of Minkowskian Relativity

scott.walter [at] univ-lorraine.fr

Oxford University Press, 1999, 91-127

FIG 1. The space-time formalism and the non-Euclidean style ( $N=144$).

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

where $c$ stands for the velocity of propagation of light in empty space. Physical laws were to be expressed with respect to a four-dimensional manifold with coordinates ${x}_{1}$, ${x}_{2}$, ${x}_{3}$, ${x}_{4}$, where ordinary Cartesian coordinates $x$, $y$, and $z$, went over into the first three, and the fourth was defined to be an imaginary time coordinate, ${x}_{4}\equiv \mathrm{it}$. When the units are chosen such that $c=1$, Minkowski remarked, the above quadratic expression passes over to the form

${x}_{1}^{2}+{x}_{2}^{2}+{x}_{3}^{2}+{x}_{4}^{2}.$ |

Implicitly, Minkowski took as his formal starting point the final section of Poincaré's memoir on the dynamics of the electron (Poincaré, 1906, § 8). He acknowledged the French mathematician's use of an imaginary temporal coordinate in the final section of his lecture, albeit in a rather oblique fashion, when he observed that Poincaré's search for a Lorentz-covariant law of gravitation involved the consideration of Lorentz-group invariants (Minkowski, 1907, 16). Poincaré used three real space coordinates and one imaginary temporal coordinate to define four-dimensional vectors for position, velocity, force, and force density. Did Minkowski then consider Poincaré to have anticipated his planned reformulation of the laws of physics in four dimensions? As mentioned above, Minkowski described his contribution in terms of a notational improvement which revealed the symmetry shared by a certain quadratic form and the Maxwell-Lorentz electromagnetic field equations. Indeed, Minkowski distinguished his work on the principle of relativity from that of Einstein, Poincaré and Max Planck on precisely this basis:

Here I will bring to the notation from the beginning that symmetry, whereby the form of the equations, as I believe, really becomes extremely transparent. This is something brought out by none of the previously-mentioned authors, not even by Poincaré himself. ${}^{12}$In other words, Minkowski presented his main result as a notational one in which the Lorentz-covariance of the electromagnetic equations appeared as never before. He insisted here upon the fact that Poincaré did not write the Maxwell-Lorentz equations in four-dimensional terms. This was hardly an oversight on Poincaré's part. The French scientist did not propose a four-dimensional vector calculus for general use, nor had he any intention of developing such a calculus for physics. While Poincaré recognized the feasibility of a translation of physics into the language of four-dimensional geometry, he said this would be "very difficult and produce few benefits." In this sense, he felt a four-dimensional vector calculus would be "much like Hertz's mechanics." ${}^{13}$ Whether or not Minkowski was aware of Poincaré's view, it is quite clear that he did not share his opinion. Once he had presented his central finding, Minkowski still had to show what a four-dimensional vector calculus has to do with a non-Euclidean manifold. When he reached the part of his lecture dealing with mechanics, Minkowski explained himself in the following way. The tip of a four-dimensional velocity vector ${w}_{1}$, ${w}_{2}$, ${w}_{3}$, ${w}_{4}$, Minkowski stipulated,

