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Editor's note: Dr. Eugene Guth was fortunate to have one-on-one meetings with Albert Einstein once in Europe and once in the USA. The meeting in Europe was thought to have focused on Einstein's Ph.D. thesis and subsequent work on the viscosity theory of suspensions and rheology. The meeting in the USA was thought to have focused more on general topics in physics and general relativity, which is the subject of the notes that follow. After Einstein's death in 1955, Dr. Guth was given access to Einstein's papers and correspondence archived at the Institute for Advanced Studies in Princeton.

In 1970, Dr. Guth published a book chapter entitled "Contribution to the History of Einstein's Geometry as a Branch of Physics," in Relativity, M. Carmeli et al., Eds. (Plenum,New York, 1970), that inter alia asserted that Einstein in fact had priority over David Hilbert, considered by many to be the leading mathematician of the day, in deriving the correct and final form of the General Relativity field equations including the tensor and trace terms. This claim by Eugene Guth spurred a significant debate among general relativity theorists, with physicist-historian Jagdish Mehra publishing a book entitled Einstein, Hilbert, and the Theory of Gravitation, in which Mehra mentions Eugene Guth's claim and rejects it. Mehra smugly noted that Hilbert's 1924 version of his 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." Mehra's book apparently received devastating critiques for faulty conclusions from one or more book reviewers.

A Wikipedia article on the General Relativity Priority Dispute is only the latest critique of Mehra's research. "His 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)." https://en.wikipedia.org/wiki/General_relativity_priority_dispute

Note that the Wikipedia article on the General Relativity Priority Dispute fails to mention the seminal article by Dr. Eugene Guth that apparently sparked the controversy, which illustrates why professionals do not consider Wikipedia articles to be trustworthy or authoritative.

As mentioned in a 1997 article published in Science magazine and captioned, "Belated Decision in the Hilbert-Einstein Priority Dispute," available at Belated_Decision_in_the_Hilbert-Einstein_Priority_Dispute the "dispute" has now been resolved in Einstein's favor, as claimed by Eugene Guth, and specifically refers to Dr. Guth's book chapter in 1970 as the aegis for giving priority to Einstein.

Similarly, French historian of science, Patrick R. Girard, in updating his 1981 Ph.D. dissertation submitted to the University of Wisconsin-Madison, Department of History of Science, into a book entitled History of Einstein's General Relativity: Conceptual Development of the Theory(2020) available at https://hal.science/tel-02893156v1/file/Girard-History%20GTR.pdf, writes as follows.

"Annexe A EINSTEIN'S PRIORITY IN THE MATTER OF THE FIELD EQUATIONS. Though the majority of physicists correctly attribute the priority of the field equations to Einstein, a priority issue between Einstein and Hilbert is sometimes raised. Thus, I shall briefly analyze that question for the sake of historical accuracy. Shortly before Einstein's communication to the Berlin Academy on 25 November 1915, David Hilbert gave a lecture to the Mathematische Gesellschaft in Göttingen in which he developed a sweeping theory yielding in particular a set of generally covariant gravitational field equations. The talk appears to have been given on 20 or 23 November. (The printed version of the lecture indicates "Presented at the session of 20 November 1915" whereas Hilbert on a postcard to Einstein dated 14 November gives 23 November as the date of the lecture. Since, in the lecture Hilbert apparently did not mention Einstein's contributions but did so extensively in the printed version, the latter was clearly a revised one. Hence, having only this document available we do not know for sure which field equations exactly were given during the lecture but can only assume them to have been the same as the ones of the printed version, which appears likely.

