A Review of

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

by Judith R. Goodstein

Published on February 17, 2019, American Mathematical Society

In this accessible book, archivist Judith Goodstein provides the biographical and his- torical backdrop to the mathematical tools that enabled Einstein to formulate his general theory of relativity. The key was tensor calculus, which was famously brought to Einstein’s attention by his long-time friend Marcel Grossmann, and which had been developed by Gregorio Ricci Curbastro and Tullio Levi-Civita. These latter two are “Einstein’s Italian mathematicians”, and the first nine chapters of Goodstein’s book detail their respective family backgrounds, education, marriages, early careers, and temperaments—the quiet, conservative Catholic Ricci, and his favorite disciple, the outgoing, secular Levi-Civita. Some of these chapters, especially Chapter 7, also in- clude the genesis of Ricci’s calculus—which Einstein and Grossman would name “tensor calculus”—and give an elementary introduction to the main ideas underlying it.

Ricci’s work languished unappreciated for many years. Critics recognized its elegance but sidelined it as a curiosity rather than a powerful technique with wide applications, while Levi-Civita’s stellar career was ultimately, and prematurely, destroyed under Mussolini’s fascist anti-Semitic laws. Meantime, the real breakthrough for Ricci and Levi-Civita came in 1913—nearly three decades after Ricci had first published on the topic. In a joint paper published in 1913, Einstein and Grossmann showed the power of tensor calculus when they used it to develop the mathematical and physical foundation that would underlie Einstein’s final 1915 general theory of relativity. In the process, Grossmann introduced the idea of contravariant, covariant and mixed tensors, as well as an index notation. The historical development of Einstein and Grossmann’s application of Ricci and Levi-Civita’s work to general relativity is the main subject of Chapters 10–12.

These are particularly fascinating chapters, which include discussion of the corre- spondence between Levi-Civita and Einstein, as well as the subsequent development and interpretation of tensor calculus by Einstein, Levi-Civita, and others. For instance, Levi-Civita elaborated the connection between covariant derivatives and parallel dis- placements in 1917, and Weyl elaborated on (and named) the Riemann tensor in 1918. These chapters, like the earlier ones, also include a host of other mathematicians and physicists—their contributions to the development of tensor calculus or relativity, and their friendships and rivalries.

Appendix A, by Michele Vallisneri, gives a more mathematically detailed (but still elementary) account of the development and application of tensor theory to general relativity. It is an excellent, very readable, short (12-page) introduction for the non- specialist.

Finally, Appendix B reprints Levi-Civita’s moving obituary for Ricci, which serves as a summary of Ricci’s work and career, while Appendix C offers the same for Levi- Civita, via Hodge’s obituary published by the Royal Society.

-Robyn Arianrhod

Copyright American Mathematical Society 2019