walter [at] univ-nancy2.fr

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. ${}^{17}$ (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 "

$x\text{'}=x+\alpha t,\hspace{0.5em}y\text{'}=y+\beta t,\hspace{0.5em}z\text{'}=z+\gamma t,\hspace{0.5em}t\text{'}=t.$ |

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 (1909: 77).

After noting verbally the distinction between these two groups, Minkowski turned to the blackboard for a graphical demonstration. He drew a diagram to demonstrate that the above transformations allowed one to draw the time axis in any direction in the half-space $t>0$. While no trace has been found of Minkowski's drawing, it may have resembled the one published later by Max Born and other expositors of the theory of relativity (see Figure 1). ${}^{20}$ This was the occasion for Minkowski to introduce a spate of neologisms (Minkowski 1909: 76-77):

${c}^{2}{t}^{2}-{x}^{2}-{y}^{2}-{z}^{2}=1,$ |

where $c$ was an unspecified, positive-valued parameter (Minkowski 1909: 77). Suppressing two dimensions in $y$ and $z$, he then showed how this unit hypersurface might be used to construct a group of transformations ${G}_{c}$, once the arbitrary displacements of the zero point were associated with rotations about the origin. The figure obtained was introduced on a transparent slide, showing two pairs of symmetric axes. ${}^{21}$

Minkowski constructed the figure using the upper branch of the two-branched unit hyperbola ${c}^{2}{t}^{2}-{x}^{2}=1$ to determine the parallelogram $\mathrm{OA}\text{'}B\text{'}C\text{'}$, from which the $x\text{'}$ and $t\text{'}$ axes were established (see Figure 2, left, and the Appendix). The relation between this diagram and the one corresponding to classical mechanics he pointed out directly: as the parameter $c$ approached infinity,

this special transformation becomes one in which the $t\text{'}$ axis can have an arbitrary upward direction, and $x\text{'}$ approaches ever closer to $x$. ${}^{22}$ (Minkowski 1909: 78)In this way, the new space-time diagram collapsed into the old one, in a lovely graphic recovery of classical kinematics. ${}^{23}$ The limit-relation between the group ${G}_{c}$ and the group corresponding to classical mechanics $({G}_{\infty})$ called forth a comment on the history of the principle of relativity. Minkowski observed that in light of this limit-relation, and

since ${G}_{c}$ is mathematically more intelligible than ${G}_{\infty}$, a mathematician would well have been able, in free imagination, to arrive at the idea that in the end, natural phenomena actually possess an invariance not with respect to the group ${G}_{\infty}$, but rather to a group ${G}_{c}$, with a certain finite, but in ordinary units of measurementTo paraphrase, it was no more than a fluke of history that a nineteenth-century mathematician did not discover the role played by the group ${G}_{c}$ in physics, given its greater mathematical intelligibility in comparison to the group ${G}_{\infty}$. In other words, the theory of relativity was not a product of pure mathematics, although it could have been. Minkowski openly recognized the role-albeit a heuristic one-of experimental physics in the discovery of the principle of relativity. All hope was not lost for pure mathematics, however, as Minkowski continued:extremely large[value of] $c$. Such a premonition would have been an extraordinary triumph for pure mathematics. ${}^{24}$ (Minkowski 1909: 78)

While mathematics displays only more staircase-wit here, it still has the satisfaction of realizing straight away, thanks to fortunate antecedents and the exercised acuity of its senses, the fundamental consequences of such a reformulation of our conception of nature. ${}^{25}$ (Minkowski 1909: 78)Minkowski conceded that, in this instance, mathematics could only display wisdom after the fact, instead of a creative power of discovery. Again he stressed the mathematician's distinct advantage over members of other scientific disciplines in seizing the deep consequences of the new theoretical view.

The existence of the invariance of the laws of nature for the group ${G}_{c}$ would now be understood as follows: from the entirety of natural phenomena we can derive, through successively enhanced approximations, an ever more precise frame of reference $x$, $y$, $z$ and $t$, space and time, by means of which these phenomena can then be represented according to definite laws. This frame of reference, however, is by no means uniquely determined by the phenomena.For anyone who might have objected that others had already pointed this out, Minkowski offered an interpretation of his theory on the space-time diagram. ${}^{30}$We can still arbitrarily change the frame of reference according to the transformations of the group termed ${G}_{c}$ without changing the expression of the laws of nature. ${}^{29}$ (Minkowski 1909: 78-79)