This assumption having usually been made, the apparent similarity of Hilbert's field equations with Einstein's final ones has led to various claims of priority. Thus, Hermann Weyl, in his 1918 book, stated that Hilbert formulated the gravitational field equations "about the same time as Einstein, though only in the framework of Mie's theory"; Wolfgang Pauli, in 1921, went further and stated "at the same time as Einstein, and independently, Hilbert formulated the generally covariant field equations." Similarly, Pascual Jordan in his book Schwerkraft und Weltall speaks of the Einstein-Hilbert equations. More recently, the claim of independent discovery has been rejected by Eugene Guth, who referred to it as a "myth" but was taken over again by Jagdish Mehra who wrote: "During the third period, 1915-16, the field equations of gravitation were formulated by Einstein, were independently derived by Hilbert ...," and further below, added: "There can be no doubt that Hilbert's derivation was entirely independent." The whole issue of the origin of the field equations was analyzed in detail by John Earman and Clark Glymour, who surprisingly left the claim of independent discovery undecided and even suggested the possibility of an influence of Hilbert on Einstein with respect to Einstein's renunciation of the Einstein-Grossmann theory and the discovery of the final field equations. I shall side with Hilbert himself and Guth and show: (a) that the priority of the field equations belongs to Einstein ; (b) that the suggestion of an influence of Hilbert on Einstein is essentially unfounded. We shall examine successively the claims of simultaneous, independent discovery and the suggestions of an influence of Hilbert on Einstein.

The expression "simultaneous discovery" implying that the discovery relates to the same object at about the same time, it must first be pointed out that Hilbert's theory as a whole was basically different from Einstein's. Whereas Einstein's theory, in 1915, dealt only with gravitation, Hilbert's theory, by attempting to synthesize Gustav Mie's and Einstein's ideas, aimed, as the title of the paper "Die Grundlagen der Physik" indicates, at deriving all of physics from two axioms. . . ." (internal citations omitted)

"Footnote 34: A recent paper has disclosed the original text of Hilbert's conference : Leo Corry et al., "Belated Decision in the Hilbert-Einstein Priority Dispute", Science, 278, 1270-1273 (1997). The paper shows that Hilbert's equations of his conference are not covariant and are not the same as those of Einstein, thus putting a definitive end to the debate in favor of Einstein."

Based on his review of the Einstein archive and recollections of his conversations with Einstein and Paul Ewald, the following notes were written by Eugene Guth in the late 1960s. They are published here to show a sense of excitement at finding Einstein deserved credit for the first correct derivation of the General Relativity field equations including the trace term.


Eugene Guth: Hilbert's interest in General Relativity (GR) was greatly stimulated by a visit of Einstein in Göttingen around June 1915. In turn, Einstein, as he wrote to Sommerfeld, was greatly impressed by Hilbert's understanding of his (Einstein's) theory. At that time the physical foundations of GR had been established by Einstein, including the correct real shift and (half of the) light deflexion. But his field equations

Γμv = *κͲμv

contained a complicated tensor Γμv and, consequently, were neither covariant nor self-consistent. Einstein and Grossmann, did consider, the possibility

Γμv = Ͳμv ,

but rejected it. They didn't see how this choice could lead to a reduction of the field equation to Poisson's equation in the Newtonian limit. Einstein even cooked-up a "physical proof" that generally covariant field equations are not possible!

Einstein did not realize at that time that general covariance implies 4 arbitrary functions for fixing a particular coordinate system. This fact can be proven without explicit use of Bianchi's or equivalent identities. Nor was he aware that the covariant divergence of Γμv has to vanish, since that of Ͳμv does. Here, of course, the Bianchi or equivalent identities come in explicitly.

Einstein summarized the status of his theory in a long paper, "The Formal Foundation of the General Theory of Relativity" published in 1914 in the "Sitzungsberichte der Preußischen Akademie der Wissenschaften zu Berlin" (Proceedings of the Prussian Academy of Sciences), p. 1030, presented Oct. 29 (1914).

Beginning as of Oct. 1915, on return from a visit to Switzerland, Einstein saw the inadequacies of his former "derivations" of the field equation. Nov. 4th, Einstein presented his first of 4 short papers to the Berlin Academy. Now the stage is set for Hilbert's entrance. Einstein received galley proofs of his note on Nov. 7th (a shining example of Prussian efficiency!) The same day he wrote a letter to Hilbert and sent him a galley proof of his note with the same mail!

The existing of letters between Einstein and Hilbert from Nov. 7 - Nov. 20 was a great surprise to me as it will be to everybody. This was a time, when, according to Freundlich, a meat hook hung from the ceiling of Einstein's study containing a large number of unanswered letters. Freundlich questions Einstein what if more letters arrive; Einstein answered with two words: "burn them."

