K. Lorentz transformations could be deduced from simple kine- Yet even in this light, Einstein’s analysis involved addi- matic assumptions, along with the postulate of the constancy tional peculiarities that could engender confusion. For ex- of the speed of light, irrespective of the exact validity of the ample, in Eq. ͑4͒ the term xЈ might seem to be the initial rest of contemporary electromagnetic theory. After the paper was published in the Annalen der Physik position of a material point in its equation of rectilinear mo- 1 29 tion. Also, Eq. ͑5͒ could be misinterpreted as consisting of in 1905, Einstein discarded the original manuscript. Late coordinate values of k, whereas the arguments in each set of in life, Einstein expressed surprise at the complexity of the parentheses belong to K, and only three in each, namely the paper. In 1943, Einstein was drafting a copy of the paper to y, z, t terms, are coordinates. Einstein compounded this am- donate for a fundraising auction. His secretary, Helen Dukas, biguity by talking in one place about xЈ as if it were the recounted that ‘‘... she would sit next to Einstein and dictate position of the light ray when it reaches clock B. Such slop- the text to him. At one point, Einstein lay down his pen, piness of expression and notation was quite common in pa- turned to Helen and asked her whether he had really said pers on theoretical physics at the time, but in the demonstra- what she had just dictated to him. When assured that he had, ¨ ¨ 30 tion of radically novel claims, it could hardly help their Einstein said, ‘Das hatte sich einfacher sagen konnen.’ ’’ intelligibility. Yet Einstein gave more than one meaning to In comparison to other parts of Einstein’s 1905 paper, his derivation of the transformation equations stands out as by some of the symbols he employed, for example, to x, xЈ, and far the most subtle mathematical argument. Later derivations t. By avoiding exact expressions for the positions of a light of the same equations, by Einstein and others, were im- ray along the X axis, his analysis scarcely distinguished be- mensely simpler.31 Hence we may understand Einstein’s re- tween the concepts of position and length. For the most part, mark, ‘‘That could have been said more simply,’’ as referring he distinguished explicitly the concepts of time intervals and to his derivation of the transformation equations. instants ͑single time coordinates͒, but he hardly drew analo- gous distinctions for the concepts of space. Also, he didn’t ACKNOWLEDGMENTS explicitly distinguish between concepts of distance and dis- placement, nor between speed and velocity; distinctions that I thank Olivier Darrigol, John Stachel, Michel Janssen, stemmed from vector theory and had been introduced pre- and Sam Schweber, as well as the reviewers, for helpful
6 Am. J. Phys., Vol. 72, No. 5, May 2004 Alberto A. Martı´nez 6 ϭ ϩ Ϫ͒͒ ͑ ͒ comments and several useful suggestions. I also thank Goran t1 t ͑x/͑c , 22d Prstic, Roger H. Stuewer, and Serge Rudaz, for comments on ϭ ͑ Ϫ͒ϩ ͑ ϩ͒ ͑ ͒ an earlier version of this work. x2 x/ c x/ c , 22e ϭ ϩ Ϫ͒͒ϩ ϩ͒͒ ͑ ͒ t2 t ͑x/͑c ͑x/͑c . 22f These values can be simplified but the above form conveys APPENDIX: ALTERNATIVE DERIVATION OF THE their physical significance. These values can now be substi- TRANSFORMATIONS tuted into Eq. ͑8͒, so that we obtain: We now reconstruct Einstein’s derivation without intro- 1 x x x x ͫ͑0,0,0,t͒ϩͩ ϩ ,0,0,tϩ ϩ ͪͬ ducing the xЈ term. Let us see what we derive instead of Eq. 2 cϪ cϩ cϪ cϩ ͑5͒ if we establish the exact values of the coordinates in K at the times of emission, reflection, and return of the light ray x x ϭͩ xϩ ,0,0,tϩ ͪ , ͑23͒ traveling between the clocks in k. Consider again the expres- cϪ cϪ sion, Eq. ͑8͒, instead of Einstein’s expression, Eq. ͑5͒, 1͓͑x ,y ,z ,t ͒ϩ͑x ,y ,z ,t ͔͒ϭ͑x ,y ,z ,t ͒. 