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Figures of Light in the Early History of Relativity (1905–1914)

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Beyond Einstein

Part of the book series: Einstein Studies ((EINSTEIN,volume 14))

Abstract

Albert Einstein’s bold assertion of the form invariance of the equation of a spherical light wave with respect to inertial frames of reference (Einstein, Annalen der Physik 322:891–921) became, in the space of 6 years, the preferred foundation of his theory of relativity. Early on, however, Einstein’s universal light-sphere invariance was challenged on epistemological grounds by Henri Poincaré, who promoted an alternative demonstration of the foundations of relativity theory based on the notion of a light ellipsoid. A third figure of light, Hermann Minkowski’s lightcone also provided a new means of envisioning the foundations of relativity. Drawing in part on archival sources, this paper shows how an informal, international group of physicists, mathematicians, and engineers, including Einstein, Paul Langevin, Poincaré, Hermann Minkowski, Ebenezer Cunningham, Harry Bateman, Otto Berg, Max Planck, Max von Laue, A. A. Robb, and Ludwig Silberstein, employed figures of light during the formative years of relativity theory in their discovery of the salient features of the relativistic worldview.

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Notes

  1. 1.

    For gradualist views of the acceptance of relativity theory see Hirosige (1968), Miller (1981), and Darrigol (1996, 2000).

  2. 2.

    On the assumption of linearity, see Brown (2005, 26), and for the kinematic background to Einstein’s first paper on relativity, see Martínez (2009). Einstein did not let kinematics decide the matter once and for all in 1905. In a letter of September 1918 written to his friend, the anti-relativist and political assassin Friedrich Adler, Einstein considered the global factor in the Lorentz transformation to be of an empirical nature, whose value had been determined (to Einstein’s satisfaction) by the results of certain electron-deflection experiments (Walter 2009, 213). Poincaré expressed his views to Lorentz by letter in May 1905; see Walter et al. (2007, §§ 38.4, 38.5).

  3. 3.

    On the compatibility argument, see Williamson (1977). Gaps in Einstein’s reasoning are apparent from a modern standpoint; see, for example, Kennedy (2005).

  4. 4.

    Sommerfeld to H.A. Lorentz, 26 Dec. 1907, in Kox (2008, § 165).

  5. 5.

    For an assessment of Baker’s rise to prominence among Cambridge geometers, see Barrow-Green and Gray (2006).

  6. 6.

    See McCrea (1978), and John Heilbron’s interview with Cunningham (Heilbron 1963).

  7. 7.

    A proponent of Einstein’s theory is understood here to be an individual seeking either to support or to extend any of the novel ideas contained in Einstein’s 1905 paper. On the British reception of relativity, see Goldberg (1970), Sánchez-Ron (1987), and Warwick (2003).

  8. 8.

    See Goldberg (1970), and Hunt (1986).

  9. 9.

    Cunningham noted a personal communication with Larmor, to the effect that while a proof of the Lorentz transformation’s validity for electron theory to second order of approximation in v∕c appeared in the latter’s Æther and Matter (Larmor 1900), Larmor had “known for some time that [the Lorentz transformation] was exact” (Cunningham 1907, 539).

  10. 10.

    Cunningham (1911) recalled this fact, without mentioning Neumann.

  11. 11.

    Cunningham’s conclusion agrees with that reached later by Fermi; see Rohrlich (2007, 17), Janssen and Mecklenburg (2006).

  12. 12.

    Cunningham (1907, 547). Both Cunningham and Planck failed to understand Bucherer’s theory, which allowed for closed links between electrons; see Darrigol (2000, 371).

  13. 13.

    According to Balàzs (1972), Bucherer’s remark shows that he was “confused about the basic problem of relativity,” in that he failed to “realize the connection of this problem with the Michelson-Morley experiment and its relation to the transformation laws.” Yet the Lorentz-FitzGerald contraction explains on its own the null result of the Michelson-Morley experiment, as Bucherer and contemporary theorists knew quite well.

  14. 14.

