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Max Born on Hermann Minkowski's Contributions to Spacetime Physics


Poincaré first published his observation that the Lorentz transformations can be viewed as rotations in a four-dimensional space but he regarded it as a mathematical construction which does not represent anything in the physical world (see Appendix). It was Minkowski who first realized (by analyzing the experimental evidence at his time, captured in the relativity principle, including the Michelson-Morley experiment) that reality is a four-dimensional world with time as the fourth dimension. Minkowski's arguments for the reality of spacetime have not been addressed so far, let alone refuted.

Two things appear to indicate that Minkowski arrived independently at what Einstein called special relativity and at the notion of spacetime, but Einstein [1] and Poincaré [2] published first. Minkowski did not publish his results earlier since he was developing the full-blown four-dimensional formalism of spacetime physics reported in 1907 and published in 1908 as a 59-page treatise (it is this formalism that we now use). As Max Born recalled (see below), Minkowski "did not publish them because he wished first to work out the mathematical structure in all its splendour."

First, physicists and mathematicians (experts in spacetime physics) now know well that that formalism (requiring 59 pages to present) (i) could not have been developed only in several months, and (ii) reveals that at that time Minkowski had the most profound understanding of the reason of why all experiments failed to detect absolute motion (uniform motion with respect to the absolute space) - Minkowski successfully decoded the profound physical message hidden in the failure of those experiments: all they failed because there is no such thing as absolute space; every observer has his / her own space and time, which in turn inescapably demonstrates that reality is an absolute four-dimensional world with time as the fourth dimension ("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" [7]). Einstein merely postulated in [1] that only time was relative, whereas Minkowski explained the physical meaning of that postulate - time is relative because observers in relative motion have different times - and remarked that neither Einstein nor Lorentz realized that observers in relative motion have different spaces as well (which is self-evident to a mathematician).

Second, the recollections of Minkowski's student Max Born also indicate (or rather confirm) that Minkowski arrived independently at the physics of flat spacetime. Born recalled that Minkowski "occasionally alluded to" his spacetime research at a seminar organized by Minkowski and Hilbert in the early summer of 1905. At the time of the seminar neither Einstein's paper [1] nor Poincaré's paper [2] were published! Annalen der Physik received Einstein's paper On the Electrodynamics of Moving Bodies on June 30, 1905; Rendiconti del Circolo matematico di Palermo received Poincaré's paper "On the Dynamics of the Electron" (in which Poincaré regarded the Lorentz transformations as rotations in a four-dimensional space with time as the fourth dimension) on July 23, 1905.

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 early 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; Rendiconti del Circolo matematico di Palermo received Poincaré's paper on July 23, 1905. Also, "Lorentz's 1904 paper (with a form of the transformations now bearing his name) was not on the syllabus" [3].

Minkowski's student Max Born, who attended the seminar in 1905, wrote [4]:

"We studied papers by Hertz, Fitzgerald, 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 [5]:

"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."

Three years later (in 1962) he was more specific [6]:

"Minkowski's first ideas about relativity were already worked out and shown" at the seminar.

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 [7]:

"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 physical 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 (which is impossible in a three-dimensional world, where there exist one (and therefore absolute) space and one (and therefore absolute) time) [8]:

"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 [10]. 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 [12]) 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 (see Appendix) and Einstein [12]. Born's recollections only confirmed that.

Appendix: Minkowski and Poincaré

Poincaré first published the important result that the Lorentz transformation had a geometric interpretation as a rotation in a "4-dimensional space. We see that the Lorentz transformation is a rotation of that space around the origin, regarded as fixed" [2]. However, unlike Minkowski, Poincaré seems to have seen nothing revolutionary in the idea of a mathematical four-dimensional space (with time as the fourth dimension) as Damour remarked [14, p. 51]:

"although the first discovery of the mathematical structure of the space-time of special relativity is due to Poincaré's great article of July 1905, Poincaré (in contrast to Minkowski) had never believed that this structure could really be important for physics. This appears clearly in the final passage that Poincaré wrote on the question some months before his death [15]."

[Note: "Poincaré's great article of July 1905" Sur la dynamique de l'électron (Rendiconti del Circolo matematico di Palermo 21 (1906) 129-176) had been received on July 23, 1905, printed on December 14-16, 1905 and published in January 1906.]

