Kinematic subtleties in Einstein’s first derivation of the Lorentz transformations

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reprinted in The Collected Papers of Albert Einstein, edited by John Stachel (Princeton U.P., Princeton, 1989), Vol. 2. Several translations of Einstein’s paper are available, including the following.

H. A. Lorentz, A. Einstein, H. Minkowski, and H. Weyl, The Principle of Relativity, translated by W. Perrett and G. B. Jeffery with notes by Arnold Sommerfeld (Dover, New York, 1952).

Anna Beck (translator), and Peter Havas (consultant), The Collected Papers of Albert Einstein (Princeton U.P., Princeton, 1989), Vol. 2.

Arthur I. Miller, Albert Einstein’s Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905–1911) (Addison–Wesley, Reading, MA, 1981) and (Springer-Verlag, New York, 1998).

Einstein’s Miraculous Year; Five Papers That Changed the Face of Physics, edited by John Stachel, with the assistance of Trevor Lipscombe, Alice Calaprice, and Sam Elworthy (Princeton U.P., Princeton, 1998). Note that the translations by Perrett and Jeffery and by Miller have some slight but significant defects.

The quotes that follow are from Carl Seelig, Albert Einstein: A Documentary Biography, translated by Mervyn Savill (Staples, London, 1956), pp. 73, 88. See also Max Flückiger, Albert Einstein in Bern: Das Ringen um ein neues Weltbild (Paul Haupt, Berne, 1974), pp. 103, 209.

Russell McCormmach, “Editor’s Foreword,” Historical Studies in the Physical Sciences, 7th Annual Volume (Princeton University Press, Princeton, 1976), p. xxxvi. McCormmach’s expression was later modified to read “ordinary algebra … modest mathematical equipment,” in Christa Jungnickel and Russell McCormmach, Intellectual Mastery of Nature (University of Chicago Press, Chicago, 1986), Vol. 2, p. 337.

For example, even William Rowan Hamilton, with his extraordinary mathematical skills, encountered difficulties as a youth studying Newton’s works; he commented that Newton wrote the Universal Arithmetick in “the same masterly manner as the Principia, and yet in many parts is rendered almost as difficult, by its conciseness and omission of intermediate steps.” Letter of 28 September, 1823, in W. R. Hamilton, Life of Sir William Rowan Hamilton, edited by R. P. Graves (Hodges, Figgis & Co., Dublin, 1882), Vol. 1, p. 149.

For example, see Miller in Ref. 1, Torretti in Ref. 12,

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Miller, Ref. 1.

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reprinted in H. A. Lorentz, Collected Papers (Martinus Nijhoff, The Hague, 1937), Vol. 4, pp. 153–214.

H. A. Lorentz, Versuch einer Theorie der elektrischen und optischen Erscheinungen in bewegten Körpern (Brill, Leiden, 1895); reprinted in H. A. Lorentz, Collected Papers, Vol. 5, pp. 1–137. Joseph Larmor, Aether and Matter: A Development of the Dynamical Relations of the Aether to Material Systems on the Basis of the Atomic Constitution of Matter; including a Discussion of the Influence of the Earth’s Motion on Optical Phenomena (Cambridge U.P., Cambridge, 1900).

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Einstein, Ref. 1, p. 898; italics in the original.

For a rigorous discussion of the connection between homogeneity and linearity, see Roberto Torretti, Relativity and Geometry (1983), reprinted in (Dover, New York, 1996), pp. 71–76. Torretti systematically clarifies the matter because he notes that at first “it is not at all clear that the linearity of the Lorentz transformations can follow from this fact alone” (p. 72), the homogeneity of space and time. Torretti shows that the linearity can be justified on physical grounds by appealing to the principle of inertia. Also, Olivier Darrigol suggests (personal communication) that by homogeneity Einstein presumably meant that a straight line should transform into a straight line, and that a rectilinear uniform motion should transform into another of the same kind.

Einstein, Ref. 1, p. 898.

Equation (4) has been identified as a transformation; see, for example, Torretti, Ref. 12, p. 58.

See for instance H. A. Lorentz, Lehrbuch der Differential- und Integralrechnung nebst einer Einführung in andere Teile der Mathematik, revised by G. C. Schmidt (Barth, Leipzig, 1900), p. 83.

Miller, Ref. 1, p. 196; see also p. 213. The claim that $x′,$ $y′,$ $z′$ are coordinates of the moving system also is made in Stachel (1998), Ref. 1, p. 160, Note 2.

Torretti, Ref. 12, pp. 57–58.

Gustav Kirchhoff, Vorlesungen über Mathematische Physik: Mechanik (Teubner, Leipzig, 1883), 3rd ed., p. 4.

Einstein, Ref. 1, p. 898.

Miller, Ref. 1, p. 397.

Miller, Ref. 1, p. 209. Moreover, throughout the decades some writers mistakenly criticized Einstein’s demonstration of the relativity of simultaneity and his derivations of the Lorentz transformations claiming that Einstein had violated his postulate of the constancy of the speed of light by employing speeds other than c in his arguments.

Some thorough books do not neglect this important distinction. See, for example, Henri Arzèlies, Relativistic Kinematics (Pergamon, Oxford, 1966), pp. 142–143.

Miller, Ref. 1, p. 209.

I thank Olivier Darrigol for elucidating this account.

Einstein, Ref. 1, p. 899.

Miller, Ref. 1, p. 212. Likewise, Fölsing comments: “At an opaque point in his deduction he [Einstein] introduces, without any warning or explanation, a slight mathematical operation whose purpose becomes obvious only if the desired result is already known. This underhand device, by means of which he rather forcibly ‘computes his way’ to the Lorentz transformations, deprives the deduction of some of its elegance and stringency.” Albrecht Fölsing, Albert Einstein. A Biography, translated by Ewald Osers (Viking/Penguin, New York, 1997), p. 188.

Miller, Ref. 1, p. 209.

Alberto A. Martı́nez, “The Neglected Science of Motion: The Kinematic Origins of Relativity,” Ph.D. dissertation, University of Minnesota, December, 2000, UMI, 2001.

As Einstein informed Julian Boyd, librarian of the Princeton University Library. See Abraham Pais, ‘Subtle is the Lord…’ The Science and the Life of Albert Einstein (Oxford U.P., New York, 1982), p. 147. Einstein’s declaration of having discarded the original manuscript appears in Flückiger, Ref. 2, p. 103.

Pais, Ref. 29, p. 147. John Stachel, having also heard the story directly from Helen Dukas, received the impression that it was at two distinct places during the reading that Einstein made this remark to her. We do not know whether he necessarily referred to something in his derivation of the transformations. Incidentally, the Collected Papers of Albert Einstein, Vol. 2 (Ref. 1, p. 309) includes notes on three slight corrections that Einstein made to a reprint copy of his 1905 paper, but such corrections are far too minor for us to assume that his later comment to Dukas referred to them.

For example:

A.

 

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reprinted in The Collected Papers of Albert Einstein, edited by John Stachel (Princeton U.P., Princeton, 1989), Vol. 2; and A. Einstein, Relativity: The Special and General Theory, translated by Robert W. Lawson (P. Smith/Henry Holt and Co., New York, 1931), Appendix I.