Albert Einstein and David Hilbert – Einstein’s General Relativity

Sometime in October 1915 Einstein dropped the Einstein-Grossman “Entwurf” theory. He adopted the postulate that his field equations were covariant with respect to arbitrary transformations of a determinant equal to 1, and on November 4, 1915 he presented to the Prussian Academy these new field equations. Starting on November 4, 1915, Einstein gradually expanded the range of the covariance of his field equations

On November 7, 1915, Einstein sent David Hilbert the proofs to his first paper of November 4, and he wanted Hilbert to look at this work. Hilbert alsoreadEinstein’s1914reviewpaper discussing his “Entwurf” theory: Hilbert found some mistake in this paper; Einstein wrote that his colleague Arnold Sommerfeld wrote him that Hilbert had objected to the 1914 “Entwurf” foundations paper

By November 10, 1915 Hilbert probably answered Einstein’s letter, telling him about his system of electromagnetic theory of matter, the unified theory of gravitation and electromagnetism, in which the source of the gravitational field is the electromagnetic field. Hilbert’s goal was to develop an electromagnetic theory of matter, which would explain the stability of the electron

Between November 4 and November 11 it seems that Einstein was influenced by Hilbert’s physical attitude towards a field theory of matter. In his addendum to the first note, published on November 11 Einstein directly referred to the supporters of the electrodynamic worldview, “One now has to remember that, in accord with our knowledge, ‘matter’ is not to be conceived as something primitively given, or physically simple. There even are those, and not just a few, who hope to be able to reduce matter to purely electrodynamic processes, which of course would have to be done in a theory more complete than Maxwell’s electrodynamics”. Einstein probably discussed the electrodynamic worldview with Hilbert and felt that he was now in competition with the latter

In the addendum to the November 4 paper, the November 11 paper, Einstein added a coordinate condition (determinant equal to 1), which allowed him to take the last step and to write the field equations of gravitation in a general covariant form. He then dropped his November 4 postulate and adopted it as a coordinate condition

The day afterwards Einstein wrote Hilbert again. He told him about the progress in his work. Hilbert replied and invited Einstein to come to Göttingen. Hilbert explained to Einstein the main points of his new unified theory of gravitation and electromagnetism, and told Einstein that he had already discussed his discovery with Sommerfeld. He wanted next to explain it to Einstein. He thus invited him to come to hear his talk on November 16. Hilbert told Einstein that the latter’s November 4 paper was entirely different from his own work

With hindsight Hilbert’s work was different from Einstein’s November 4 paper in that, Hilbert eventually endeavored to derive generally covariant field equations for the combined gravitational and electromagnetic fields without explicitly writing down these equations. Hilbert accepted Einstein’s 1914 Hole Argument against general covariance (after Einstein had silently dropped it). Hilbert was thus finally obliged to supplement his generally covariant field equations by four non-generally covariant field equations based on rather dubious energy considerations, which Hilbert would eventually drop later when he would publish his paper (after Einstein presented his final form of field equations to the Prussian Academy on November 25). Einstein replied and told Hilbert he could not come, but requested a copy of his work. In response, Hilbert perhaps sent a copy of the lecture he had given on the subject on November 16, or else a copy of a manuscript of the paper he would present five days later on November 20 to the Royal Society in Göttingen

Einstein was already less patient after he had received Hilbert’s work. He replied to Hilbert on November 18 telling him that his work agrees – as far as he could see – exactly with what he had found in the last few weeks and have already presented to the Prussian Academy. Einstein was in competition with Hilbert and appeared to have been still influenced by his unified theory of matter, gravitation and electromagnetism until November 18. Indeed on Thursday, November 18, Einstein presented to the Prussian Academy his solution to the longstanding problem of the precession of the perihelion of Mercury, on the basis of his November 11 General theory of relativity

The day afterwards Hilbert sent a polite letter in which he congratulated Einstein on overcoming the perihelion motion. He was quite astonished that Einstein calculated so rapidly the precession of Mercury’s perihelion. In fact the basic calculation has already been done two years earlier with Michele Besso in the Einstein-Besso manuscript. Einstein transferred the basic framework of the calculation from the Einstein-Besso manuscript, and corrected it according to his November field equations

In November 1915 Einstein could calculate so rapidly the precession of Mercury’s perihelion for another reason. Einstein’s November 11 field equations for the metric tensor are the field equations for the gravitational field in the November 18 paper. The added coordinate condition, determinant equal to 1 (from Einstein’s November 11 paper), implied by the assumption of an electromagnetic origin of matter, was essential for Einstein’s calculation of the precession of Mercury’s perihelion

The November 11 field equations are non-linear partial differential equations of the second rank, and there is no general solution to these equations. Solving the field equations give the components of the metric tensor. In his November 18 paper Einstein tried to find approximate solutions

What happened during the week of November 18–25, 1915? After or while working on the solution of the problem of the Perihelion of Mercury, Einstein could resolve the final difficulties in his November 11 theory. It took him an extra week to arrive at the November 25 field equations. On November 26 Einstein wrote his close friend Heinrich Zangger, however, only one colleague has really understood it [his theory], and he is seeking to clearly “nostrify” it (Abraham’s expression).This colleague was David Hilbert

Recall that on November 19 Hilbert sent Einstein a letter in which he congratulated him on overcoming the perihelion motion. Hilbert ended his letter by asking Einstein to continue and keep him up to date on his latest advances. Hilbert did not tell Einstein about the important talk he was giving the day afterwards. Hilbert presented on November 20 a paper to the Göttingen Academy of Sciences, “The Foundations of Physics”, including his version to the gravitational field equations of general relativity. Five days later on November 25, Einstein presented to the Prussian Academy his version to the gravitational field equations

At the end of the day it appears that Einstein did not “nostrify” Hilbert. After November 18 Einstein was no more influenced by Hilbert’s theory of matter, and he was thus not in competition with him anymore. His new field equations of November 25 with the new trace term are related to his work of November 4, and appear to have sprung from it

In two papers (here and here) I derive Einstein’s November 25, 1915 field equations from Einstein’s November 4, 1915 field equations and connect between the two. In his 1916 review paper, “The Foundation of the General Theory of Relativity” Einstein connected between his  November 4 and November 25 field equations and I follow his derivation

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About Gali Weinstein

Einstein scholar. My interests are: Einstein's theory of special relativity, Einstein's theory of general relativity, history of special relativity and history of general relativity.
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