Here is "the final passage that Poincaré wrote on the question some months before his death" [15]:

"Everything happens as if time were a fourth dimension of space, and as if four-dimensional space resulting from the combination of ordinary space and of time could rotate not only around an axis of ordinary space in such a way that time were not altered, but around any axis whatever. . .

What shall be our position in view of these new conceptions? Shall we be obliged to modify our conclusions? Certainly not; we had adopted a convention because it seemed convenient and we had said that nothing could constrain us to abandon it. Today some physicists want to adopt a new convention. It is not that they are constrained to do so; they consider this new convention more convenient; that is all. And those who are not of this opinion can legitimately retain the old one in order not to disturb their old habits. I believe, just between us, that this is what they shall do for a long time to come."

Poincaré even appeared to have thought that the spacetime convention would be disadvantageous [16]:

"It quite seems, indeed, that it would be possible to translate our physics into the language of geometry of four dimensions. Attempting such a translation would be giving oneself a great deal of trouble for little profit, and I will content myself with mentioning Hertz's mechanics, in which something of the kind may be seen. Yet, it seems that the translation would always be less simple than the text, and that it would never lose the appearance of a translation, for the language of three dimensions seems the best suited to the description of our world, even though that description may be made, in case of necessity, in another idiom."

Poincaré believed that our physical theories are only convenient descriptions of the world and therefore it is really a matter of convenience and our choice which theory we would use. As Damour stressed it [14, p. 52], it was

"the sterility of Poincaré's scientific philosophy: complete and utter "conventionality" ... which stopped him from taking seriously, and developing as a physicist, the space-time structure which he was the first to discover."

What makes Poincaré's failure to comprehend the profound physical meaning of the relativity principle and the geometric interpretation of the Lorentz transformations especially sad is that it is perhaps the most cruel example in the history of physics of how an inadequate philosophical position can prevent a scientist, even as great as Poincaré, from making a discovery. However, this sad example can serve a noble purpose. Science students and young scientists can study it and learn from it because scientists often think that they do not need any philosophical position for their research [17]:

"Scientists sometimes deceive themselves into thinking that philosophical ideas are only, at best, decorations or parasitic commentaries on the hard, objective triumphs of science, and that they themselves are immune to the confusions that philosophers devote their lives to dissolving. But there is no such thing as philosophy-free science; there is only science whose philosophical baggage is taken on board without examination."


[1] A. Einstein, On the Electrodynamics of Moving Bodies, Annalen der Physik 17 (1905), pp. 891-921. Received on June 30, 1905, Published on September 26, 1905 (original German publication Zur Elektrodynamik bewegter Körper).

[2] H. Poincaré, On the Dynamics of the Electron, (Rendiconti del Circolo matematico di Palermo 21 (1906) pp. 129-176. Received on July 23, 1905, printed on December 14-16, 1905 and published in January 1906 (original French publication Sur la dynamique de l'électron).

[3] 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.

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

[5] 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.

[6] Quoted from S. Walter, Hermann Minkowski's approach to physics, Math Semesterber (2008) 55, pp. 213-235, p. 222.

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

[8] H. Minkowski, Space and Time, new translation in [9, p. 114].

[9] 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).

[10] 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.

[11] 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.

[12] Minkowski arrived at the concept of spacetime by revealing the physical meaning of the equivalence of the times of observers in relative motion, whereas, by contrast, Einstein postulated that equivalence. At that time Einstein had apparently had difficulty realizing the depth of Minkowski's ideas and his 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" [13].

[13] 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.

[14] T. Damour, Once Upon Einstein, Translated by E. Novak (A. K. Peters, Wellesley 2006).

[15] H. Poincaré, Mathematics and Science: Last Essays (Dernières Pensées), Translated by J.W. Bolduc (Dover, New York 1963) pp. 23-24.

[16] H. Poincaré, Science and Method, In: The Value of Science: Essential Writings of Henri Poincaré (Modern Library, New York 2001) p. 438.

[17] D. C. Dennett, Darwin's Dangerous Idea: Evolution and the Meanings of Life (Simon and Schuster, New York 1996) p. 21.