2 0 0 0 0 2 2 2 2 1 1 1 1 1 xЈ xЈ ͫ͑0,0,0,t͒ϩͩ 0,0,0,tϩ ϩ ͪͬ Each of the values of the coordinates in K can be established 2 cϪ cϩ as follows. At the time t0 if the origins of K and k coincide, ϭ xЈ and a light ray is emitted, its coordinates in K are x0 y 0 ϭͩ Ј ϩ ͪ ϭ ϭ ϭ ͑ x ,0,0,t Ϫ . z0 0 and t0 t. At this time the point where a clock is c ⌶ ͒ fixed to the axis is located at a distance x from the origin Are the two expressions equivalent? Einstein’s formulation is of K. As the light ray travels along the X axis, moves away algebraically simpler, because of his use of xЈ, but concep- with velocity . Once the ray traverses the distance x, the tually it is ambiguous for the reasons mentioned. To further point is no longer there, and the ray must traverse an extra Ϫ distinguish the two approaches, we seek the difference be- distance (t1 t0) to reach . Thus the coordinates in K of tween the next result Einstein obtained and what follows the signal upon its arrival at are otherwise. ϭ ϩ Ϫ ͒ ͑ ͒ To obtain the differential equation expressing the depen- x1 x ͑t1 t0 , 19a dence of on x and t, as in Einstein’s approach, we may ϭ ͑ ͒ y 1 0, 19b make a series expansion of each term in . For example, because ϭ ͑ ͒ z1 0, 19c x c ϭ ϩ͑ ͑ Ϫ͒͒ ͑ ͒ xϩ ϭx , ͑24͒ t1 t x/ c . 19d cϪ cϪ At this time t1 the light signal and the point coincide at a then for : ϩ Ϫ 1 distance x (t1 t0) from the origin of K. Now the ray 2 ͑hx͒2ץ hxץ travels in the opposite direction, while the origin of k moves ͑xhc,tϩhx͒ϭ͑xhc,t͒ϩ ϩ ϩ , ¯ !t2 2ץ t 1ץ toward it with the velocity . Because at the time t1 the origin of k was at a distance x from the ray, the ray traverses ͑25͒ less than this distance to meet it. Thus the coordinates in K of the ray when it reaches k’s origin are where we have let hϭ1/(cϪ). Now, if we differentiate the function and the series with respect to x, we obtain ϭ Ϫ ͒ϩ Ϫ ͒ ͑ ͒ x2 ͑t1 t0 ͑t2 t1 , 20a 2ץ ץ ץ ͑xhc,tϩhx͒ץ y ϭ0, ͑20b͒ ϭ hcϩ hϩ xh2ϩ ͑26͒ ¯ t2ץ tץ xץ xץ 2 ϭ ͑ ͒ z2 0, 20c If we neglect terms of second order and higher, we find ץ ץ ץ t ϭtϩ͑x/͑cϪ͒͒ϩ͑x/͑cϩ͒͒. ͑20d͒ 2 1 c 1 ϭ ͩ ͪ ϩ ͩ ͪ . ͑27͒ Ϫ ץ Ϫ ץ ץ Ϫ These values can be restated by expressing (t1 t0) and (t2 x x c t c Ϫ t1) in terms of x, , and c: By carrying out the same procedure for the other side of Eq. ͑ ͒ Ϫ ͒ϭ Ϫ͒ ͑ ͒ 23 , we obtain ͑t1 t0 x/͑c , 21a ץ ץ ͑ Ϫ ͒ϭ ͑ ϩ͒ ͑ ͒ 1 2 c 2c t2 t1 x/ c . 21b ͫ ͑0͒ϩ ͩ ͪ ϩ ͩ ͪͬ t c2Ϫ2ץ x c2Ϫ2ץ 2 Equations ͑21a͒ and ͑21b͒ are equivalent to Eqs. ͑6a͒ and 1ץ cץ 6b͒. So, the successive values of the x and t coordinates in K͑ ϭ ͩ ͪ ϩ ͩ ͪ , ͑28͒ t cϪץ x cϪץ are ϭ ͑ ͒ x0 0, 22a so that we now obtain ϭ ͑ ͒ ץ ץ t0 t, 22b ϩ ϭ0. ͑29͒ tץ x 2ץ ͒ ͑ ϭ ϩ Ϫ͒ x1 x x/͑c , 22c c
7 Am. J. Phys., Vol. 72, No. 5, May 2004 Alberto A. Martı´nez 7 Compare this result to Einstein’s: ϭyͱ1Ϫ2/c2. ͑38͒ The same procedure for the light ray transmitted along the Zץ ץ ϩ ϭ0. ͑30͒ axis yields an equivalent result, so that the resulting four tץ xЈ c2Ϫ2ץ equations are From Eq. ͑30͒ he obtained the algebraic expression for , Eq. ϭ͑tϪx/c2͒, ͑39a͒ ͑15͒. But following the same procedure we derive from Eq. ͑29͒ the different equation: ϭ͑xϪt͒, ͑39b͒ ϭ͑ Ϫ 2͒ ͑ ͒ t x/c 31 ϭyͱ1Ϫ2/c2, ͑39c͒ ͑where the value of  has not yet been determined͒, instead ϭzͱ1Ϫ2/c2. ͑39d͒ of Einstein’s result, Eq. ͑15͒, ͑ ͒ ϭ ͱ Ϫ2 2 Equation 39 now suggests that 1/ 1 /c , which ϭaͩ tϪ xЈͪ , may be demonstrated by using the same arguments Einstein c2Ϫ2 used to establish that ͑͒ϭ1, or any simpler arguments ϭ Ј showing, say, that y. Hence