Scott A. Walter
scott.walter [at] univ-lorraine.fr,
Université de Lorraine and LHSP–Archives Poincaré (CNRS, UMR 7117)

H. Goenner, J. Renn, J. Ritter, and T. Sauer (eds.), *The
Expanding Worlds of General Relativity*, Birkhäuser, 1999, 45–86

- Introduction
- 1 Minkowski’s authority in mathematics and physics
- 2 The Cologne lecture
- 3 Responses to the Cologne lecture
- 4 Concluding remarks
- 5 Appendix: Minkowski’s space-time diagram and the Lorentz transformations
- Acknowledgments
- References

The importance of the theory of relativity for twentieth-century
physics, and the appearance of the Göttingen mathematician Hermann
Minkowski at a turning point in its history have both attracted
significant historical attention. The rapid growth in scientific and
philosophical interest in the principle of relativity has been linked
to the intervention of Minkowski by Tetu Hirosige, who identified
Minkowski’s publications as the turning point for the theory of
relativity, and gave him credit for having clarified its fundamental
importance for all of physics
(Hirosige 1968, 46;
1976, 78).
Lewis Pyenson has placed Minkowski’s work in the context of a mathematical
approach to physics popular in Göttingen, and attributed its success
to the prevalence of belief in a neo-Leibnizian notion of
pre-established harmony between pure mathematics and physics
(Pyenson 1985;
1987, 95).
The novelty to physics of the aesthetic canon
embodied in Minkowski’s theory was emphasized by Peter
Galison (1979),
and several scholars have clarified technical and epistemological
aspects of Minkowski’s theory.^{1}^{1}On Minkowski’s role in the
history of relativity see also Illy (1981) and Pyenson
(1987). Many
references to the primary and secondary literature on the theory of
relativity may be found in Miller
(1981) and Paty
(1993). Pauli
(1958)
remains an excellent guide to the primary literature.
In particular, the introduction of sophisticated mathematical
techniques to theoretical physics by Minkowski and others is a theme
illustrated by Christa Jungnickel and Russell
McCormmach.^{2}^{2}McCormmach (1976);
Jungnickel & McCormmach
(1986, II, 334–347).

In what follows, we address another aspect of Minkowski’s role in the history of the theory of relativity: his disciplinary advocacy. Minkowski’s 1908 Cologne lecture ‘‘Raum und Zeit’’ (Minkowski 1909) may be understood as an effort to extend the disciplinary frontier of mathematics to include the principle of relativity. We discuss the tension created by a mathematician’s intrusion into the specialized realm of theoretical physics, and Minkowski’s strategy to overcome disciplinary obstacles to the acceptance of his work. The effectiveness of his approach is evaluated with respect to a selection of responses, and related to trends in bibliometric data on disciplinary contributions to non-gravitational theories of relativity through 1915.

At the time of the meeting of the German Association in late September 1908, Minkowski was recognized as an authority on the theory of relativity nowhere outside of the university town of Göttingen. The structure and content of Minkowski’s lecture, we will see later, was in many ways a function of a perceived deficit of credibility. In order to understand this aspect of Minkowski’s lecture, we first examine how Minkowski became acquainted with the electrodynamics of moving bodies.

Around 1907, Minkowski’s scientific reputation rested largely upon his
contribution to number theory.^{3}^{3}Minkowski published his lectures on Diophantine analysis in
Minkowski (1907a).
Yet Minkowski was also the author of an article on capillarity
(Minkowski
1907b) in
the authoritative Encyklopädie der mathematischen Wissenschaften,
granting him a credential in the domain of mechanics and mathematical
physics. In addition, Minkowski had lectured on capillarity, potential
theory, and analytical mechanics, along with mathematical subjects such as
Analysis Situs and number theory at Zurich Polytechnic, where Einstein,
Marcel Grossmann and Walter Ritz counted among his students; he also
lectured on mechanics and electrodynamics (among other subjects) in
Göttingen, where he held the third chair in mathematics, created for him
at David Hilbert’s request in 1902.^{4}^{4}Copies of Minkowski’s
manuscript notes of these lectures
are in the Niels Bohr Library, Minkowski Papers, Boxes 7, 8 and 9.