    See Einstein (1907, § 3); reed. in Stachel et al. (1989, vol. 2, Doc. 47). Cunningham’s paper appeared in the October 1907 issue of the Philosophical Magazine, and Einstein’s review article was submitted for publication in Johannes Stark’s Jahrbuch der Radioaktivität und Elektronik on 4 December 1907.

  15. 15.

    An alternative approach, advanced by Born (1909), involved redefining the notion of a rigid body in Minkowski spacetime. On related developments, see Maltese and Orlando (1995).

  16. 16.

    Sommerfeld insisted in his lectures on electrodynamics that a Lorentz transformation does not change a “Lichtkugel” into a “Lichtellipsoid” (Sommerfeld 1948, 236).

  17. 17.

    See the edition of Henri Vergne’s notes of Poincaré’s 1906–1907 lectures at the Paris Faculty of Science (Poincaré 1953) and his 1912 lectures at the École supérieure des postes et télégraphes (Poincaré 1913), along with the two articles (Poincaré 1908a, 1909). The article of 1908 was reedited by Poincaré in Science et méthode (Poincaré 1908b); the light ellipse is described on p. 239, but the diagram was suppressed from this version, presumably by the editor, Gustave Le Bon.

  18. 18.

    See Lémeray (1912), communicated to the Paris Academy of Sciences on 9 December 1912 and the retraction (ibid., p. 1572). It is not clear whether Lémeray meant to attribute a flattened light ellipsoid or an elongated light ellipsoid to Einstein. Several years later, the Swiss physicist Édouard Guillaume (1921) referred to an “ellipsoïde de Poincaré.” Guillaume corresponded with Einstein on this topic; see Kormos Buchwald et al. (2006, Doc. 241).

  19. 19.

    See Langevin’s notes of Poincaré’s lectures, Fonds Langevin, box 123, Bibliothèque de l’École supérieure de physique et de chimie industrielle, Paris.

  20. 20.

    For details on Langevin’s paper, see Miller (1973).

  21. 21.

    See Langevin’s notebook, box 123, and letter to his wife of 26 September 1904, box 3, Fonds Langevin, Library of the École supérieure de physique et de chimie industrielle, Paris.

  22. 22.

    On Hertz’s solution, see Darrigol (2000, 251).

  23. 23.

    Henri Vergne, notebook 2, François Viète Center, University of Nantes.

  24. 24.

    Poincaré’s fantasy was extended by Tolman (1914) via dimensional analysis, in the form of a “principle of similitude,” a view that attracted sharp criticism from Bridgman (1916).

  25. 25.

    The notion of an absolutely resting frame remained an abstraction for Poincaré. In 1912, he upheld the conventionality of spacetime and expressed a preference for Galilei spacetime over Minkowski spacetime (Walter 2009).

  26. 26.

    See Poincaré (1901, 536), where the value is off by a factor of ten. In a later essay, Poincaré (1904, 312) supplied the “correct” value of the correction factor α for terrestrial observers and an ether at rest with respect to the Sun, where α = (ℓ − ℓ′)∕ℓ = 1 − γ −1.

  27. 27.

    The published version of the notes differs markedly from the original, suggesting that their editor, the astronomer Marguerite Chopinet, disagreed with their content; cf. Poincaré (1953, 219).

  28. 28.

    “Alors je prends une surface rigoureusement spherique. Je la mesure avec mon mètre: dans la direction du mouvement mon mètre sera contracté de α; sa longueur vraie sera devenue 1∕α. Donc mon diamètre dans le sens du mouvement aura pour longueur mesurée α. Dans le sens perpendiculaire la longueur mesurée sera 1. Donc une sphère paraîtra un ellipsoïde allongé dans le sens du mouvement.”

  29. 29.

    Using the relations specified in Figure 1.2, we have

    $$\displaystyle \begin{aligned} a(1 - e^2) = a(1 - (1 - b^2/a^2)) = a(1 - (1 - c^2 t^2/a^2)) = ac^2 t^2/a^2 = ct/\gamma. \end{aligned}$$

    Rearranging the latter expression in terms of t, we find t = aγ(1 − e 2)∕c, and substituting the value of a(1 − e 2) from (1.4), we obtain Poincaré’s expression (1.5) for apparent time t′.

