General relativity priority dispute

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. 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 g^{\mu\nu};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." • 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 g_{\mu\nu} 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" • Hilbert nominated Einstein for the third Bolyai prize in 1915 'for the high mathematical spirit behind all his achievements' ==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 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. In their 1997 Science paper, Corry, Renn and Stachel quote the above passage and comment that "the arguments by which Einstein is exculpated are rather weak, turning on his slowness in fully grasping Hilbert's mathematics", and so they attempted to find more definitive evidence of the relationship between the work of Hilbert and Einstein, basing their work largely on a recently discovered pre-print of Hilbert's paper. A discussion of the controversy around this paper is given below. Those who contend that Einstein's paper was motivated by the information obtained from Hilbert have referred to the following sources: • The correspondence between Hilbert and Einstein mentioned above. More recently, it became known that Einstein was also given notes of Hilbert's 16 November talk about his theory. • Einstein's 18 November paper on the perihelion motion of Mercury, which still refers to the incomplete field equations of 4 and 11 November. (The perihelion motion depends only on the vacuum equations, which are unaffected by the trace term that was added to complete the field equations.) Reference to the final form of the equations appears only in a footnote added to the paper, indicating that Einstein had not known the final form of the equations on 18 November. This is not controversial, and is consistent with the well-known fact that Einstein did not complete the field equations (with the trace term) until 25 November. • Letters of Hilbert, Einstein, and other scientists may be used in attempts to make guesses about the content of Hilbert's letter to Einstein, which is not preserved, or of Hilbert's lecture in Göttingen on 16 November. Those who contend that Einstein's work takes priority over Hilbert's, or that both authors worked independently have used the following arguments: • Hilbert modified his paper in December 1915, and the 18 November version sent to Einstein did not contain the final form of the field equations. The extant part of the printer proofs does not have the explicit field equations. This is the point of view defended by Corry, Renn, Stachel, and Sauer. • Sauer (1999) and Todorov (2005) agree with Corry, Renn and Satchel that Hilbert's proofs show that Hilbert had originally presented a non-covariant theory, which was dropped from the revised paper. Corry et al. quote from the proofs: "Since our mathematical theorem ... can provide only ten essentially independent equations for the 14 potentials [...] and further, maintaining general covariance makes quite impossible more than ten essential independent equations [...] then, in order to keep the deterministic characteristic of the fundamental equations of physics [...] four further non-covariant equations ... [are] unavoidable." (proofs, pages 3 and 4. Corry et al.) Hilbert derives these four extra equations and continues "these four differential equations [...] supplement the gravitational equations [...] to yield a system of 14 equations for the 14 potentials g^{\mu\nu}, q_s: the system of fundamental equations of physics". (proofs, page 7. Corry et al.). Hilbert's first theory (16 November lecture, 20 November lecture, 6 December proofs) was titled "The fundamental equations of Physics". In proposing non-covariant fundamental equations, based on the Ricci tensor but restricted in this way, Hilbert was following the causality requirement that Einstein and Grossmann had introduced in the Entwurf papers of 1913. • One may attempt to reconstruct the way in which Einstein arrived at the field equations independently. This is, for instance, done in the paper of Logunov, Mestvirishvili and Petrov quoted below. Renn and Sauer investigate the notebook used by Einstein in 1912 and claim he was close to the correct theory at that time. ==Scholars==

This section cites notable publications where people have expressed a view on the issues outlined above. Albrecht Fölsing on the Hilbert-Einstein interaction (1993) From Fölsing's 1993 (English translation 1998) and at the time the scientific advisor of the Institute for High Energy Physics), is author of a book about Poincaré's relativity theory and coauthor, with Mestvirishvili and Petrov, of an article rejecting the conclusions of the Corry/Renn/Stachel paper. They discuss both Einstein's and Hilbert's papers, claiming that Einstein and Hilbert arrived at the correct field equations independently. Specifically, they conclude that: :Their pathways were different but they led exactly to the same result. Nobody "nostrified" the other. So no "belated decision in the Einstein–Hilbert priority dispute", about which [Corry, Renn, and Stachel] wrote, can be taken. Moreover, the very Einstein–Hilbert dispute never took place. :''All is absolutely clear: both authors made everything to immortalize their names in the title of the gravitational field equations. But general relativity is Einstein's theory.'' Rowe argues that Wuensch's book offers nothing but tendentious, unsubstantiated, and in many cases highly implausible, speculations. == In popular works by famous physicists ==

Wolfgang Pauli's Encyclopedia entry for the theory of relativity pointed out two reasons physicists did not consider Hilbert's derivation equivalent to Einstein's: 1) it required accepting the stationary-action principle as a physical axiom and more important 2) it was based on Mie unified field theory. In his 1999 article for Time Magazine which featured Einstein Man of the Century Stephen Hawking wrote: Kip Thorne concludes, in remarks based on Hilbert's 1924 paper, that Hilbert regarded the general theory of relativity as Einstein's: However, Kip Thorne also stated, "Remarkably, Einstein was not the first to discover the correct form of the law of warpage [. . . .] Recognition for the first discovery must go to Hilbert" based on "the things he had learned from Einstein's summer visit to Göttingen." This last point is also mentioned by Corry et al. == Insignificance of the dispute ==

As noted by the historians John Earman and Clark Glymour, "questions about the priority of discoveries are often among the least interesting and least important issues in the history of science." There was no real controversy between Einstein and Hilbert themselves: And: ==See also==