In Göttingen, Minkowski took an interest in a subject strongly associated
with the work of many of his new colleagues: electron theory. An early
manifestation of this interest was Minkowski’s co-direction of a seminar
on the subject with his friend Hilbert, plus Gustav Herglotz and Emil
Wiechert, which met during the summer semester of 1905.^{5}^{5}On
the Göttingen electron theory seminar, see
Pyenson (1985, 102).
While Lorentz’s 1904 paper (with a form of the transformations now bearing
his name) was not on the syllabus, and Einstein’s 1905 paper had not yet
appeared, one of the students later recalled that Minkowski had hinted
that he was engaged with the Lorentz transformations.^{6}^{6}Undated
manuscript,
Niedersächsische Staats- und
Universitätsbibliothek, Nachlaß Hilbert 570/9;
Born (1959, 682).

Minkowski was also busy with his article on capillarity, however, and for
the next two years there is no trace of his engagement with the theory of
relativity. In October 1907, Minkowski wrote to Einstein to request an
offprint of his Annalen article on the electrodynamics of moving
bodies, for use in his seminar on the partial differential equations of
physics, jointly conducted by Hilbert.^{7}^{7}Minkowski to Einstein,
9 October 1907, in Klein et al., eds. (1993, Doc. 62);
course listing in Physikalische Zeitschrift 8 (1907): 712.
Fragmentary notes by Hermann Mierendorff from this seminar show a
discussion of Lorentz’s electrodynamics of moving media, see
Niedersächsische Staats- und Universitätsbibliothek, Nachlaß Hilbert 570/5;
Pyenson (1985, 83).
During the same semester, Minkowski introduced
the principle of relativity into his lectures on the theory of functions
(‘‘Funktionentheorie.’’ Minkowski Papers: Box 9, Niels Bohr Library).
During the following Easter vacation, he gave a short series of lectures
on ‘‘New Ideas on the Basic Laws of Mechanics’’ for the benefit of science
teachers.^{8}^{8}‘‘Neuere Ideen über die Grundgesetze der Mechanik,’’ held in
Göttingen from 21 April to 2 May, see L’Enseignement Mathématique
10 (1908): 179.

In what seem to be notes to these holiday lectures, Einstein’s knowledge
of mathematics was subject to criticism. Minkowski reminded his audience
that he was qualified to make this evaluation, since Einstein had him to
thank for his education in mathematics. From Zurich Polytechnic, Minkowski
added, a complete knowledge of mathematics could not be obtained.^{9}^{9}Undated manuscript, Niedersächsische Staats- und
Universitätsbibliothek, Math. Archiv 60: 4, 52. Minkowski’s uncharitable
assessment of mathematics at Zurich Polytechnic belied the presence on the
faculty of his friend Adolf Hurwitz, a member of the mathematical elite,
and a lecturer of great repute. Graduates included Marcel Grossmann,
L.-Gustave du Pasquier and Minkowski’s doctoral student Louis Kollros, all
of whom were called upon to teach university mathematics upon completion
of their studies. In recollections of his years as Einstein’s classmate,
Kollros wrote that there was ‘‘almost too much mathematics’’ at Zürich
Polytechnic
(Kollros 1956, 273).
Minkowski’s remark that Einstein’s
mathematical knowledge was incomplete may have been based on the fact
that, unlike his classmates, Einstein did not elect to pursue graduate
studies in mathematics, after obtaining the diploma from Polytechnic.