we arrive again at the final which may be restated by including the value he gave to x , form of the transformation equations. that is, xЈϭxϪt: a͒ Ϫ2 Electronic mail: [email protected] x t 1 ͑ ͒ ϭaͩ tϪ ͪ . ͑32͒ A. Einstein, ‘‘Zur Elektrodynamik bewegter Ko¨rper,’’Ann. Phys. Leipzig c2Ϫ2 17, 895–921 ͑1905͒; reprinted in The Collected Papers of Albert Einstein, edited by John Stachel ͑Princeton U.P., Princeton, 1989͒, Vol. 2. Several Notice how Eq. ͑31͒ resembles the transformation equation translations of Einstein’s paper are available, including the following. H. for that Einstein only subsequently obtained: ϭ(t A. Lorentz, A. Einstein, H. Minkowski, and H. Weyl, The Principle of Ϫ 2 Relativity, translated by W. Perrett and G. B. Jeffery with notes by Arnold x/c ). Sommerfeld ͑Dover, New York, 1952͒. Anna Beck ͑translator͒, and Peter The only distinction between Eq. ͑31͒ and the final trans- Havas ͑consultant͒, The Collected Papers of Albert Einstein ͑Princeton formation is that we have yet to establish that  has the same U.P., Princeton, 1989͒, Vol. 2. Arthur I. Miller, Albert Einstein’s Special value in both. This great similarity indicates that, for this Theory of Relativity: Emergence (1905) and Early Interpretation (1905– point onward, the algebraic derivation of the function in 1911) ͑Addison–Wesley, Reading, MA, 1981͒ and ͑Springer-Verlag, New ͒ terms of x instead of xЈ leads more directly to the results York, 1998 . Einstein’s Miraculous Year; Five Papers That Changed the Face of Physics, edited by John Stachel, with the assistance of Trevor sought by Einstein. All that remains is to show that this ap- Lipscombe, Alice Calaprice, and Sam Elworthy ͑Princeton U.P., Princeton, proach agrees with Einstein’s by completing the derivation 1998͒. Note that the translations by Perrett and Jeffery and by Miller have of the transformation equations. Again, we will proceed in some slight but significant defects. direct analogy to Einstein’s 1905 procedure. 2The quotes that follow are from Carl Seelig, Albert Einstein: A Documen- To express the quantities , , in terms of x, y, z, we need tary Biography, translated by Mervyn Savill ͑Staples, London, 1956͒,pp. to express in equations that light propagates with a speed c 73, 88. See also Max Flu¨ckiger, Albert Einstein in Bern: Das Ringen um ein neues Weltbild ͑Paul Haupt, Berne, 1974͒, pp. 103, 209. when measured in the moving system. For a light ray emitted 3Russell McCormmach, ‘‘Editor’s Foreword,’’ Historical Studies in the at the time ϭ0 in the direction of increasing , ϭc,so Physical Sciences, 7th Annual Volume ͑Princeton University Press, Princ- that from Eq. ͑31͒ we obtain: eton, 1976͒, p. xxxvi. McCormmach’s expression was later modified to read ‘‘ordinary algebra ... modest mathematical equipment,’’ in Christa ϭ͑ctϪx/c͒. ͑33͒ Jungnickel and Russell McCormmach, Intellectual Mastery of Nature ͑University of Chicago Press, Chicago, 1986͒,Vol.2,p.337. Because x corresponds to the distance light travels in K in a 4For example, even William Rowan Hamilton, with his extraordinary math- time t, xϭct, so that ematical skills, encountered difficulties as a youth studying Newton’s ϭ͑ Ϫ ͒ ͑ ͒ works; he commented that Newton wrote the Universal Arithmetick in x t . 34 ‘‘the same masterly manner as the Principia, and yet in many parts is To obtain the equations relating and to y and z,we rendered almost as difficult, by its conciseness and omission of interme- proceed in an analogous manner by considering rays moving diate steps.’’ Letter of 28 September, 1823, in W. R. Hamilton, Life of Sir William Rowan Hamilton, edited by R. P. Graves ͑Hodges, Figgis & Co., along the two other axes. First, for the H axis: Dublin, 1882͒, Vol. 1, p. 149. 