# How was Einstein led to make a contact with Differential Geometry for his theory of General Relativity?

General Relativity was developed with Differential Geometry as the tool.

How was Einstein led to make a contact with Differential Geometry for his theory of General Relativity? Who suggested him to use Differential Geometry?

Einstein himself told the story in his Kyoto address of 1922, which I quote from Pais's biography titled Subtle is the Lord:

"If all systems are equivalent, then Euclidean geometry cannot hold in all of them. To throw out geometry and keep laws is equivalent to describing thoughts without words. We must search for words before we can express thoughts. What must we search for at this point? This problem remained insoluble to me until 1912, when I suddenly realized that Gauss's theory of surfaces holds the key for unlocking this mystery. I realized that Gauss's surface coordinates had a profound significance. However, I did not know at that time that Riemann had studied the foundations of geometry in an even more profound way. I suddenly remembered that Gauss's theory was contained in the geometry course given by Geiser when I was a student...

I realized that the foundations of geometry have physical significance. My dear friend the mathematician Grossmann was there when I returned from Prague to Zurich. From him I learned for the first time about Ricci and later about Riemann. So I asked my friend whether my problem could be solved by Riemann's theory, namely, whether the invariants of the line element could completely determine the quantities I had been looking for".

"I had the decisive idea of the analogy between the mathematical problem of the theory and the Gaussian theory of surfaces only in 1912, however, after my return to Zurich, without being aware at that time of the work of Riemann, Ricci, and Levi-Civita. This was first brought to my attention by my friend Grossmann when I posed to him the problem of looking for generally covariant tensors whose components depend only on derivatives of the coefficients $$[g_{\mu\nu}]$$ of the quadratic fundamental invariant $$[g_{\mu\nu}dx^\mu dx^\nu]$$".

Pais, who personally interviewed Einstein for the book, gives further details. Grossman and Einstein studied together at the ETH (Swiss Federal Institute of Technology in Zurich) from 1896 to 1900. After seven years of teaching high school Grossman was hired as a full professor of geometry at ETH in 1907. Interestingly enough, that was the same year that Einstein had "the happiest thought of my life" that "the relativity postulate has to be extended to coordinate systems which, relative to each other, are in nonuniform motion", i.e. the idea of general covariance. But Einstein was in Prague at the time, and his original attempts at relativistic gravity involved a scalar "c-field". Grossman was lobbying ETH for offering Einstein a position. He succeeded in 1912, and Einstein accepted, over Lorentz backed invitation to Utrecht. Here is what happened next, in Pais's words:

"In order to appreciate what happened in August 1912, it is essential to know that before his arrival in Zurich Einstein had already concluded that the description of gravitation in terms of the single scalar c-field of the Prague days had to go and that a new geometry of physical space-time was needed. I am convinced that he arrived in Zurich with the knowledge that not just one but ten gravitational potentials were needed... It must have been at that time that he said, 'Grossmann, Du musst mir helfen, sonst werd' ich verrückt!' G., you must help me or else I'll go crazy! With Grossmann's help the great transition to Riemannian geometry must have taken place during the week prior to August 16, as is indicated by Einstein's letter to Hopf.

These conclusions are in harmony with my own recollections of a discussion with Einstein in which I asked him how the collaboration with Grossmann began. I have a vivid though not verbatim memory of Einstein's reply: he told Grossmann of his problems and asked him to please go to the library and see if there existed an appropriate geometry to handle such questions. The next day Grossmann returned (Einstein told me) and said that there indeed was such a geometry, Riemannian geometry. It is quite plausible that Grossmann needed to consult the literature since, as we have seen, his own field of research was removed from differential geometry."

In 1912-13 Einstein and Grossmann worked on the first theory where gravity is represented by the metric tensor, the Entwurf (sketch in German), but they mistakenly concluded that the gravitational field could not be generally covariant. Brown in Reflections on Relativity opines that "this accident may have been all that prevented Grossmann from being perceived as a co-creator of general relativity", but quotes Einstein's contrary opinion from a 1915 letter to Sommerfeld:

"Grossmann will never lay claim to being co-discoverer. He only helped in guiding me through the mathematical literature but contributed nothing of substance to the results."