This frank assessment of Einstein’s skills in mathematics, Minkowski
explained, was meant to establish his right to evaluate Einstein’s work,
since he did not know how much his authority carried with respect to ‘‘the
validity of judgments in physical things,’’ which he wanted ‘‘now to
submit.’’ A pattern was established here, in which Minkowski would first
suggest that Einstein’s work was mathematically incomplete, and then call
upon his authority in mathematics in order to validate his judgments in
theoretical physics. While Minkowski implicitly recognized Einstein’s
competence in questions of physics, he did not yet appreciate how much
Europe’s leading physicists admired the work of his former student.^{10}^{10}In a letter of 18 October 1908, Minkowski wrote to Robert Gnehm of
his satisfaction in learning—during the Cologne meeting of scientists
and physicians—how much Einstein’s work was admired by the likes of
Walther Nernst, Max Planck and H. A. Lorentz
(Seelig 1956, 131–132).
Even in his fief of Göttingen, Minkowski knew he could not expect any
authority to be accorded to him in theoretical physics, yet this awareness
of his own lack of credentials in physics did not prevent him from
lecturing on the principle of relativity.

While the scientific world had no real means of judging Minkowski’s competence in theoretical physics due to the paucity of relevant publications, Minkowski himself did not consider his knowledge in physics to be extensive. It is for this reason that he sought an assistant capable of advising him on physical matters, and when Max Born—a former student from the electron theory seminar—wrote him from Breslau (now Wrocław, Poland) for help with a technical problem, he found a suitable candidate.

Initially attracted to mathematics, Born heard lectures by Leo
Königsberger in Heidelberg, and Adolf Hurwitz in Zürich, and later
considered Hurwitz’s private lectures as the high point of his student
career. In Göttingen, Born obtained a coveted position as Hilbert’s
private assistant, and began a doctoral dissertation on Bessel functions
under Hilbert’s direction. When he abandoned the topic, as Born recalled
in old age, Hilbert laughed and consoled him, saying he was much better
in physics.^{11}^{11}Transcript of an oral interview with Thomas S. Kuhn, 18 October 1962,
Archives for History of Quantum Physics, p. 5.
In the same year, Born attended Hilbert and Minkowski’s electron theory
seminar, along with Max Laue and Jakob Laub, among others
(Born 1959, 682;
Pyenson 1985, 102).
Profoundly influenced by what he learned in this
seminar, and deeply devoted to both Hilbert and Minkowski, Born was not
permitted to write a dissertation on electron theory, although the idea
appealed to him
(Born 1959, 684).
Felix Klein obliged him to write a
dissertation on elasticity theory, but in order to avoid having ‘‘the great
Felix’’ as an examiner, Born took up Karl Schwarzschild’s suggestion to
prepare for the oral examination in astronomy (Born
1906;
1968, 20–21).
After defending his doctoral dissertation on 14 January 1907, Born spent six
months in Cambridge with Joseph Larmor and J. J. Thomson before returning to
Breslau, where the young theoretical physicists Stanislaus Loria and Fritz
Reiche brought Einstein’s 1905 Annalen paper on relativity to his
attention
(Born 1959, 684).

In studying relativity with Reiche, as Born recounted later, he
encountered some difficulties. He formulated these in a letter to
Minkowski, seeking his former teacher’s advice. Minkowski’s response
to Born’s letter was a great surprise, for instead of the requested
technical assistance, Minkowski offered him the possibility of an
academic career. Minkowski wrote that he had been working on the same
problem as Born, and that he ‘‘would like to have a young collaborator
who knew something of physics, and of optics in particular’’
(Born 1978, 130).^{12}^{12}According to another version, the manuscript sent
to Minkowski showed a new way of calculating the electromagnetic
mass of the electron, described by Born as a combination of
‘‘Einstein’s ideas with Minkowski’s mathematical methods’’
(Born 1968, 25).
Besides mathematics, Born had studied physics in
Göttingen, attending Voigt’s ‘‘stimulating’’ lectures on optics and an
advanced course on optical experimentation
(Born 1968, 21). It was
just this background in optics that Minkowski lacked, and he looked to
Born to guide him through unknown territory. In return, Minkowski
promised Born he would open the doors to an academic career. The
details were to be worked out when they met at the meeting of the
German Association of Scientists and Physicians, later that year in
Cologne
(Born 1978, 130).^{13}^{13}Minkowski’s premature death
prevented him from personally fulfilling his obligation to Born, but
his Göttingen colleagues accorded Born the venia legendi in
theoretical physics, on Voigt’s recommendation
(Born 1978, 136).

