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PDF of Lorentz Transforms 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- ␰ϭxϪ␯t, ͑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.
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