We can, for example, also designate time [as] $t\text{'}$, according to the figure described. However, in connection with this, space must then necessarily be defined by the manifold of three parameters $x\text{'}$, $y$, $z$, on which physical laws would now be expressed by means of $x\text{'}$, $y$, $z$, $t\text{'}$ in exactly the same way as with $x$, $y$, $z$, $t$. Then from here on, we would no longer haveThe emphasis on space was no accident, as Minkowski presented the notion of "endlessly many spaces" as his personal contribution, in analogy to Einstein's concept of relative time. The grandiose announcement of the end of space and time served as a frame for the enunciation of Minkowski's principle of relativity. ${}^{32}$ Rhetorical gestures such as this directed attention to Minkowski's theory; its acceptance by the scientific community, however, may be seen to depend largely upon the presence of two elements: empirical adequacy, claimed by Minkowski at the opening of the lecture, and the perception of an advantage over existing theories. Minkowski went on to address in turn the work of two of his predecessors, Lorentz and Einstein. Before discussing Minkowski's exposé of their work, however, we want to consider briefly the work of a third precursor, whose name was not mentioned at all in this lecture: Henri Poincaré.spacein the world, but endlessly many spaces; analogously, endlessly many planes exist in three-dimensional space. Three-dimensional geometry becomes a chapter of four-dimensional physics. You realize why I said at the outset: space and time are to sink into the shadows; only a world in and of itself endures. ${}^{31}$ (Minkowski 1909: 79)

Concerning the credit to be accorded to individual authors, stemming from the foundations of Lorentz's ideas, Einstein developed the principle of relativity more distinctly [and] at the same time applied it with particular success to the treatment of special problems in the optics of moving media, [and] ultimately [was] also the first to draw conclusions concerning the variability of mechanical mass in thermodynamic processes. A short while later, and no doubt independently of Einstein, Poincaré extended [the principle of relativity] in a more mathematical study to Lorentz electrons and their status in gravitation. Finally, Planck sought the basis of a dynamics grounded on the principle of relativity. ${}^{36}$ (Minkowski 1907b: 16-17)Following their appearance in this short history of the principle of relativity, the theoretical physicists Lorentz, Einstein and Max Planck all made it into Minkowski's Cologne lecture, but the more mathematical Poincaré was left out. At least one theoretical physicist felt Minkowski's exclusion of Poincaré in "Raum und Zeit" was unfair: Arnold Sommerfeld. In the notes he added to a 1913 reprint of this lecture, Sommerfeld attempted to right the wrong by making it clear that a Lorentz-covariant law of gravitation and the idea of a four-vector had both been proposed earlier by Poincaré. Among the mathematicians following the developments of electron theory, many considered Poincaré as the founder of the new mechanics. For instance, the editor of

You undoubtedly know the pamphlet by Minkowski, "Raum und Zeit," published after his death, as well as the ideas of Einstein and Lorentz on the same question. Now, M. Fredholm tells me that you have touched upon similar ideas before the others, while expressing yourself in a less philosophical, more mathematical manner. ${}^{37}$ (Mittag-Leffler 1909)It is unknown if Poincaré ever received this letter. Like Sommerfeld, Mittag-Leffler and Fredholm reacted to the omission of Poincaré's name from Minkowski's lecture. The absence of Poincaré from Minkowski's speech was remarked by leading scientists, but what did Poincaré think of this omission? His first response, in any case, was silence. In the lecture Poincaré delivered in Göttingen on the new mechanics in April 1909, he did not see fit to mention the names of Minkowski and Einstein (Poincaré 1910a). Yet where his own engagement with the principle of relativity was concerned, Poincaré became more expansive. In Berlin the following year, for example, Poincaré dramatically announced that already back in 1874 (or 1875), while a student at the École polytechnique, he and a friend had experimentally confirmed the principle of relativity for optical phenomena (Poincaré 1910b: 104). ${}^{38}$ Less than five years after its discovery, the theory of relativity's prehistory was being revealed by Poincaré in a way that underlined its empirical foundations-in contradistinction to the Minkowskian version. If Poincaré expressed little enthusiasm for the new mechanics unleashed by the principle of relativity, and had doubts concerning its experimental underpinnings, he never disowned the principle. ${}^{39}$ In the spring of 1912, Poincaré came to acknowledge the wide acceptance of a formulation of physical laws in four-dimensional (Minkowski) space-time, at the expense of the Lorentz-Poincaré electron theory. His own preference remained with the latter alternative, which did not require an alteration of the concept of space (Poincaré 1912: 170). In the absence of any clear indication why Minkowski left Poincaré out of his lecture, a speculation or two on his motivation may be entertained. If Minkowski had chosen to include some mention of Poincaré's work, his own contribution may have appeared derivative. Also, Poincaré's modification of Lorentz's theory of electrons constituted yet another example of the cooperative role played by the mathematician in the elaboration of physical theory. ${}^{40}$ Poincaré's "more mathematical" study of Lorentz's electron theory demonstrated the mathematician's dependence upon the insights of the theoretical physicist, and as such, it did little to establish the independence of the physical and mathematical paths to the Lorentz group. The metatheoretical goal of establishing the essentially mathematical nature of the principle of relativity was no doubt more easily attained by neglecting Poincaré's elaboration of this principle.