In April 1908, Minkowski published a technically accomplished paper on the electromagnetic processes in moving bodies (‘‘Die Grundgleichungen für die electromagnetischen Vorgänge in bewegten Körpern,’’ hereafter Grundgleichungen). In this essay, Minkowski wrote the empty-space field equations of relativistic electrodynamics in four-dimensional form, using Arthur Cayley’s matrix calculus. He also derived the equations of electrodynamics of moving media, and formulated the basis of a mechanics appropriate to four-dimensional space with an indefinite squared interval. Minkowski’s study represented the first elaboration of the principle of relativity by a mathematician in Germany.

Soon after its publication, the Grundgleichungen sustained restrained comment from Minkowski’s former students Albert Einstein and Jakob Laub (1908b, 1908a). These authors rejected out of hand the four-dimensional apparatus of Minkowski’s paper, the inclusion of which, they wrote, would have placed ‘‘rather great demands’’ on their readers (Einstein & Laub 1908b, 532). No other reaction to Minkowski’s work was published before the Cologne meeting.

By the fall of 1908, Minkowski had spoken publicly of his views on relativity on several occasions, but never outside of Göttingen. The annual meeting of the German Association was Minkowski’s first opportunity to speak on relativity before an elite international audience of physicists, mathematicians, astronomers, chemists and engineers. At no other meeting could a scientist in Germany interact with other professionals working in disciplines outside of his own.

The organization of the various disciplinary sections of the annual
meeting of the German Association fell to the corresponding
professional societies
(Forman 1967, 156).
For example, the German
Physical Society organized the physics section, and the German Society
of Mathematicians managed the mathematics section. For the latter
section, the theme of discussion was announced in late April by the
society’s president, Felix Klein. In a call for papers, Klein
encouraged authors to submit works especially in the area of
mechanics. Prior to the announcement, however, Klein must have already
arranged at least one contribution in mechanics, since he added a
teaser, promising an ‘‘expert aspect’’ of a recent investigation in
this area.^{14}^{14}Jahresbericht der deutschen
Mathematiker-Vereinigung 17 (1908): 61, dated 26 April
1908. It is tempting to identify this as a forward reference to
Minkowski’s lecture, a draft of which predates Klein’s communication
by a few days. The lecture was to be the first talk out of seven in
the mathematics section, which doubled as a session of the German
Society of Mathematicians.^{15}^{15}Most of the lectures in the first
section were published in volume 18 of the Jahresbericht der
deutschen Mathematiker-Vereinigung. Shortly after the end of the
First World War, the German Physical Society also held session at
meetings of the German Association (see
Forman 1967, 156).

The Göttingen archives contain four distinct manuscript drafts
of Minkowski’s Cologne lecture, none of which corresponds precisely to
either of the two printed versions of the lecture in the original
German.^{16}^{16}Niedersächsische Staats- und Universitätsbibliothek,
Math. Arch. 60: 2 and 60: 4. An early draft is dated 24 April 1908
(60: 4, folder 1, p. 66.); the other drafts are undated. Unless
stipulated otherwise, we refer here to the longer essay published
posthumously in both the Physikalische Zeitschrift and the Jahresbericht der deutschen Mathematiker-Vereinigung in early 1909.

From the outset of his lecture, Minkowski announced that he would reveal a radical change in the intuitions of space and time:

Gentlemen! The conceptions of space and time which I would like to develop before you arise from the soil of experimental physics. Therein lies their strength. Their tendency is radical. From this hour on, space by itself and time by itself are to sink fully into the shadows and only a kind of union of the two should yet preserve autonomy.

