Albert Einstein's discovery of the gravitational field equations of general relativity and David Hilbert's almost simultaneous derivation of the theory using an elegant variational principle, during a period when the two corresponded frequently, has led to numerous historical analyses of their interaction. The analyses came to be called a priority dispute.
Einstein and Hilbert The events of interest to historians of the dispute occurred in late 1915. At that time Albert Einstein, now perhaps the most famous modern scientist, had been working on gravitational theory since 1912. He had "developed and published much of the framework of general relativity, including the ideas that gravitational effects require a tensor theory, that these effects determine a non-Euclidean geometry, that this metric role of gravitation results in a redshift and in the bending of light passing near a massive body." While David Hilbert never became a celebrity, he was seen as a mathematician unequaled in his generation, with an especially wide impact on mathematics. When he met Einstein in the summer of 1915, Hilbert had started working on an axiomatic system for a unified field theory, combining the ideas of Gustav Mie's on electromagnetism with Einstein's general relativity. As the historians referenced below recount, Einstein and Hilbert corresponded extensively throughout the fall of 1915, culminating in lectures by both men in late November that were later published. The historians debate consequences of this friendly correspondence on the resulting publications.
Undisputed facts
The following facts are well established and referable:
The proposal to describe gravity by means of a pseudo-Riemannian metric was first made by Einstein and Marcel Grossmann in the so-called Entwurf theory published 1913.M. Grossmann, Entwurf einer verallgemeinerten Relativitatstheroie und einer Theorie der Gravitation: II. Mathematischer Teil (I. Physikalischer Teil von A. Einstein), B. G Teubner, Leipzig and Berlin 1913, p. 36. Grossmann identified the contracted Riemann tensor as the key for the solution of the problem posed by Einstein. This was followed by several attempts of Einstein to find valid field equations for this theory of gravity. David Hilbert invited Einstein to the University of Göttingen for a week to give six two-hour lectures on general relativity, which he did in June–July 1915. Einstein stayed at Hilbert's house during this visit. Hilbert started working on a combined theory of gravity and electromagnetism, and Einstein and Hilbert exchanged correspondence until November 1915. Einstein gave four lectures on his theory on 4, 11, 18 and 25 November in Berlin, published as [Ein15a], [Ein15b], [Ein15c], [Ein15d]. 4 November: Einstein published non-covariant field equations and on 11 November returned to the field equations of the "Entwurf" papers, which he now made covariant by the assumption that the trace of the energy-momentum tensor was zero, as it was for electromagnetism. Einstein sent Hilbert proofs of his papers of 4 and 11 November. (Sauer 99, notes 63, 66) 15 November: Invitation issued for the 20 November meeting at the academy in Göttingen. "Hilbert legt vor in die Nachrichten: Grundgleichungen der Physik". (Sauer 99, note 73) 16 November: Hilbert spoke at the Göttingen Mathematical Society "Grundgleichungen der Physik" (Sauer 99, note 68). Talk not published. 18 November: Einstein replied a letter from Hilbert, saying as far as he (Einstein) could tell, Hilbert's system was equivalent to the one he (Einstein) had found in the preceding weeks. (Sauer 99, note 72). Einstein also told Hilbert in this letter that he (Einstein) had "considered the only possible generally covariant field equations three years earlier", adding that "The difficulty was not to find generally covariant equations for the ; this is easy with the help of the Riemann tensor. What was difficult instead was to recognize that these equations form a generalization, and that is, a simple and natural generalization of Newton's law" (A. Einstein to D. Hilbert, 18 November, Einstein Archives Call No. 13-093). Einstein also told Hilbert in that letter that he (Einstein) had calculated the correct perihelion advance for Mercury, using covariant field equations based on the assumption that the trace of the energy momentum tensor vanished as it did for electromagnetism. 18 November: Einstein presented the calculation of the perihelion advance to Prussian Academy. 20 November: Hilbert lectured at the Göttingen Academy. The content of his presentation and of the proofs of the paper later published on the presentation are at the heart of the dispute among historians (see below). 25 November: In his last lecture, Einstein submitted the correct field equations. The published paper (Einstein 1915d) appeared on 2 December and did not mention Hilbert Hilbert starts his paper by citing Einstein: "The vast problems posed by Einstein as well as his ingeniously conceived methods of solution, and the far-reaching ideas and formation of novel concepts by means of which Mie constructs his electrodynamics, have opened new paths for the investigation into the foundations of physics."David Hilbert "The Foundations of Physics (First Communication)" Originally published as "Die Grundlagen der Physik. (Erste Mitteilung)" in Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen. Math.-phys. Klasse. 1916. Issue 8, p. 395–407. Presented in the session of 20 November 1915. Hilbert's paper took considerably longer to appear. He had galley proofs that were marked "December 6" by the printer in December 1915. Most of the galley proofs have been preserved, but about a quarter of a page is missing. The extant part of the proofs contains Hilbert's action from which the field equations can be obtained by taking a variational derivative, and using the contracted Bianchi identity derived in theorem III of Hilbert's paper, though this was not done in the extant proofs. Hilbert rewrote his paper for publication (in March 1916), changing the treatment of the energy theorem, dropping a non-covariant gauge condition on the coordinates to produce a covariant theory, and adding a new credit to Einstein for introducing the gravitational potentials into the theory of gravity. In the final paper, he said his differential equations seemed to agree with the "magnificent theory of general relativity established by Einstein in his later papers"D. Hilbert, Nac. Ges. Wiss. Goettingen 1916, 395, cited in [Cor97]. Hilbert nominated Einstein for the third Bolyai prize in 1915 'for the high mathematical spirit behind all his achievements' The 1916 paper was rewritten and republished in 1924 [Hil24], where Hilbert wrote: Einstein [...] kehrt schließlich in seinen letzten Publikationen geradewegs zu den Gleichungen meiner Theorie zurück. (Einstein [...] in his most recent publications, returns directly to the equations of my theory.)[Hil24] page 2
Historians on Hilbert's point of view Historians have discussed Hilbert's view of his interaction with Einstein.
