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Hermann Minkowski




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Max Born on Hermann Minkowski's Contributions to Spacetime Physics

Max Born's recollections seem to indicate (or rather confirm) that Minkowski arrived independently on what Einstein called special relativity and on the notion of spacetime, but Einstein and Poincaré published first while Minkowski was developing the full-blown four-dimensional formalism of spacetime reported in 1907 and published in 1908 as a 59-page treatise, which alone indicates that Minkowski developed his own ideas.

By 1905 Hermann Minkowski was already internationally recognized as an exceptional mathematical talent. At that time he became interested in the electron theory and especially in an unresolved issue at the very core of fundamental physics - at the turn of the nineteenth and twentieth century Maxwell's electrodynamics had been interpreted to show that light is an electromagnetic wave, which propagates in a light-carrying medium (the luminiferous ether), but its existence was put into question since Michelson's interference experiments failed to detect the Earth's motion in that medium.

Minkowski's documented involvement with the electrodynamics of moving bodies began in the summer of 1905 when he and his friend David Hilbert co-directed a seminar in Göttingen on the electron theory. The paper of Minkowski's student - Albert Einstein - on special relativity was not published at that time; Annalen der Physik received the paper on June 30, 1905. Poincaré's longer paper "Sur la dynamique de l'électron" (in which Poincaré regarded the Lorentz transformations as rotations in a four-dimensional space with time as the fourth dimension) was not published either; it appeared in 1906. Also, "Lorentz's 1904 paper (with a form of the transformations now bearing his name) was not on the syllabus" [1].

Minkowski's student Max Born, who attended the seminar in 1905, wrote [2]:

"We studied papers by Hertz, Fitzcerald, Larmor, Lorentz, Poincaré, and others but also got an inkling of Minkowski's own ideas which were published only two years later."

Born also recalled what Minkowski had specifically said during the seminar in 1905 [3]:

"I remember that Minkowski occasionally alluded to the fact that he was engaged with the Lorentz transformations, and that he was on the track of new interrelationships."

Again Born wrote in his autobiography about what he had heard from Minkowski after Minkowski's lecture "Space and Time" given on September 21, 1908 [4]:

"He told me later that it came to him as a great shock when Einstein published his paper in which the equivalence of the different local times of observers moving relative to each other were pronounced; for he had reached the same conclusions independently but did not publish them because he wished first to work out the mathematical structure in all its splendour. He never made a priority claim and always gave Einstein his full share in the great discovery."

These facts and especially the depth of the ideas developed in Minkowski's publications are the best proof that in the period 1905-1908 Minkowski had found a truly revolutionary resolution of the difficult issues surrounding the electrodynamics of moving bodies - that the relativity principle implies that the Universe is a four-dimensional world with time as the fourth dimension: the relativity principle restates the experimental fact that physical phenomena are the same in all inertial reference frames; Minkowski managed to decode the profound message hidden in that experimental fact - physical phenomena are the same for all inertial observers because each observer describes them in terms of his own space and time [5]:

"Hereafter we would then have in the world no more the space, but an infinite number of spaces analogously as there is an infinite number of planes in three-dimensional space. Three-dimensional geometry becomes a chapter in four-dimensional physics."

For example, each inertial observer measures the speed of light in his own space using his own time, which naturally explains why the speed of light is the same in all inertial frames.

Unfortunately, Minkowski had never indicated exactly when he arrived at that discovery, but Born's recollections show that Minkowski was already discussing his ideas at the seminar in the summer of 1905 - note that at that time Einstein's 1905 paper was not published; Minkowski asked Einstein to send him the 1905 paper hardly on October 9, 1907 [7]. So, Minkowski's insight had occurred sufficiently long before his December 1907 lecture "The Fundamental Equations for Electromagnetic Processes in Moving Bodies" when he presented the fully developed mathematical formalism of the four-dimensional physics of spacetime introduced by him, because such a revolutionary four-dimensional formalism (published in 1908 as a 59-page treatise [8]) could not have been created in just several months. It appears Minkowski needed two years - from 1905 to 1907 - to develop the mathematics of spacetime.

It is precisely the complexity of this novel mathematical apparatus specifically developed to describe spacetime (or the World as Minkowski called it) which indicates that Minkowski had developed his own ideas at which he arrived independently of Poincaré and Einstein [9]. Born's recollections only confirmed that.


[1] S. Walter, Minkowski, Mathematicians, and the Mathematical Theory of Relativity, in H. Goenner, J. Renn, J. Ritter, T. Sauer (eds.), The Expanding Worlds of General Relativity, Einstein Studies, volume 7, (Birkhäuser, Basel 1999) pp. 45-86, p. 46.

[2] M. Born, Physics in My Generation 2nd ed. (Springer-Verlag, New York 1969) p. 101

[3] Quoted from T. Damour, What is missing from Minkowski's "Raum und Zeit" lecture, Annalen der Physik 17 No. 9-10 (2008), pp. 619-630, p. 626.

[4] M. Born, My Life: Recollections of a Nobel Laureate (Scribner, New York 1978) p. 131.

[5] H. Minkowski, Space and Time, new translation in [6, p. 114].

[6] H. Minkowski, Space and Time: Minkowski's Papers on Relativity, translated by Fritz Lewertoff and Vesselin Petkov; edited by V. Petkov (Minkowski Institute Press, Montreal 2012).

[7] Postcard: Minkowski to Einstein, October 9, 1907, in: M.J. Klein, A. J. Kox, and R. Schulmann (eds) The Collected Papers of Albert Einstein, Volume 5: The Swiss Years: Correspondence, 1902-1914 (Princeton University Press, Princeton 1995), p. 62.

[8] H. Minkowski, Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern, Nachrichten der K. Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-physikalische Klasse (1908) S. 53-111; reprinted in H. Minkowski, Zwei Abhandlungen über die Grundgleichungen der Elektrodynamik, mit einem Einführungswort von Otto Blumenthal (Teubner, Leipzig 1910) S. 5-57, and in Gesammelte Abhandlungen von Hermann Minkowski, ed. by D. Hilbert, 2 vols. (Teubner, Leipzig 1911), vol. 2, pp. 352-404.

[9] At that time Einstein's reaction at Minkowski's four-dimensional physics was rather hostile. Sommerfeld's recollection of what Einstein said on one occasion provides an indication of Einstein's initial attitude towards the work of his mathematics professor on the foundations of spacetime physics: "Since the mathematicians have invaded the relativity theory, I do not understand it myself any more" [10].

[10] A. Sommerfeld, To Albert Einstein's Seventieth Birthday. In: Albert Einstein: Philosopher-Scientist. P. A. Schilpp, ed., 3rd ed. (Open Court, Illinois 1969) pp. 99-105, p. 102.