is always a point on the surfaceMany mathematicians in Minkowski's audience probably recognized in (1) the equation of a pseudo-hypersphere of unit imaginary radius, and in (2) its real counterpart, the two-sheeted unit hyperboloid. Formally equivalent, both hypersurfaces provide a basis for a well-known model of non-Euclidean space of constant negative curvature, popularized by Helmholtz. ${}^{15}$ Although Minkowski did not bother to unfold the geometry of velocity vectors, in the hypersurfaces (1) and (2), we have the premises of an explanation for Minkowski's description of the world as being-in a certain sense-a four-dimensional non-Euclidean manifold. The conjugate diameters of the hyperboloid (2), Minkowski went on to explain, give rise to a geometric image of the Lorentz transformation. Any point on (2) can be taken to lie on the $t-$diameter, and this change of axes corresponds to an orthogonal transformation of both the time and space coordinates which, as Minkowski observed, is a Lorentz transformation. Thus the three-dimensional hyperboloid (2) embedded in Minkowski's four-dimensional space affords an interpretation of the Lorentz transformation. Over the years, Minkowski's terminology has generated significant confusion among commentators. In one sense, it appropriately underlined both the four-dimensionality of Minkowski's planned calculus, and the hyperbolic geometry of velocity vectors. Yet the label is flawed, for the following reason: although both the pseudo-hypersphere (1) and the two-sheeted unit hyperboloid (2) may be considered models of non-Euclidean space, neither one constitutes a four-dimensional manifold. Minkowski was surely aware of this ambiguity when he maintained that the label was only true "in a certain sense." In any case, Minkowski never again referred to a manifold as both four-dimensional and non-Euclidean. Along with the problematic label, the geometric interpretation of velocity vectors likewise vanishes from view in Minkowski's subsequent writings. Felix Klein, for one, regretted the change; in his opinion, Minkowski later hid from view his "innermost mathematical, especially invariant-theoretical thoughts" on the theory of relativity (1926-1927, vol. 2, 75). Six months after his lecture to the Göttingen Mathematical Society, Minkowski published his first essay on the principle of relativity. Entitled "The Basic Equations of Electromagnetic Processes in Moving Bodies," it presented a new theory of the electrodynamics of moving media, incorporating formal insights of the relativity theories introduced earlier by Einstein, Poincaré and Planck. For example, it took over the fact that the Lorentz transformations form a group, and that Maxwell's equations are covariant under this group. Minkowski also shared Poincaré's view of the Lorentz transformation as a rotation in a four-dimensional space with one imaginary coordinate, and his five four-vector expressions. These insights Minkowski developed and presented in an original, four-dimensional approach to the Maxwell-Lorentz vacuum equations, the electrodynamics of moving media, and in an appendix, Lorentz-covariant mechanics. In the sophistication of its mathematical expression, Minkowski's paper rivalled that of Poincaré, acknowledged by many to be the world's leading mathematician. One aspect of the principle of relativity, according to Minkowski, made it an excellent object of mathematical study:

or, if you wish, on

${w}_{1}^{2}+{w}_{2}^{2}+{w}_{3}^{2}+{w}_{4}^{2}=-1$ $(1)$

and represents at the same time the four-dimensional vector from the origin to this point, and this also corresponds to null velocity, to rest, a genuine vector of this sort. Non-Euclidean geometry, of which I spoke earlier in an imprecise fashion, now unfolds for these velocity vectors. ${}^{14}$

${t}^{2}-{x}^{2}-{y}^{2}-{z}^{2}=1,$ $(2)$

To the mathematician, accustomed to contemplating multi-dimensional manifolds, and also to the conceptual layout of the so-called non-Euclidean geometry, adapting the concept of time to the application of Lorentz transformations can give rise to no real difficulty. ${}^{16}$Understanding the concept of time in the theory of relativity, in other words, represented no challenge for mathematicians because of their experience in handling similar concepts from $n$-dimensional and non-Euclidean geometry. This is not the only time Minkowski encouraged mathematicians to study the principle of relativity in virtue of its mathematical or geometrical form. His Cologne lecture was to go even further in this direction, by suggesting that the essence of the principle of relativity was purely mathematical (Walter, 1999, § 2.1). In the passage quoted above, Minkowski's claim is a less general one, to the effect that mathematicians' familiarity with non-Euclidean geometry would allow them to handle the concept of time in the Lorentz transformations. Yet apart from this disciplinary aside, the subject of non-Euclidean geometry is conspicuously absent from all that Minkowski published on relativity theory. On the other hand, Minkowski retained the geometric interpretation of the Lorentz transformations that had accompanied the now-banished non-Euclidean interpretation of velocity vectors. In doing so, he elaborated the notion of velocity as a rotation in four-dimensional space. He introduced a formula for the frame velocity $q$ in terms of the tangent of an imaginary angle $i\psi $, such that

