The Non-Euclidean Style of Minkowskian Relativity

The history of relativity is structured for most commentators by two landmark discoveries due to Albert Einstein: the special theory (1905) and the general theory of relativity (1915). To get from one theory to the other, we know that Einstein relied on a certain number of fundamental concepts, such as the equivalence principle, and a few key mathematical techniques, for instance, the absolute tensor calculus of Gregorio Ricci-Curbastro and Tullio Levi-Civita. Einstein also had need of a third theory and technique, elaborated by his former mathematics professor, Hermann Minkowski (1864–1909), although he did not recognize this for several years. In this paper, we examine the fortunes of Minkowski’s space-time theory from 1908 to 1916. Our focus is on the emergence of Minkowski’s four-dimensional formalism as a standard technique in theoretical physics, and we investigate one aspect of this history in some detail: the reformulation and reinterpretation of the laws of special relativity in the language of non-Euclidean geometry. The related work done on the space-time theory, or what we call the “non-Euclidean style” of Minkowskian relativity, provides an example of the geometrization of physics brought about by Minkowski and his followers. In order to situate our topic in a broader scientific context, we first describe the status of applications of non-Euclidean geometry in physics around the turn of the century. Next, we present a quantitative overview of publications on Minkowskian relativity for the period 1908– 1915. We then review Minkowski’s appeal to non-Euclidean geometry, and link this to the mixed reception of his work. There follows a comparative study of the emergence and development of the non-Euclidean style in selected works by Arnold Sommerfeld, Alfred A. Robb, Vladimir Varičak, Gilbert N. Lewis, Edwin B. Wilson and Émile Borel.