  30. 30.

    The context of Poincaré’s invitation to Göttingen is discussed in Walter (2018).

  31. 31.

    On Poincaré’s engagement with electrotechnology, and wireless telegraphy in particular, see Galison (2003), Gray (2013), and Walter (2017).

  32. 32.

    One may wonder why the watch in Poincaré’s thought experiment runs fast, and not slow, as would be required by time dilation in an Einsteinian or Minkowskian context. An explanation is at hand, if we focus on the first observer’s experience. At first, he believes he has a certain velocity, say 200 km/s. An exchange of telemetry data with the second observer convinces him that he is moving slower than he thought previously. One way for him to account for this revision is to admit that his watch is running fast. Other explanations for the fast watch can be imagined; see Walter (2014).

  33. 33.

    “…un théorème de géométrie très simple montre que le temps apparent que la lumière mettra à aller de A en B, c’est-à-dire la différence entre le temps local en A au moment du départ de A, et le temps local en B au moment de l’arrivée en B, que ce temps apparent, dis-je, est le même que si la translation n’existait pas, ce qui est bien conforme au principe de relativité.”

  34. 34.

    An excerpt of the Revue article was included in Poincaré’s Science et méthode (Poincaré 1908b), neglecting mathematical details, such as Poincaré’s discussion of relative velocity.

  35. 35.

    For a sketch of the French reception of relativity, see Walter (2011).

  36. 36.

    Despite Lorentz’s embrace of what Louis du Pasquier called the “principle of light-wave sphericity,” the Swiss mathematician later wrote that Lorentz rejected this principle (Du Pasquier 1922, 68).

  37. 37.

    On Guillaume’s collaboration with Einstein, see Einstein’s letter to Jacob Laub, 20 March 1909, in Klein et al. (1993, Doc. 143).

  38. 38.

    A displacement from one point to another on the light ellipse corresponds to a Lorentz transformation in this interpretation. The radii from a focus to any two points of the ellipse are related by a rotation and, in general, a dilation or a contraction.

  39. 39.

    On Poincaré’s models of hyperbolic geometry, see Gray (1989) and Zahar (1997).

  40. 40.

    For background, see Walter (1999a, 2008).

  41. 41.

    On the rise of Göttingen as a scientific center, see Manegold (1970) and Rowe (1989).

  42. 42.

    See Frank (1947, 206). Miller (1976, 918) emphasizes the relative simplicity of the mathematical tools deployed by Einstein in his relativity paper, in comparison to those Poincaré brought to bear on similar problems. Renn (2007, 69) observes that Einstein’s uncanny aptitude for informal analysis of complex problems served him well in both special and general relativity.

  43. 43.

    Minkowski’s visually intuitive approach to relativity is explored at length by Galison (1979).

  44. 44.

    On Minkowski’s use of hyperbolic geometry in this lecture, see Reynolds (1993).

  45. 45.

    “Werden jedoch vier Raumpunkte, die nicht in einer Ebene liegen, zu einer und derselben Zeit t 0 aufgefaßt, so ist es nicht mehr möglich, durch eine Lorentz-Transformation eine Abänderung des Zeitparameters vorzunehmen, ohne daß der Charakter der Gleichzeitigkeit dieser vier Raum-Zeitpunkt verloren” (Minkowski 1908, 69).

  46. 46.

    See Born (1909, 9; 1959, 503). For further references and details on Minkowski’s distortion and its reception, see Walter (1999a).

  47. 47.

    NSUB Handschriftenabteilung. The demonstration missing from the published text of Minkowski’s lecture was later supplied by Arnold Sommerfeld, in an editorial note to his friend’s lecture. The annotated version of the lecture appeared in an anthology of papers on the theory of relativity edited by Blumenthal (1913). According to Rowe (2009, 37), Sommerfeld was the driving force behind the latter anthology.

  48. 48.