Furthermore, Einstein in a letter to Sommerfeld on Nov. 28, Einstein apologized for not answering earlier Sommerfeld's "friendly and interesting letter." He had last month one of the most exciting, hardest, but also most successful times of his life. He just could not think to write.

Einstein has two reasons to communicate with Hilbert. First, Hilbert showed great understanding of Einstein's work on GR on Einstein's visit in Göttingen. 2. Einstein heard from Sommerfeld, the way Einstein puts it, that "Hilbert also found a hair in Einstein's soup," which made it unedible for him. (This comment probably relates to Einstein's 1914 Berlin Academy paper.) Einstein adds that he, 4 weeks ago, convinced himself that his previous derivations were unsound and he hopes Hilbert will like his present field equations.

In this note Einstein [Editor: the page of notes stop]

If Hilbert felt not enough appreciated in GR, he could have suggested to Blumenthal and/or Sommerfeld to include his papers (2) in a new edition of LEM!!!

_______________________________


[Editor's note: Eugene Wigner served as David Hilbert's Assistant for Physics in 1927.]

Dear Eugene (Wigner),

The Hilbert-Einstein correspondence I looked at in Princeton, after our short conversation, shows the last stages of the genesis of General Relativity (GR) as a much more dramatic event than either Mehra or I thought.

In Nov. 1915, Einstein was so engrossed in his reformulation of GR that he didn't answer 'til the 28th several letters of Sommerfeld, but he did have an almost daily correspondence with Hilbert! On the 7th he wrote to Hilbert that the same mail will bring him the proofs of a paper in which he changed the field equations, after he himself four weeks ago found, that his method of proof was "trügerisch." Sommerfeld wrote to Einstein that after Hilbert found a hair in Einstein's soup, Einstein was wondering whether Hilbert will like this new solution. (This was the Nov 4 paper)

Hilbert's answer seems to have been lost. On the 12th, Einstein writes that he made new progress postulating that √ -q = 1, leads to general covariance. This refers to the paper of Nov. 8. He also mailed Hilbert two copies of his 1914 Berlin Academy paper. He adds that if [Ͳμμ = 0] is generally true, then gravity should play a fundamental role in building-up of matter.


_______________________________________


Eugene Guth: From EINSTEIN'S GEOMETRY AS A BRANCH OF PHYSICS, pp.183-84, in Relativity, Plenum Press (1970)
 
Apparently, Weyl, in the first textbook of general relativity (first edition published in 1918),41 originated the story that the field equations of general relativity have been established also by Hilbert simultaneously with and independently of Einstein. This remark was taken over by Pauli [Guth was a postdoc under Pauli] in his well-known article on relativity42and later on by several other text book writers. The remark of Weyl and Pauli does not correspond with the historical truth. In the first place, these two great experts did not look up carefully the references in Hilbert's paper. Otherwise, they would have noticed that Hilbert quotes all of Einstein's communications of November, 1915, in the Proceedings of the Berlin Academy. In particular, he quotes the paper on p. 844 which contains the final form of the GR field equations.43

It is interesting, however, how this "myth" has arisen. For the following remarks I am indebted to Professor P. P. Ewald who was, in 1915, Hilbert's "assistant for physics." His story is reinforced by the recently published exchange of letters between Einstein and Sommerfeld.37 In a letter dated July 15, 1915, Einstein says about his visit in Göttingen that he had great joy there; everything he said was understood in detail. He was very enthusiastic about Hilbert. Clearly Einstein gave a talk in Göttingen. Hilbert, whose absent-mindedness was legendary, started to think about the problem on the basis of what Einstein said, who at that time did not have yet the correct form of his field equations. I heard from Ewald that Hilber, very likely in the fall of 1915, gave a talk in Göttingen presenting the correct equations with referring to Einstein. However, by that time, Einstein must have had also the correct equations, certainly before Hilbet did. Sommerfeld heard about Hilbert's talk in Göttingen and suggested that Hilbert write a letter apologizing to Einstein, which Hilbert of course did. At any rate, he never claimed having been the independent discoverer of the field equations of GR.