$q=-i\mathrm{tan}i\psi =({e}^{\psi}-{e}^{-\psi})/({e}^{\psi}+{e}^{-\psi}).$ |

Minkowski could very well have expressed frame velocity in the equivalent form $q=\mathrm{tanh}\psi ,$ where the angle of rotation is real instead of imaginary, and all four space-time coordinates are real. He did not do so, but used the imaginary rotation angle $i\psi $ to express the special Lorentz transformation in the trigonometric form:

${x}_{1}\text{'}={x}_{1},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{x}_{2}\text{'}={x}_{2},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{x}_{3}\text{'}={x}_{3}\mathrm{cos}i\psi +{x}_{4}\mathrm{sin}i\psi ,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{x}_{4}\text{'}=-{x}_{3}\mathrm{sin}i\psi +{x}_{4}\mathrm{cos}i\psi .$ |

The use of circular functions here underscores the fact that a special Lorentz transformation is equivalent to a rotation in the $({x}_{3}{x}_{4})-$plane. Likewise, by expressing velocity in terms of an imaginary rotation, Minkowski may have betrayed his knowledge of the formal connection between the composition of Lorentz transformations and relative velocity addition, remarked earlier by Einstein on different grounds (Einstein, 1905, § 5). However, Minkowski neither mentioned the law of velocity addition, nor expressed it in formal terms. Minkowski's preference for circular functions may be understood in relation to his project to express the laws of physics in four-dimensional terms. Four-dimensional vector algebra is a natural extension of the ordinary vector analysis of Euclidean space when the time coordinate is multiplied by $\sqrt{-1}$. Expressing the Lorentz transformation as a hyperbolic rotation would have obscured the connection for physicists. Mathematicians, on the other hand, had little use for vector analysis, and were unlikely to be put off by the use of hyperbolic functions. Judging from his correspondence, Minkowski was not shy of hyperbolic geometry. In a postcard sent to his former teacher and friend, the Zürich mathematician Adolf Hurwitz (1859-1919), Minkowski described the "quintessence" of his relativity paper as the "Principle of the Hyperbolic World." ${}^{17}$ The link to ordinary Euclidean space from four-dimensional space-time was one of the themes Minkowski stressed in his lecture to the scientists assembled in Cologne for the annual meeting of the German Association in September, 1908. Under the new space-time view, Minkowski announced, "Three-dimensional geometry becomes a chapter of four-dimensional physics." ${}^{18}$ In the same triumphant spirit, Minkowski suggested that his new four-dimensional understanding of the laws of physics deserved its own label. The "Principle of the Hyperbolic World" that he had tried on Hurwitz was shelved in favor of the more ecumenical "Postulate of the Absolute World" (1909, 82). Although Minkowski explained this to mean that only the four-dimensional world in space and time is given by phenomena (1909, 82), one suspects an inside joke with Hurwitz, since in the German mathematical community, hyperbolic geometry was sometimes referred to as absolute geometry. Although the lecture on "Space and Time" was read to the mathematics section of the Cologne meeting, it recapitulated a selection of Minkowski's results so that they could be understood by those with little mathematical training. Notably, it displayed the fundamental four-vectors of relativistic mechanics, while neglecting the finer points of his matrix calculus. Likewise, the above-mentioned interpretation of Lorentz transformations with respect to a real hyperboloid (2) resurfaced in a two-dimensional version lending itself to graphical illustration. Minkowski's space-time diagram (see Figure 2) refers to an invariant hyperbola in the $(\mathrm{xt})-$plane, ${t}^{2}-{x}^{2}=1$, which is just equation (2) without the $y$ and $z$ coordinates. The diagonals describing the equations $x=t$ and $x=-t$ correspond here to the paths of light rays in empty space that pass through the origin of the coordinate system; in four-dimensional space-time these rays form an invariant hypercone. While Minkowski did not bother to show this himself, from the geometrical relations of the diagram one may derive the special Lorentz transformation. ${}^{19}$