5 ϭcϭc͑tϪx/c2͒. ͑35͒ For example, see Miller in Ref. 1, Torretti in Ref. 14, or Robert B. Will- iamson, ‘‘Logical economy in Einstein’s ‘On the Electrodynamics of Mov- Now, as observed from the system K, a ray of light propagat- ing Bodies,’ ’’ Stud. Hist. Philos. Sci. 8, 46–60 ͑1977͒, or Harvey R. ing along the H axis of k travels a diagonal path with a speed Brown and Adolfo Maia, Jr., ‘‘Light-speed constancy versus light-speed ͱ 2Ϫ2 invariance in the derivation of relativistic kinematics,’’ Br. J. Philos. Sci. c , such that when 44, 381–407 ͑1993͒. 6Miller, Ref. 1. y 7 ϭt, xϭt, ͑36͒ H. A. Lorentz, ‘‘De l’influence du mouvement de la terre sur les phe´- ͱc2Ϫ2 nome`nes lumineux,’’ Versl. Gewone Vergad. Afd. Koninklijke Akademie van Wetenschappen te Amsterdam 2, 297–372 ͑1886͒; also in Archives because the H axis has moved a distance t as the ray travels Ne´erlander 21,19–84͑1887͒; reprinted in H. A. Lorentz, Collected Pa- pers ͑Martinus Nijhoff, The Hague, 1937͒, Vol. 4, pp. 153–214. to the point . If we substitute this value of x into the ex- 8 pression for , Eq. ͑35͒, we find: H. A. Lorentz, Versuch einer Theorie der elektrischen und optischen Er- scheinungen in bewegten Ko¨rpern ͑Brill, Leiden, 1895͒; reprinted in H. A. 2 Lorentz, Collected Papers, Vol. 5, pp. 1–137. Joseph Larmor, Aether and ϭctͩ 1Ϫ ͪ . ͑37͒ Matter: A Development of the Dynamical Relations of the Aether to Ma- c2 terial Systems on the Basis of the Atomic Constitution of Matter; including a Discussion of the Influence of the Earth’s Motion on Optical Phenomena We substitute the value of t to find: ͑Cambridge U.P., Cambridge, 1900͒. H. A. Lorentz, ‘‘Electromagnetische
8 Am. J. Phys., Vol. 72, No. 5, May 2004 Alberto A. Martı´nez 8 verschijnselen in een stelsel dat zich met willekeurige snelheid, kleiner 20Miller, Ref. 1, p. 397. dan die van het licht, beweegt,’’ Koninklijke Akademie van Wetenschap- 21Miller, Ref. 1, p. 209. Moreover, throughout the decades some writers pen te Amsterdam. Wis-en Natuurkundige Afdeeling. Verslagen van de mistakenly criticized Einstein’s demonstration of the relativity of simulta- Gewone Vergaderingen 12, 986–1009 ͑1904͒; reissued as ‘‘Electromag- neity and his derivations of the Lorentz transformations claiming that Ein- netic phenomena in a system moving with any velocity smaller than that of stein had violated his postulate of the constancy of the speed of light by light,’’ Koninklijke Akademie van Wetenschappen te Amsterdam, Section employing speeds other than c in his arguments. of Sciences, Proceedings 6,809–831͑1904͒. Henri Poincare´, ‘‘Sur la dy- 22Some thorough books do not neglect this important distinction. See, for namique de l’e´lectron,’’ Comptes rendus de l’Acade´mie des Sciences example, Henri Arze`lies, Relativistic Kinematics ͑Pergamon, Oxford, ͓Paris͔ 40, 1504–1508 ͑1905͒; reprinted in H. Poincare´, Oeuvres 9, 489– 1966͒, pp. 142–143. 493 ͑1954͒. Henri Poincare´, ‘‘Sur la dynamique de l’e´lectron,’’ Rendiconti 23Miller, Ref. 1, p. 209. del Circolo matematico di Palermo 21, 129–176 ͑1906͒; reprinted in Oeu- 24I thank Olivier Darrigol for elucidating this account. vres 9,494–550͑1954͒. 25Einstein, Ref. 1, p. 899. 