of first clearly recognizing that the time of one electron is just as good as that of the other, that is to say, that $t$ and $t\text{'}$ are to be treated identically. ${}^{43}$ (Minkowski 1909: 81)This interpretation of Einstein's notion of time with respect to an electron was not one advanced by Einstein himself. We will return to it shortly; for now we observe only that Minkowski seemed to lend some importance to Einstein's contribution, because he went on to refer to him as having deposed the concept of time as one proceeding unequivocally from phenomena. ${}^{44}$

Neither Einstein nor Lorentz rattled the concept of space, perhaps because in the above-mentioned special transformation, where the plane of $x\text{'}t\text{'}$ coincides with the plane of $\mathrm{xt}$, an interpretation is [made] possible by saying that the $x$-axis of space maintains its position. ${}^{45}$ (Minkowski 1909: 81-82)This was the only overt justification offered by Minkowski in support of his claim to have surpassed the theories of Lorentz and Einstein. His rather tentative terminology [

Proceeding beyond the concept of space in a corresponding way is likely to be appraised as only another audacity of mathematical culture. Even so, following this additional step, indispensable to the correct understanding of the group ${G}_{c}$, the termWhere Einstein had deposed the concept of time (and time alone, by implication), Minkowski claimed in a like manner to have overthrown the concept of space, as Galison has justly noted (1979: 113). Furthermore, Minkowski went so far as to suggest that his "additional step" was essential to a "correct understanding" of what he had presented as the core of relativity: the group ${G}_{c}$. He further implied that the theoretical physicists Lorentz and Einstein, lacking a "mathematical culture," were one step short of the correct interpretation of the principle of relativity. Having disposed in this way of his precursors, Minkowski was authorized to invent a name for his contribution, which he called the postulate of the absolute world, or world-postulate for short (1909: 82). It was on this note that Minkowski closed his essay, trotting out the shadow metaphor one more time:relativity postulatefor the requirement of invariance under the group ${G}_{c}$ seems very feeble to me. ${}^{46}$ (Minkowski 1909: 82)

The validity without exception of the world postulate is, so I would like to believe, the true core of an electromagnetic world picture; met by Lorentz, further revealed by Einstein, [it is] brought fully to light at last. ${}^{47}$ (Minkowski 1909: 88)According to Minkowski, Einstein clarified the physical significance of Lorentz's theory, but did not grasp the true meaning and full implication of the principle of relativity. Minkowski marked his fidelity to the Göttingen electron-theoretical program, which was coextensive with the electromagnetic world picture. When Paul Ehrenfest asked Minkowski for a copy of the paper going by the title "On Einstein-Electrons," Minkowski replied that when used in reference to the

${x}^{2}+{y}^{2}+{z}^{2}={c}^{2}{t}^{2},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}{\xi}^{2}+{\eta}^{2}+{\zeta}^{2}={c}^{2}{\tau}^{2}.$ |

Einstein initially presented this equivalence as proof that his two postulates were compatible; later he recognized that the Lorentz transformations followed from this equivalence and a requirement of symmetry (Einstein 1905: 901; 1907: 419). At the same time, he made no further comment on the geometric significance of this invariance and maintained at least a semantic distinction between kinematics and geometry. ${}^{54}$ Minkowski chose to fold one into the other, regarding ${c}^{2}{t}^{2}-{x}^{2}-{y}^{2}-{z}^{2}$ as a

$\frac{2\hspace{0.5em}\stackrel{\u203e}{\mathrm{AB}}}{{t}_{A}\text{'}-{t}_{A}}=c.$ |

Essentially the same presentation of time and simultaneity was given by Einstein in his 1908 review paper, except in this instance he chose to refer to one-way light propagation (1907: 416). In summary, by the time of the Cologne lecture, Einstein had defined clock synchronicity using both round-trip and one-way light travel between points in an inertial frame. Furthermore, we know for certain that Minkowski was familiar with both of Einstein's papers. The formal equivalence of Einstein's theory with that of Minkowski is not an issue, since Minkowski adopted unequivocally the validity of the Lorentz transformations, and stated just as clearly that the constant appearing therein was the velocity of propagation of light in empty space. The issue is Minkowski's own knowledge of this equivalence, in other words, his recognition of either an intellectual debt to Einstein, or of the fact that he independently developed a partially or fully equivalent theory of relativity. In what follows, we examine some old and new evidence concerning Minkowski's grasp of Einstein's time concept. Insofar as meaning may be discerned from use, Minkowski's use of the concepts of time and of simultaneity was equivalent to that of Einstein. In the Cologne lecture, for example, Minkowski demonstrated the relativity of simultaneity, employing for this purpose his space-time diagram (1909: 83). A more detailed exposé of the concept-without the space-time diagram-had appeared in the