It need scarcely be emphasized that this new view of the concept of time makes the most serious demands upon the capacity of abstraction and the imaginative power of the physicist. It surpasses in boldness everything achieved so far in speculative investigations of nature, and even in philosophical theories of knowledge: non-Euclidean geometry is child's play in comparison. ${}^{30}$Planck certainly meant to underline Einstein's accomplishment and its significant philosophical consequences. As John Heilbron observes, Planck was a key figure in securing acceptance of Einstein's work in Germany. ${}^{31}$ The comparison of non-Euclidean geometry to child's play, however, was most likely a rejoinder to Minkowski's remark on time, according to which

Sommer maintains that [Minkowski's] speech in Cologne was simply grand; when reading it, however, I always get a slight brain-shiver, now (that) space and time appear conglomerated together in a gray, miserable chaos. ${}^{36}$Thus unlike his mathematical colleague Julius Sommer (1871-1943), Max Wien was not inspired by the idea of referring the laws of physics to a space-time manifold. And while Wien appeared to admit the validity of Minkowskian relativity, his willingness to develop the theory and investigate its experimental consequences was undoubtedly compromised by its perceived abstraction. At the September, 1909, meeting of the German Association of Natural Scientists and Physicians in Salzburg, Arnold Sommerfeld (1868-1951) attempted to spark physicists' interest in Minkowski's formalism. A former assistant to Felix Klein, Sommerfeld succeeded Ludwig Boltzmann in the Munich chair of theoretical physics in 1906. ${}^{37}$ At first skeptical of Einstein's theory, Sommerfeld found Minkowski's space-time theory highly persuasive, and following Minkowski's death, became its most distinguished advocate in physics (Walter 1999, § 3.1). In his Salzburg talk, Sommerfeld insisted upon the practical advantage in problem solving offered by the space-time view:

Minkowski's profound space-time view not only facilitates the general construction of the relative-theory in (a) systematic way, but also proves successful as a convenient guide in specific problems. ${}^{38}$As an example of the advantage of the Minkowskian approach, Sommerfeld selected the case of Einstein's "famous addition theorem," according to which velocity parallelograms do not close. This "somewhat strange" result, Sommerfeld suggested, became "completely clear" (

$\beta =\frac{1}{i}\mathrm{tan}({\phi}_{1}+{\phi}_{2})=\frac{1}{i}\frac{\mathrm{tan}{\phi}_{1}+\mathrm{tan}{\phi}_{2}}{1-\mathrm{tan}{\phi}_{1}\mathrm{tan}{\phi}_{2}}=\frac{{\beta}_{1}+{\beta}_{2}}{1+{\beta}_{1}{\beta}_{2}},$ |

where $\beta =v/c$, and the subscripts correspond to two systems in uniform parallel motion. The formal concision and conceptual simplicity of Sommerfeld's derivation were widely appreciated; some years later even Einstein adopted the method (Miller 1981, 281, note 4). For the more general case of two inertial systems moving in different directions, Sommerfeld interpreted the imaginary rotation angle $\phi $ as an arc of a great circle on a sphere of imaginary radius. Here the relative velocity of an arbitrary point with respect to any two systems of reference in uniform motion is found by constructing a triangle on the surface of the sphere, the sides of which follow from the cosine law (see Figure 3). ${}^{39}$ Sommerfeld did not mention non-Euclidean geometry in so many words, yet the surface of a hemisphere of imaginary radius was a well-known model of hyperbolic geometry, as mentioned earlier. Sommerfeld's spherical-trigonometric formulae employing an imaginary angle can be rewritten in terms of

... the formulæ of the theory of relativity are not only essentially simplified, but it also allows a geometric interpretation that is wholly analogous to the interpretation of the classical theory in Euclidean geometry. ${}^{47}$The non-Euclidean style, in other words, was the one most appropriate to the theory of relativity. A similar claim had been made on behalf of the space-time formalism by Minkowski and Sommerfeld, as we saw earlier. However, where Minkowski and Sommerfeld accompanied this claim with a display of new physical relations, Varicak arrived empty-handed. He effectively promoted the cause of non-Euclidean geometry in relativity by showing how to express relativistic formulae with hyperbolic functions, and yet he did not offer any new physical insights.

Abraham, Max. "Sull'elettrodinamica di Minkowski,"