Who was a bigger mathematician, Einstein or Newton?



Newton was a bigger mathematician than Einstein.
In 1912 while trying to derive the correct form of the field equations for his new gravitation theory, Einstein was in search of his friend from school, Marcel Grossmann, who was a professor of mathematics in Zurich. He told Grossmann one day: "Grossmann, you have to help me, or I shall go crazy!" and Grossman had looked through the literature, and managed to show Einstein that the mathematical tools he needed were tensor calculus and differential geometry developed by Riemann, Ricci, Levi-Civita and Christoffel.

In 1912 Einstein first derived generally-covariant field equations (under unimodular transformations) but he abandoned the generally covariant field equations with "heavy heart" and began to search for non-generally covariant field equations. Einstein thought that the Ricci tensor in his field equations should reduce in the weak field limit to his static gravitational field equation and then to the Newtonian limit. But in 1912 he couldn't extract the Newtonian limit. In 1915 he managed to obtain the Newtonian limit but in 1913 he chose non-generally covariant field equations instead, equations scholars name “Entwurf” field equations.

In 1914 From the Entwurf field equations and the conservation law of energy-momentum he obtained four coordinate conditions by which he could restrict the coordinate system. He reasoned that the field equations hold for every coordinate system that is adapted to the four coordinate conditions; that is to say, the field equations hold for adapted coordinate systems. So the covariance of the Entwurf field equations was far-reaching in adapted coordinate systems. Thereafter, Einstein proved that his gravitational tensor was a covariant tensor for adapted coordinate systems, and coordinate transformations between two adapted coordinate systems were arbitrary non-linear transformations

In an exchange of letters with Einstein during spring 1915, Tulio Levi-Civita, as one of the founding fathers of tensor calculus, presented his objections to Einstein's above proof. In his correspondence with Einstein Levi-Civita demonstrated to Einstein that his gravitational tensor could not be a covariant tensor for adapted coordinate systems. Einstein tried to find ways to save his proof and in most of the letters he stubbornly repeated the same arguments over and over again. But Levi-Civita the mathematician finally showed Einstein that he attempted the impossible.

Indeed, in November 1915 Einstein gave up his Entwurf field equations and returned to his generally covariant field equations from 1912.
On the other hand, two years later in the summer of 1917, Levi-Civita criticized Einstein's use of the pseudo-tensor for the energy components of the gravitational field.

In 1915 Einstein had failed to appreciate the problems with his 1914 Entwurf gravitational tensor, and strongly resisted accepting Levi-Civita's criticism.

Ahha… now it was Levi-Civita’s turn to be stubborn and he could not admit a pseudo-tensor and therefore did not accept Einstein's claims about the pseudo-tensor for the energy components of the gravitational field, insisting instead that it had to be a tensor!

This time Einstein was right because if the energy components of the gravitational field were a tensor, then the energy-momentum principle would not exclude any physical process. So Einstein the physicist knew that the energy components of the gravitational field could not be a tensor.

The controversy between Einstein and Levi-Civita was an actual dispute between two opposing worldviews: that of Einstein the theoretical physicist and that of Levi-Civita, the mathematician and one of the founders of tensor calculus.

Thus Einstein was a genius physicist and Levi-Civita was a genius mathematician. In fact Einstein wasn’t a mathematician, he was a genius theoretical physicist and being a genius theoretical physicist (and not a genius mathematician) led him to his key discoveries.

Newton was a genius physicist and a genius mathematician. Newton is the primary inventor of calculus. I am not going to discuss the Leibniz-Newton priority dispute here.

Post was written by Gali Weinstein, 
PhD foundations of physics, history and philosophy of physics.
Tel Aviv, israel.