Einstein's Italian Mathematicians: Ricci, Levi-Civita, and the Birth of General Relativity

Gregorio Ricci Curbastro (1853–1925) and Tullio Levi-Civita (1873–1941) were Italian geometers who worked to extend the ideas of curved space initiated by Gauss and Riemann; they called their subject the absolute differential calculus. The main exposition of their work was a landmark 1900 paper in the Math. Annalen titled “Méthodes de calcul différential absolu et leurs applications”. Marcel Grossmann (1878–1936) and Albert Einstein (1879–1955) then combined the Italians’ work with vector analysis to create the new subject of tensor analysis, as well as the beginnings of differential geometry. Tensor analysis was presented in the second half of a 1913 paper by Grossmann and Einstein titled “Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation” (Outline of a generalized theory of relativity and of a theory of gravitation), known as the Entwurf for short. (The concept of tensor already existed in a very limited way in mechanics and elasticity, but Grossmann and Einstein expanded it greatly.)

Einstein had been struggling to incorporate gravitation into his theory of special relativity, and finally decided that he needed some new kind of geometry to make it work. He asked his friend and collaborator Grossmann to research available geometries. Grossmann came back with Ricci and Levi-Civita’s paper, and with Einstein further developed the theory and published the Entwurf. The gravitational part of the paper turned out to be flawed and unsatisfactory, and Einstein spent an additional two years working out the correct physical model (the field equations) for gravitation, finally completing the general theory of relativity in late 1915.

Einstein’s publication of a general relativity paper in late 1914 (still following the Entwurf model) attracted Levi-Civita's attention, and he started a correspondence with Einstein about some points of the paper that he disagreed with. Einstein finally met Levi-Civita and Ricci in person in 1921 when he visited Italy to give lectures on general relativity.

The present book combines several parts. The main part, by Goodstein, is a (mostly non-scientific) biography of the two Italians. It concentrates on their families and their careers, and in Ricci’s case a lot of academic politics. Levi-Civita was a student of Ricci and championed and helped extend his work. There is a good collection of photographic portraits of all the people involved in the story. There is a brief but very clear and useful appendix by physicist Michele Vallisneri titled “From Ricci’s absolute differential calculus to Einstein’s theory of general relativity”, that explains Ricci and Levi-Civita’s work in technical terms and shows how Grossmann and Einstein adapted it. There are also two technical obituaries: Ricci’s obituary by Levi-Civita (appearing here in English for the first time) and Levi-Civita’s obituary by W. V. D. Hodge.

Another recent book related to this subject is Claudia Graf-Grossmann’s biography Marcel Grossmann. This covers Grossmann’s whole life, but includes his collaboration with Einstein and has an appendix that reprints their Entwurf paper. It also has a detailed technical appendix by Tilman Sauer about Grossmann’s work, including the Entwurf.

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Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.