First of all I would like to indicate how, [starting] from the mechanics accepted at present, one could arrive through purely mathematical considerations at changed ideas about space and time.^{17}^{17}‘‘M. H.! Die Anschauungen über Raum und Zeit, die ich Ihnen entwickeln möchte, sind auf experimentell-physikalischem Boden erwachsen. Darin liegt ihre Stärke. Ihre Tendenz ist eine radikale. Von Stund’ an sollen Raum für sich und Zeit für sich völlig zu Schatten herabsinken und nur noch eine Art Union der beiden soll Selbständigkeit bewahren. Ich möchte zunächst ausführen, wie man von der gegenwärtig angenommen Mechanik wohl durch eine rein mathematische Überlegung zu veränderten Ideen über Raum und Zeit kommen könnte.’’ (Minkowski 1909, 75)

The evocation of experimental physics was significant in the
first sentence of Minkowski’s lecture, and it was deceptive. In what
followed, Minkowski would refer to experimental physics only once, to
invoke the null result of Albert A. Michelson’s optical experiment to
detect motion with respect to the luminiferous ether. Otherwise,
Minkowski kept his promise of a ‘‘rein mathematische’’ exposé,
devoid of experimental considerations. A purely theoretical presentation
enabled Minkowski to finesse the recent well-known experimental results
purporting to disconfirm relativity theory, obtained by Walter
Kaufmann.^{18}^{18}The empirical adequacy of the ‘‘Lorentz-Einstein’’ theory had been
challenged by Walter Kaufmann in 1905, on the basis of his measurements of
the magnetic deflection of cathode rays (see Miller
1981 and Hon
1995).
Two days after Minkowski’s lecture, Alfred Bucherer announced to the
physical section the results of his deflection experiments, which
contradicted those of Kaufmann and confirmed the expectations of the
Lorentz-Einstein theory
(Bucherer 1908).
In the discussion of this
lecture, Minkowski expressed joy in seeing the ‘‘monstrous’’ rigid electron
hypothesis experimentally defeated in favor of the deformable electron of
Lorentz’s theory (see
Bucherer 1908, 762).

Less illusory than the mention of experimental physics was Minkowski’s announcement of a radical change in conceptions of space and time. That this revelation was local and immediate, is signaled by the phrase ‘‘from this hour on’’ [von Stund’ an]. Here it was announced that a union of space and time was to be revealed, and for the first time. This was a rhetorical gesture (all of the results presented in the Cologne lecture had been published in the Grundgleichungen), but it was an effective one, because the phrase in question became emblematic of the theory of relativity in broader circles.

It may be noted from the outset that the claims Minkowski made for his theory fell into two categories. In one category were Minkowski’s claims for scientific priority, which concerned the physical, mathematical and philosophical aspects of his theory of relativity. In what follows, we will concentrate on the second category of claims, which were metatheoretical in nature. The latter claims concerned the theory’s type, not its constituent elements. Claims of the second sort, all having to do with the geometric nature of the theory, reinforced those of the first sort. The opening remarks provide an example: the allusion to changed ideas about space and time belongs to the first sort, while the claim of a purely mathematical development is of the second kind.

In order to demonstrate the difference between the old view of space and
time and the new one, Minkowski distinguished two transformation groups
with respect to which the laws of classical mechanics were covariant.^{19}^{19}Minkowski introduced the use of covariance with respect to
the Lorentz transformations in Minkowski
(1908, 473). In the Cologne
lecture, the term invariant was employed in reference to both
covariant and invariant expressions.
Considering first the same zero point in time and space for two systems in
uniform translatory motion, he noted that the spatial axes $x$, $y$,
$z$ could undergo an arbitrary rotation about the origin. This
corresponded to the invariance in classical mechanics of the sum of
squares ${x}^{2}+{y}^{2}+{z}^{2}$, and was a fundamental characteristic of
physical space, as Minkowski reminded his audience, that did not concern
motion. Next, the second group was identified with the transformations:

$${x}^{\prime}=x+\alpha t,{y}^{\prime}=y+\beta t,{z}^{\prime}=z+\gamma t,{t}^{\prime}=t.$$ |

Thus physical space, Minkowski pointed out, which one supposed to be at rest, could in fact be in uniform translatory motion; from physical phenomena no decision could be made concerning the state of rest (Minkowski 1909, 77).