9W. Voigt, ‘‘Ueber das Doppler’ sche Princip,’’ Ko¨nigliche Gesellschaft der 26Miller, Ref. 1, p. 212. Likewise, Fo¨lsing comments: ‘‘At an opaque point Wissenschaften und der Georg-Augusts-Universita¨tzuGo¨ttingen. Nach- in his deduction he ͓Einstein͔ introduces, without any warning or expla- richten 14, 41–51 ͑1887͒; reprinted in Phys. Z. 16, 381–386 ͑1915͒. nation, a slight mathematical operation whose purpose becomes obvious 10P. Frank, ‘‘Die Stellung des Relativita¨tsprinzips im System der Mechanik only if the desired result is already known. This underhand device, by und der Elektrodynamik,’’ Sitzungsber. Kaiserlichen Akademie der Wis- means of which he rather forcibly ‘computes his way’ to the Lorentz senschaften. Mathematisch-Naturwissenschaftliche Klasse ͓Wien͔ Section transformations, deprives the deduction of some of its elegance and strin- IIa 118 ͑4͒, 373–446 ͑1909͒. gency.’’ Albrecht Fo¨lsing, Albert Einstein. A Biography, translated by 11Einstein, Ref. 1, p. 898; italics in the original. Ewald Osers ͑Viking/Penguin, New York, 1997͒,p.188. 12For a rigorous discussion of the connection between homogeneity and 27Miller, Ref. 1, p. 209. linearity, see Roberto Torretti, Relativity and Geometry ͑1983͒, reprinted in 28Alberto A. Martı´nez, ‘‘The Neglected Science of Motion: The Kinematic ͑Dover, New York, 1996͒, pp. 71–76. Torretti systematically clarifies the Origins of Relativity,’’ Ph.D. dissertation, University of Minnesota, De- matter because he notes that at first ‘‘it is not at all clear that the linearity cember, 2000, UMI, 2001. of the Lorentz transformations can follow from this fact alone’’ ͑p. 72͒, the 29As Einstein informed Julian Boyd, librarian of the Princeton University homogeneity of space and time. Torretti shows that the linearity can be Library. See Abraham Pais, ‘Subtle is the Lord...’ The Science and the Life justified on physical grounds by appealing to the principle of inertia. Also, of Albert Einstein ͑Oxford U.P., New York, 1982͒, p. 147. Einstein’s dec- Olivier Darrigol suggests ͑personal communication͒ that by homogeneity laration of having discarded the original manuscript appears in Flu¨ckiger, Einstein presumably meant that a straight line should transform into a Ref. 2, p. 103. straight line, and that a rectilinear uniform motion should transform into 30Pais, Ref. 29, p. 147. John Stachel, having also heard the story directly another of the same kind. from Helen Dukas, received the impression that it was at two distinct 13Einstein, Ref. 1, p. 898. places during the reading that Einstein made this remark to her. We do not 14Equation ͑4͒ has been identified as a transformation; see, for example, know whether he necessarily referred to something in his derivation of the Torretti, Ref. 12, p. 58. transformations. Incidentally, the Collected Papers of Albert Einstein, Vol. 15See for instance H. A. Lorentz, Lehrbuch der Differential- und Integral- 2 ͑Ref. 1, p. 309͒ includes notes on three slight corrections that Einstein rechnung nebst einer Einfu¨hrung in andere Teile der Mathematik, revised made to a reprint copy of his 1905 paper, but such corrections are far too by G. C. Schmidt ͑Barth, Leipzig, 1900͒,p.83. minor for us to assume that his later comment to Dukas referred to them. 16Miller, Ref. 1, p. 196; see also p. 213. The claim that xЈ, yЈ, zЈ are 31For example: A. Einstein, ‘‘U¨ ber das Relativita¨tsprinzip und die aus dem- coordinates of the moving system also is made in Stachel ͑1998͒, Ref. 1, p. selben gezogenen Folgerungen,’’ Jahrbuch der Radioactivita¨t und Elek- 160, Note 2. tronik 4, 411–462 ͑1907͒; reprinted in The Collected Papers of Albert 17Torretti, Ref. 12, pp. 57–58. Einstein, edited by John Stachel ͑Princeton U.P., Princeton, 1989͒,Vol.2; 18Gustav Kirchhoff, Vorlesungen u¨ber Mathematische Physik: Mechanik and A. Einstein, Relativity: The Special and General Theory, translated by ͑Teubner, Leipzig, 1883͒, 3rd ed., p. 4. Robert W. Lawson ͑P. Smith/Henry Holt and Co., New York, 1931͒,Ap- 19Einstein, Ref. 1, p. 898. pendix I.
9 Am. J. Phys., Vol. 72, No. 5, May 2004 Alberto A. Martı´nez 9