For the uniformly moving electron, Lorentz had called the combination $t\text{'}=(-\mathrm{qx}+t)/\sqrt{1-{q}^{2}}$ the local time of the electron, and used this concept to understand the contraction hypothesis. Lorentz himself told me conversationally in Rome that it was to Einstein's credit to have recognized thatAccording to Minkowski's account, Lorentz employed the phrase in question to characterize Einstein's new concept of time. In fact, what Lorentz had called local time was not the above expression, but $t\text{'}=t/\beta -\beta \mathrm{vx}/{c}^{2}$. When combined with a Galilean transformation, the latter expression is equivalent to the one Minkowski called Lorentz's local time. Minkowski must have recognized his mistake, because in the final, printed version of "the time of one electron is just as good as that of the other, i.e., that $t$ and $t\text{'}$ are equivalent. [Italics added] ${}^{61}$

Starting in 1909, publication numbers increased rapidly until 1912, when the attention of theoretical physicists shifted to quantum theory and theories of gravitation. The annual publication total also declined then for non-theoretical physicists, but remained stable for mathematicians until the outbreak of war in 1914. A comparison of the relative strength of disciplinary involvement with the theory of relativity can be made for a large group of contributors, if we categorize individuals according to the discipline they professed in the university. Factoring in the size of the teaching staff in German universities in 1911, and taking into consideration only research published by certified teaching personnel (more than half of all authors in 1911 Germany), we find the greatest penetration of relativity theory among theoretical physicists, with one out of four contributing at least one paper on this subject (Table 1, col. 5). Professors of mathematics and of non-theoretical physics largely outnumbered professors of theoretical physics in German universities, and consequently, the penetration of relativity theory in the former fields was significantly lower than the ratio for theoretical physics. The number of contributors for each of the three groups was roughly equivalent, yet theoretical physicists wrote three papers for every one published by their counterparts in mathematics or non-theoretical physics (Table 1, col. 4).

Discipline | Instructors | Relativists | Pubs. | Rel./Instr. |

Theoretical Physics | 23 | 6 | 21 | 26% |

Non-Theoretical Physics | 100 | 6 | 8 | 6% |

Mathematics | 86 | 5 | 7 | 6% |

for university instructors in 1911 Germany.

The troublesome calculations through which Lorentz (1895 and 1904) and Einstein (1905) prove their validity independent of the coordinate system, and [for which they] had to establish the meaning of the transformed field vectors, become irrelevant in the system of the Minkowski "world." ${}^{69}$ (Sommerfeld 1910a: 224)Sommerfeld depicted the technical difficulty inherent to Lorentz's and Einstein's theories as a thing of the past. Inasmuch as Minkowski appealed to mathematicians to study the theory of relativity in virtue of its essential mathematical nature, Sommerfeld encouraged physicists to take up Minkowski's theory in virtue of its new-found technical simplicity. The pair of

What a joy to be a mathematician today, when mathematics is seen sprouting up everywhere and blossoming, when it is shown ever more to advantage in application in the natural sciences as well as in the philosophical direction, and stands to reconquer its former central position. ${}^{78}$ (Hilbert 1909b)Minkowski's theory of relativity was no doubt a prime example for Hilbert of the reconquest of physics by mathematicians. So far we have encountered the responses to Minkowski's work by his Göttingen colleagues, who of course had a privileged acquaintance with his approach to electrodynamics. In this respect, most mathematicians were in a position closer to that of our third and final illustration of mathematical responses to the Cologne lecture, from Guido Castelnuovo. This case, however, is chosen primarily for its bearing on Minkowski's interpretation of Einsteinian kinematics, and should not be taken as definitive of mathematical opinion of his work outside of Göttingen. Castelnuovo was a leading figure in algebraic geometry, a professor of mathematics at the University of Rome and president of the Italian Mathematical Society. In an article published in