    As seen above, Poincaré also derived the Lorentz transformation from the assumption of Lorentz contraction of concrete rods, and the isotropy of light propagation for inertial observers. He later considered (apparent) time deformation as a consequence of the principle of relativity and Lorentz contraction; see (Poincaré 1913, 44).

  49. 49.

    Tanner (1917, 571). I thank J. Barrow-Green for pointing me to this source.

  50. 50.

    Robb was admitted to the Society on 27 Nov. 1905 (Proceedings of the Cambridge Philosophical Society 16, 1912, p. 16).

  51. 51.

    Robb to Larmor, 6 March 1904, Larmor Papers, St. John’s College Library; Lorentz (1909, 115). Voigt sent Lorentz a copy of Robb’s dissertation; see Lorentz to Voigt, 18 Dec. 1904, in Kox (2008, § 121).

  52. 52.

    In a letter to Larmor of 18 Jan. 1911, the Cambridge mathematician A. E. H. Love wrote that he had “noted explicitly in writing” to Robb that one of his formulas was from Lobachevski geometry, and that “space might be saved by bringing this fact in” (Larmor Papers, St. John’s College Library). On Robb’s use of hyperbolic geometry, see Walter (1999b).

  53. 53.

    For Robb the “appearance” of contraction was a necessary consequence of light time-of-flight measurements. Robb, Einstein, and their contemporaries focused on the instantaneous form of moving objects, in an approach distinct from the one adopted in the late 1950s. The latter studies characterized what Penrose (1959) referred to as the “photographic” appearance of a moving object.

  54. 54.

    Robb (1911, 1), original emphasis. Cf. Poincaré, Science and Hypothesis (Poincaré 1905a, 50).

  55. 55.

    For appreciations of Robb’s geometry, see Briginshaw (1979) and Cat (2016).

  56. 56.

    LMS Council Minutes, 10 Nov. 1910, LMS archives.

  57. 57.

    Love to Larmor, 18 Jan. 1911, op. cit. Sedleian Chair of Natural Philosophy at Oxford since 1899, Love was Secretary (i.e., managing editor) of the LMS from 1890 to 1910.

  58. 58.

    Jahrbuch über die Fortschritte der Mathematik 43, 1911, p. 559. A succinct summary of Robb’s index concept is provided by Barrow-Green and Gray (2006).

  59. 59.

    LMS Council Minutes, 9 Feb. 1911, LMS archives.

  60. 60.

    Cunningham (1914, 87–89); for an analysis of the procedure, see Newman and Price (2010).

  61. 61.

    See Cunningham (1910, 79). As for Bateman, he credited Cunningham with the discovery of the conformal transformations of the equations of electrodynamics; see Bateman (1910c, 224).

  62. 62.

    L’Enseignement mathématique 10 (1908), 336; Bateman to Hilbert, 1909, Nachlass Hilbert 13, Handschriftenabteilung, NSUB Göttingen.

  63. 63.

    See Minkowski (1909), where the Lorentz transformation is attributed to a paper published in 1887 by Voigt. Minkowski described the covariance of the differential equation of light-wave propagation as the “impetus and true motivation” for assuming the covariance of all laws of physics with respect to the transformations of the Lorentz group (p. 80).

  64. 64.

    See Bateman (1908, 629), read 8 Sept. 1908. No mention is made in this paper of the source of the transformations, but a subsequent work by Bateman credits Cunningham with the “discovery of the formulæ of transformation in the case of an inversion in the four-dimensional space” and cites papers by Hargreaves and Minkowski employing a four-dimensional space with one imaginary axis (Bateman 1909, 224, communicated 9 Oct. 1908). Minkowski’s paper (Minkowski 1908, published 5 April 1908) was cited by both Cunningham and Bateman. Remarked first by Whittaker (1951, vol. 2, 195), the significance of Minkowski’s spacetime theory for the contributions of Cunningham and Bateman is contested by Warwick (2003, 423 n. 49). On the “light-geometric approach” to the foundations of relativity by Cunningham and Bateman, see Jammer (1979, 222).

  65. 65.