It is interesting for the change of fashions and/or different tastes that Pauli states Hilbert's "presentation though would not seem to be acceptable to physicists for two reasons. First, the existence of a variational principle is introduced as an axion. Secondly, of more importance, the field equations are not derived for an arbitrary system of matter, but are specifically based on Mie's theory of matter." On the other hand, Whittaker44 states that Hilbert's procedure introducing a world-function to be used in conjunction with a variational principle represents "a distinct advance on Einstein's methods." At any rate, at present we are inclined to prefer or at least not look down upon the derivation of any physical law from a variational principle. Perhaps it is interesting to point out that historically already Ricci and Levi-Civita mentioned what turned out to be later the Lagrangean of GR as of some possible physical interest, but, of course, they did not think of the connections with gravity.45

Even though Hilbert's work was not independent of Einstein's, it was very clarifying. For example, Hilbert recognized first that the general solutions of the field equations must contain four arbitrary functions. Thus, there must be four identities between the ten field equations for the ten unknown gμv. The apparent contradiction with the causality principle is resolved by the remark that the many possible solutions of the field equations are physically completely equivalent and only formally different. Later on, Hilbert gave lectures on GR; Laue, in his textbook on GR, thanks Hilbert for putting his lecture notes at Laue's disposal. Hilbert also emphasized that not only must the general laws of physics be covariant, but every result of the theory that has a physical significance must be covariant with respect to arbitrary transformations of the coordinate system. Therefore, e.g., a statement that the particle is at rest for an interval of time of definite duration cannot have physical significance because this statement is not covariant.46

References  
  1. Einstein had written his "Notes" and his Autobiography many years after 1915. The recently published exchange of letters between Einstein and Sommerfeld [Briefwechsel Einstein/Sommerfeld (edited with comments by A. Hermann) Schwabe et Co., Basel/Stuttgart (1969); l.c. p.32], contains a letter Einstein wrote to Sommerfeld on the 28th of November, 1915. There he gives detailed arguments why his previous field equations were not correct:

    2. [1] He proved that the gravitational field on a uniformly rotating system does not obey the field equations.

    [2] The motion of the parihelion of the Mercury turns out to be 18" instead of 45" per century.

    [3] The covariance consideration in his paper for the last year (1914) does not yield the Hamilton function H. It admits, properly generalized, an arbitrary H. Therefore, the covariance with regards to a particular class of coordinate systems turned out to be useless.
Einstein also says that he has considered with Grossman three years ago [cf. also Ref 38, l.c. p. 778; (not two as stated in his "Notes")], the equation (III.7) (without the second term on the left-hand side). However, he concluded, erroneously, that it doesn't reduce to the Newtonian approximation.

  1. A. Einstein, S. B. Berlin, Akad. Wiss. p. 778, 779 (1915).
  1. H. Weyl, Raum, Zeit, Materie, Springer, Berlin, 1st edition (1918), 5th edition (1923).
  1. W. Pauli; Theory of Relativity, Pergamon, London (1958), l.c. p. 145, Ref. 277, originally in: Encykl. Math. Wiss, V. 19, Teubner, Leipzig (1921).
  1. D. Hilbert, Nachr. Ges. Wiss. Göttingen, p. 395 (1915); presented Nov. 20. Perhaps the presentation date of this paper which was five days before November 25, the presentation date of Einstein's final paper, ref. 40, confused Weyl and Pauli.
  1. E. T. Whittaker, ref. 6, l.c. p. 171.
  1. I am indebted to P. Ruffini for this remark.
  1. D. Hilbert, Math. Ann. 92, 1 (1924).
_________________________________________

Prof. John A. Wheeler Princeton University
Department of Physics, Joseph Henry Laboratories
Jadwin Hall
Post Office Box 708
Princeton, NJ 08540


John A. Wheeler
High Island
S. Bristol, Maine
04568

Saturday 1 Aug 1970

Dear Eugene -

Charles Miesner and Kip Thorne and I are working away here in S. Bristol on our intended book on general relativity. In this connection, I have just now reread your "Einstein's Geometry as a Branch of Physics" in the Cincinnati book. Such a treat! I do hope you go on with a series of articles on various phases of the history of the subject! On pag. 1984, last ¶, you say Laue thanks Hilbert for putting his notes on general relativity at Laue's disposal. What should I do, what can I do, to get hold of these Hilbert notes?

Every good wish!

Sincerely,

John


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