The statement that the velocity of light is always equal to 1 for any observer is equivalent to the statement that a change in the temporal axis also brings a change to the spatial axes. ${}^{79}$ (Castelnuovo 1911: 78)In light of our earlier reconstruction of Minkowski's argument, it would seem that Castelnuovo denied the possibility of the interpretation imputed to Einstein by Minkowski, in which a rotation of the temporal axis left the spatial axis unchanged; in Castelnuovo's view, Einstein's theory required that the temporal and spatial axes rotate together. From a disciplinary standpoint, it is remarkable that Castelnuovo claimed to be giving an authentic account of

In truth, this change could be perceived solely by [an observer moving with the speed of light]. Yet if our senses were sufficiently acute, certain differences in the details of the presentation of phenomena would not escape us. ${}^{80}$ (Castelnuovo 1911: 78)Despite his destruction of the basis to Minkowski's priority claim, Castelnuovo acknowledged the cogency of his geometric approach, while recognizing the change in the concept of space brought about by Einsteinian relativity. The perception of the aforementioned rotation of the spatial axes concomitant with a rotation of the temporal axis required either the adoption of Minkowski's point of view, or the results of experimental physics. Of course, this was a paraphrase of Minkowski; we saw earlier how he conceded that the results of experimental physics had led to the discovery of the principle of relativity, and argued that pure mathematics could have done as well without Michelson's experiment. For Castelnuovo, the acceptance of Minkowski's metatheoretical view of the mathematical essence of the principle of relativity apparently did not conflict with a rejection of his theoretical claim on a new view of space.

$x=\nu \ell \text{'}+\rho x\text{'}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\text{and}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\ell =\lambda \ell \text{'}+\mu x\text{'}.$ |

On a Minkowski diagram (where the units are selected so that $c=1$) we draw the invariant curves ${\ell}^{2}-{x}^{2}=\ell \text{'}{}^{2}-x\text{'}{}^{2}=\pm 1$ (see Figure 4).

Next, we mark two points in the coordinate system $S(x,\ell )$, $P=(0,1)$ and $Q=(1,0)$, located at the intersections of the $\ell $-axis and $x$-axis with these hyperbolae. Another system $S\text{'}$ translates uniformly at velocity $v=c\beta $ with respect to $S$, such that the origin of $S\text{'}$ appears to move according to the expression $x=\beta \ell $. This line is taken to be the $\ell \text{'}$-axis. From the expression for the hyperbolae, it is evident that the $x\text{'}$-axis and the $\ell \text{'}$-axis are mutually symmetric, and form the same angle ${\mathrm{tan}}^{-1}\beta $ with the $x$-axis and the $\ell $-axis, respectively. The two points in $S$ are denoted here as $P\text{'}=(0,1)$ and $Q\text{'}=(1,0)$ and marked accordingly, at the intersections of the hyperbolae with the respective axes. The $\ell \text{'}$-axis, $x=\beta \ell $, intersects the hyperbola ${\ell}^{2}-{x}^{2}=1$ at $P\text{'}$. Using this data, we solve for the coefficients $\nu $ and $\lambda $:

$\nu =\frac{\beta}{\sqrt{1-{\beta}^{2}}}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\text{and}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\lambda =\frac{1}{\sqrt{1-{\beta}^{2}}}.$ |

Applying the same reasoning to the $x\text{'}$-axis $(x=\ell \hspace{0.5em}\beta )$, we solve for the coefficients $\rho $ and $\mu $, evaluating the expressions for $x$ and $\ell $ at the intersection of the $x\text{'}$-axis with the hyperbola ${\ell}^{2}-{x}^{2}=-1$, at the point labeled $Q\text{'}$, and we find

$\rho =\frac{1}{\sqrt{1-{\beta}^{2}}}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\text{and}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\mu =\frac{\beta}{\sqrt{1-{\beta}^{2}}}.$ |

Substituting these coefficients into the original expressions for $x$ and $\ell $, we obtain the following transformations:

$x=\frac{x\text{'}+\beta \ell \text{'}}{\sqrt{1-{\beta}^{2}}}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\text{and}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\ell =\frac{\ell \text{'}+\beta x\text{'}}{\sqrt{1-{\beta}^{2}}}.$ |

The old form of the special Lorentz transformations is recovered by substituting $\ell =\mathrm{ct}$ and $\beta =v/c$,

$x=\frac{x\text{'}+\mathrm{vt}\text{'}}{\sqrt{1-{v}^{2}/{c}^{2}}}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\text{and}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}t=\frac{t\text{'}+\mathrm{vx}\text{'}/{c}^{2}}{\sqrt{1-{v}^{2}/{c}^{2}}}.$ |

Invoking the property of symmetry, the transformations for $x\text{'}$ and $t\text{'}$ may be calculated in the same fashion as above, by starting with $S\text{'}$ instead of $S$.