Einstein the Stubborn: Correspondence between Einstein and Levi-Civita

TL;DR

This paper examines the correspondence between Einstein and Levi-Civita, highlighting the scientific debate and development of Einstein's gravitational tensor and the recognition of its flaws leading to the final formulation of General Relativity.

Contribution

It provides a detailed historical analysis of the correspondence, revealing how Einstein's understanding evolved through mathematical debate and peer critique.

Findings

Levi-Civita objected to Einstein's initial proof of tensor covariance.

Einstein initially defended his proof but was convinced of its flaw by Levi-Civita.

Hilbert identified the main flaw in Einstein's 1914 derivation.

Abstract

Before developing his 1915 General Theory of Relativity, Einstein held the "Entwurf" theory. Tullio Levi-Civita from Padua, one of the founders of tensor calculus, objected to a major problematic element in this theory, which reflected its global problem: its field equations were restricted to an adapted coordinate system. Einstein proved that his gravitational tensor was a covariant tensor for adapted coordinate systems. In an exchange of letters and postcards that began in March 1915 and ended in May 1915, Levi-Civita presented his objections to Einstein's above proof. Einstein tried to find ways to save his proof, and found it hard to give it up. Finally Levi-Civita convinced Einstein about a fault in his arguments. However, only in spring 1916, long after Einstein had abandoned the 1914 theory, did he finally understand the main problem with his 1914 gravitational tensor. In autumn 1915 the G\"ottingen brilliant mathematician David Hilbert found the central flaw in Einstein's 1914 derivation. On March 30, 1916, Einstein sent to Hilbert a letter admitting, "The error you found in my paper of 1914 has now become completely clear to me".