Walter Isaacson points out that Hilbert's publication on his derivation of the equations of general relativity included the text: "The differential equations of gravitation that result are, as it seems to me, in agreement with the magnificent theory of general relativity established by Einstein."
Wuensch points out that Hilbert refers to the field equations of gravity as "meine Theorie" ("my theory") in his 6 February 1916 letter to Schwarzschild. This, however, is not at issue, since no one disputes that Hilbert had his own "theory", which Einstein criticized as naive and overly ambitious. Hilbert's theory was based on the work of Mie combined with Einstein's principle of general covariance, but applied to matter and electromagnetism as well as gravity.
Mehra and Bjerknes point out that Hilbert's 1924 version of the article contained the sentence "... und andererseits auch Einstein, obwohl wiederholt von abweichenden und unter sich verschiedenen Ansätzen ausgehend, kehrt schließlich in seinen letzten Publikationen geradenwegs zu den Gleichungen meiner Theorie zurück" - "Einstein [...] in his last publications ultimately returns directly to the equations of my theory.".[Hil24] English translation from Bje03a, p. 17; These statements of course do not have any particular bearing on the matter at issue. No one disputes that Hilbert had "his" theory, which was a very ambitious attempt to combine gravity with a theory of matter and electromagnetism along the lines of Mie's theory, and that his equations for gravitation agreed with those that Einstein presented beginning in Einstein's 25 November paper (which Hilbert refers to as Einstein's later papers to distinguish them from previous theories of Einstein). None of this bears on the precise origin of the trace term in the Einstein field equations (a feature of the equations that, while theoretically significant, does not have any effect on the vacuum equations, from which all the empirical tests proposed by Einstein were derived).
Sauer says "the independence of Einstein's discovery was never a point of dispute between Einstein and Hilbert ... Hilbert claimed priority for the introduction of the Riemann scalar into the action principle and the derivation of the field equations from it," (Sauer mentions a letter and a draft letter where Hilbert defends his priority for the action functional) "and Einstein admitted publicly that Hilbert (and Lorentz) had succeeded in giving the equations of general relativity a particularly lucid form by deriving them from a single variational principle". Sauer also stated, "And in a draft of a letter to Weyl, dated 22 April 1918, written after he had read the proofs of the first edition of Weyl's 'Raum-Zeit-Materie' Hilbert also objected to being slighted in Weyl's exposition. In this letter again 'in particular the use of the Riemannian curvature [scalar] in the Hamiltonian integral' ('insbesondere die Verwendung der Riemannschen Krümmung unter dem Hamiltonschen Integral') was claimed as one of his original contributions. SUB Cod. Ms. Hilbert 457/17."
Did Einstein develop the field equations independently? While Hilbert's paper was submitted five days earlier than Einstein's, it only appeared in 1916, after Einstein's field equations paper had appeared in print. For this reason, there was no good reason to suspect plagiarism on either side. In 1978, an 18 November 1915 letter from Einstein to Hilbert resurfaced, in which Einstein thanked Hilbert for sending an explanation of Hilbert's work. This was not unexpected to most scholars, who were well aware of the correspondence between Hilbert and Einstein that November, and who continued to hold the view expressed by Albrecht Fölsing in his Einstein biography: In November, when Einstein was totally absorbed in his theory of gravitation, he essentially only corresponded with Hilbert, sending Hilbert his publications and, on November 18, thanking him for a draft of his article. Einstein must have received that article immediately before writing this letter. Could Einstein, casting his eye over Hilbert's paper, have discovered the term which was still lacking in his own equations, and thus 'nostrified' Hilbert?
In the very next sentence, after asking the rhetorical question, Folsing answers it with "This is not really probable...", and then goes on to explain in detail why. [Einstein's] eventual derivation of the equations was a logical development of his earlier arguments—in which, despite all the mathematics, physical principles invariably predominated. His approach was thus quite different from Hilbert's, and Einstein's achievements can, therefore, surely be regarded as authentic.