Sometime in October 1915 Einstein dropped the Einstein-Grossman theory. Starting on November 4, 1915, Einstein gradually expanded the range of the covariance of his field equations. On November 11, 1915 Einstein was able to write the field equations of gravitation in a general covariant form, but there was a coordinate condition (there are no equations here so I cannot write it down here).
On November 18, 1915, Einstein presented to the Prussian Academy his paper, “Explanation of the Perihelion Motion of Mercury from the General Theory of Relativity”. Einstein reported in this talk that the perihelion motion of Mercury is explained by his theory. In this paper, Einstein tried to find approximate solutions to his November 11, 1915 field equations. He intended to obtain a solution, without considering the question whether or not the solution was the only possible unique solution.
Einstein’s field equations are non-linear partial differential equations of the second rank. This complicated system of equations cannot be solved in the general case, but can be solved in particular simple situations. The first to offer an exact solution to Einstein’s November 18, 1915 field equations was Karl Schwarzschild, the director of the Astrophysical Observatory in Potsdam. On December 22, 1915 Schwarzschild wrote Einstein from the Russian front. Schwarzschild set out to rework Einstein’s calculation in his November 18 1915 paper of the Mercury perihelion problem. He first responded to Einstein’s solution for the first order approximation from his November 18, 1915 paper, and found another first-order approximate solution. Schwarzschild told Einstein that the problem would be then physically undetermined if there were a few approximate solutions. Subsequently, Schwarzschild presented a complete solution. He said he realized that there was only one line element, which satisfied the conditions imposed by Einstein on the gravitational field of the sun, as well as Einstein’s field equations from the November 18 1915 paper.
“Raffiniert ist der Herrgott, aber boshaft ist er nicht” (Einstein might have already said….), because the problem with Schwarzschild’s line element was that a mathematical singularity was seen to occur at the origin! Oh my, Einstein abhorred singularities.
Actually, Schwarzschild “committed another crime”: he did not satisfy the coordinate condition from Einstein’s November 11 or November 18, 1915 paper. Schwarzschild admitted that his coordinates were not “allowed” coordinates, with which the field equations could be formed, because these spherical coordinates did not have determinant 1. Schwarzschild chose then the non-“allowed” coordinates, and in addition, a mathematical singularity was seen to occur in his solution. But Schwarzschild told Einstein: Don’t worry, “The equation of [Mercury’s] orbit remains exactly as you obtained in the first approximation”! See my paper from 2012.
Einstein replied to Schwarzschild on December 29, 1915 and told him that his calculation proving uniqueness proof for the problem is very interesting. “I hope you publish the idea soon! I would not have thought that the strict treatment of the point- problem was so simple”. Subsequently Schwarzschild sent Einstein a manuscript, in which he derived his solution of Einstein’s November 18, 1915 field equations for the field of a single mass. Einstein received the manuscript by the beginning of January 1916, and he examined it “with great interest”. He told Schwarzschild that he “did not expect that one could formulate so easily the rigorous solution to the problem”. On January 13, 1916, Einstein delivered Schwarzschild’s paper before the Prussian Academy with a few words of explanation. Schwarzschild’s paper, “On the Gravitational Field of a Point-Mass according to Einstein’s Theory” was published a month later.
In March 1916 Einstein submitted to the Annalen der Physik a review article on the general theory of relativity, “The Foundation of the General Theory of Relativity”. The paper was published two months later, in May 1916. The 1916 review article was written after Schwarzschild had found the complete exact solution to Einstein’s November 18, 1915 field equations. Even so, in his 1916 paper, Einstein preferred NOT to base himself on Schwarzschild’s exact solution, and he returned to his first order approximate solution from his November 18, 1915 paper.
A comment regarding Einstein’s calculations in his November 18, 1915 paper of the Mercury perihelion problem and Einstein’s 1916 paper. In his early works on GTR, in order to obtain the Newtonian results, Einstein used the special relativistic limit and the weak field approximation, and assumed that space was flat (see my paper). Already in 1914 Einstein had reasoned that in the general case, the gravitational field was characterized by ten space-time functions of the metric tensor. g were functions of the coordinates. In the case of special relativity this reduces to g44 = c2, where c denotes a constant. Einstein took for granted that the same degeneration occurs in the static gravitational field, except that in the latter case, this reduces to a single potential, where g44 = c2 is a function of spatial coordinates, x1, x2, x3.
Later that year David Hilbert (with a vengeance from 1915?…) arrived at a line-element similar to Schwarzschild’s one, and he concluded that the singularity disappears only if we accept a world without electricity. Such an empty space was inacceptable by Einstein who was apparently much attracted by Mach’s ideas! (later termed by Einstein “Mach’s Principle”). Okay, Einstein, said Hilbert: If there is matter then another singularity exists, or as Hilbert puts it: “there are places where the metric proves to be irregular”…. (See my paper from 2012).
Strange Days at Blake Holsey High: a student is sucked into the black hole…