    For example, see Warwick (2003, 421) and Bromwich (1901).

  66. 66.

    von Laue (1961, XVIII–XXI); von Laue to Margot Einstein, 23 Oct. 1959, cited by Holton (1965, 39).

  67. 67.

    Laue (1911b); Janssen and Mecklenburg (2006).

  68. 68.

    von Laue (1952).

  69. 69.

    See Max Born’s review in Physikalische Zeitschrift (Born 1912).

  70. 70.

    On Laue’s portrayal of Einstein’s contribution, see Staley (1998). Laue’s contributions to relativity are detailed by Norton (1992) and Rowe (2008).

  71. 71.

    Planck’s argument, which builds on that of Einstein (see above, § 1.2), has inspired many textbook authors. For an example employing a spherical array of photomultipliers at rest in two inertial frames in relative motion, see Rosser (1967, 76).

  72. 72.

    Laue’s use of primes in his light-sphere diagram is peculiar, but is reproduced intact in Figure 1.12, in keeping with the first four editions of his textbook (up to 1921). In the sixth edition (von Laue 1955, 29), A, B, and C are all unprimed, and the primed symbols are as expected: O′ and t′.

  73. 73.

    The transformations of the 15-parameter group of conformal transformations G 15 correspond to what Bateman called the “spherical wave transformations.” On the Bateman-Cunningham discovery of the covariance of Maxwell’s equations under G 15, see Rowe (1999, 211), and Kastrup (2008).

  74. 74.

    Wiechert to Lorentz, 9 March 1912, in Kox (2008, 359); Wiechert (1911, 756).

  75. 75.

    Born in Tiflis (Tbilisi, Georgia), von Ignatowsky earned his Ph.D. in physics at the University of Giessen in 1909 and found employment with the Leitz optical firm in Wetzlar (Klein et al. 1993, 251).

  76. 76.

    On von Ignatowsky’s transformation see Jammer (1979, 215), Torretti (1996, 76), Brown (2005, 105), and Darrigol (2014, 139).

  77. 77.

    On the relation between Lorentz contraction and the Heaviside ellipsoid, see Hunt (1988).

  78. 78.

    “Nun dürfen wir aber unter einem Ruhekoordinatensystem nicht etwa nur ein mathematisches Gebilde verstehen, sonder müssen uns dabei eine materielle Welt mit ihren Beobachtern und synchronem Uhren denken.”

  79. 79.

    Berg went to work for the Siemens-Halske engineering firm in Berlin, where he co-discovered element 75 (Rhenium) with Walter Noddack and Ida Tacke.

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Acknowledgements

Key points of this paper were elaborated in discussions with Olivier Darrigol, Tilman Sauer, June Barrow-Green, and David Rowe; their help is much appreciated. The paper benefits from the expert assistance of Kathryn McKee and Fiona Colbert of St. John’s College, whom I thank most kindly. Citations of the Joseph Larmor Correspondence are by permission of the Master and Fellows of St. John’s College, Cambridge. Permission to quote from the Council Minutes of the London Mathematical Society is gratefully acknowledged. I thank the Niedersächsiche Staats- und Universitätsbibliothek Göttingen for authorizing publication of the diagram in Figure 1.9. I am grateful for the support of the Dibner Rare Book and Manuscript Library and to its staff members Lilla Vekerdy and Kirsten van der Veen for their able assistance during my residence in 2013. A preliminary version of the paper was published in 2011 on PhilSci-Archive.

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    Scott A. Walter

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  1. Institut für Mathematik, Johannes Gutenberg-Universität, Mainz, Germany

    David E. Rowe

  2. Institut für Mathematik, Johannes Gutenberg-Universität, Mainz, Germany

    Tilman Sauer

  3. Centre François Viète, Université de Nantes, Nantes Cedex, France

    Scott A. Walter

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Walter, S.A. (2018). Figures of Light in the Early History of Relativity (1905–1914). In: Rowe, D., Sauer, T., Walter, S. (eds) Beyond Einstein. Einstein Studies, vol 14. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-7708-6_1

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