The authors of this article “focus on Ricci’s rotation coefficients and their role in Levi-Civita’s papers on the unified field theory”. In Section 2, they start by tracing the history of these coefficients from Ricci Curbastro to Levi-Civita and his students and colleagues by commenting the corresponding papers in detail. The following sections are devoted to Levi-Civita’s work on unified field theory in the late 1920s inspired by Einstein’s 1928 attempt to elaborate a unified geometrical field theory of gravitation and electromagnetism based on the notion ‘Fernparallelismus’ (‘teleparallelism’ in English). In Section 3, after discussing some geometrical fundamentals of Einstein’s tetrad theory, it follows a detailed commentary of a 1929 note Levi-Civita’s published in German in the Sitzungsberichte of the Prussian Academy [JFM 55.0499.01]. In this note, Levi-Civita analysed mainly mathematical aspects of Einstein’s theory, with the physically interesting result, emphasized by the authors: “Maxwell’s electromagnetic equations and the gravitational equations were obtained exactly, whereas Einstein had deduced them only as a first order approximation.” The fourth section considers the immediate influence of Levi-Civita’s 1929 papers, in which Ricci’s coefficients play an important role, on the debate on unified field theories. It is shown Levi-Civita’s network of mathematicians and physicists through his correspondence that highlights his role in the Italian scientific community.
All in all, this paper presents an informative overview about contributions of the Italian school of mathematicians around Levi-Civita to exciting developments in differential geometry and gravitational theory at the time when Einstein’s theory of general relativity had just been founded. But it should also be mentioned in passing that sometimes one finds a bit irritating formulations. So, formulations which could arouse the impression that mathematical relations, well known to specialists in mathematics and physics, were new mathematical insights of historians. Or when, uncritically, the authors agree with the opinion of M. Gasperini (p. 120) that – unfortunately, thought as a kind of knighting – the tetrad formalism “naturally leads to the formulation of general relativity as a gauge theory for a local symmetry group, thus putting gravity on the same footing of the other fundamental (strong, weak and electromagnetic) interactions”. Here, the authors are on a wrong track. (For a criticism of this standpoint and a correct idea of the foundations of gravitational gauge theory, see [M. Blagojević (ed.) and F. W. Hehl (ed.), Gauge theories of gravitation. A reader with commentaries. Hackensack, NJ: World Scientific; London: Imperial College Press (2013; Zbl 1282.83046)].)