Relativity priority dispute
Albert Einstein presented the theories of special relativity and general relativity in publications that either contained no formal references to previous literature, or referred only to a small number of his predecessors for fundamental results on which he based his theories, most notably to the work of Hendrik Lorentz for special relativity, and to the work of Carl F. Gauss, Bernhard Riemann, and Ernst Mach for general relativity. Subsequently, claims have been put forward about both theories, asserting that they were formulated, either wholly or in part, by others before Einstein. At issue is the extent to which Einstein and various other individuals should be credited for the formulation of these theories, based on priority considerations.
The candidates for credit
Concerning special relativity, the most important names that are mentioned in discussions about the distribution of credit are Albert Einstein, Hendrik Lorentz, Henri Poincaré, and Hermann Minkowski. Consideration is also given to numerous other scientists for either anticipations of some aspects of the theory, or else for contributions to the development or elaboration of the theory. These include Woldemar Voigt, August Föppl, Joseph Larmor, Emil Cohn, Friedrich Hasenöhrl, Max Planck, Max von Laue, Gilbert Newton Lewis and Richard Chase Tolman, and others. In addition, polemics exist about alleged contributions of others such as Olinto De Pretto who according to some mathematical scholars did not create relativity but was the first to use the equation; Also Einstein's first wife Mileva Marić, although her contribution is not considered to have any foundation according to serious scholars.
Concerning general relativity, there is a controversy about the amount of credit that should go to Einstein, Marcel Grossmann, and David Hilbert. Many others (such as Gauss, Riemann, William Kingdon Clifford, Ricci, Gunnar Nordström and Levi-Civita) contributed to the development of the mathematical tools and geometrical ideas underlying the theory of gravity. Also polemics exist about alleged contributions of others such as Paul Gerber.
Undisputed and well-known facts
The following facts are well established and referrable:
- In 1889, ([Poi89]), Henri Poincaré argued that the ether might be unobservable, in which case the existence of the ether is a metaphysical question, and he suggested that some day the ether concept would be thrown aside as useless. However, in the same book (Ch. 10) he considered the ether a "convenient hypothesis" and continued to use the concept also in later books in 1908 ([Poi08], Book 3) and 1912 ([Poi13], Ch. 6).
- In 1895, Poincaré argued that results like those obtained by Michelson and Morley (Michelson–Morley experiment) show that it seems to be impossible to detect the absolute motion of matter or the relative motion of matter in relation to the ether. In 1900 [Poi00] he called this the Principle of Relative Motion, i.e., that the laws of movement should be the same in all inertial frames. Alternative terms used by Poincaré were "relativity of space" and "principle of relativity". In 1904 he expanded that principle by saying: "The principle of relativity, according to which the laws of physical phenomena must be the same for a stationary observer as for one carried along in a uniform motion of translation, so that we have no means, and can have none, of determining whether or not we are being carried along in such a motion." However, he also stated that we do not know if this principle will turn out to be true, but that it is interesting to determine what the principle implies.
- In ([Poi00]), Poincaré published a paper in which he said that radiation could be considered as a fictitious fluid with an equivalent mass of . He derived this interpretation from Lorentz's 'theory of electrons' which incorporated Maxwell's radiation pressure.
- Poincaré had described a synchronization procedure for clocks at rest relative to each other in [Poi00] and again in [Poi04]. So two events, which are simultaneous in one frame of reference, are not simultaneous in another frame. It is very similar to the one later proposed by Einstein. However, Poincaré distinguished between "local" or "apparent" time of moving clocks, and the "true" time of resting clocks in the ether. In [Poi02] he argued that "some day, no doubt, the ether will be thrown aside as useless".
- Lorentz' paper [Lor04] containing the transformations bearing his name appeared in 1904.
- Albert Einstein in [Ein05c] derived the Lorentz equations by using the principle of constancy of velocity of light and the relativity principle. He was the first to argue that those principles (along with certain other basic assumptions about the homogeneity and isotropy of space, usually taken for granted by theorists) are sufficient to derive the theory. See Postulates of special relativity. He said: "The introduction of a luminiferous ether will prove to be superfluous inasmuch as the view here to be developed will not require an absolutely stationary space provided with special properties, nor assign a velocity-vector to a point of the empty space in which electromagnetic processes take place." * Einstein's Elektrodynamik paper [Ein05c] contains no formal references to other literature. It does mention, in §9, part II, that the results of the paper are in agreement with Lorentz's electrodynamics. Poincaré is not mentioned in this paper, although he is cited formally in a paper on special relativity written by Einstein the following year.
- In 1905 Einstein was the first to suggest that when a material body lost energy (either radiation or heat) of amount , its mass decreased by the amount .
- Hermann Minkowski showed in 1907 that the theory of special relativity could be elegantly described using a four-dimensional spacetime, which combines the dimension of time with the three dimensions of space.
- Einstein in 1920 returned to a concept of aether having no state of motion.
- The proposal to describe gravity by means of a pseudo-Riemannian metric was first made by Einstein and 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 2-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 Nov 4, Nov 11, Nov 18 and Nov 25 in Berlin, published as [Ein15a], [Ein15b], [Ein15c], [Ein15d].
- November 4, Einstein published non-covariant field equations and on November 11 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 Nov 4 and Nov 11. (Sauer 99, notes 63, 66)
- Nov 15 Invitation issued for Nov 20 meeting at the Academy in Göttingen. "Hilbert legt vor in die Nachrichten: Grundgleichungen der Physik". (Sauer 99, note 73)
- Nov 16 Hilbert spoke at the Göttingen Mathematical Society "Grundgleichungen der Physik" (Sauer 99, note 68). Talk not published.
- Nov 16 or Nov 17 Hilbert sent Einstein some information about his talk of Nov 16 (letter lost)
- Nov 18 Einstein replies to Hilbert's letter (received by Hilbert Nov 19) 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 Nov, 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.
- Nov 18 Einstein presents the calculation of the perihelion advance to Prussian Academy.
- Nov 20 Hilbert lectured to the Göttingen Academy. The proofs of his paper show that Hilbert proposed a non-covariant set of equations as the fundamental equations of physics. Thus he wrote "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. quoted by Corry et al.). Hilbert then 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 : the system of fundamental equations of physics". (proofs, page 7, quoted by Corry et al.).
- In his last lecture on Nov 25 Einstein submitted the correct field equations. The published paper (Einstein 1915d) appeared on December 2, and it did not mention Hilbert.
- 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 Mar 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"
- The events of late November through December 1915 caused bad feelings from Einstein towards Hilbert. In a November 25 letter to Zangger, Einstein accused Hilbert (without mentioning his name) of attempts to appropriate ('nostrify') his theory. On Dec 4, Hilbert nominated Einstein for election as a corresponding member of the Göttingen Mathematical Society. In a December 20 letter to Hilbert, Einstein proposed to settle the dispute.
- 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.)
The following things seem to be unclear, unknown or disputed:[according to whom?]
- To what degree Einstein was familiar with Poincaré's work
- It is known that Einstein was familiar with [Poi02], but it is not known to what extent he was familiar with other work of Poincaré in 1905. However it is known that he knew [Poi00] in 1906, because he quoted it in [Ein06].
- Lorentz' paper [Lor04] containing the transformations bearing his name appeared in 1904. The question is whether Einstein was familiar in 1905 with either this paper itself or a review of it (which appeared in the Annalen der Physik).
- To what degree Einstein was following other physicists' work at the time. Some authors claim that Einstein worked in relative isolation and with restricted access to the physics literature in 1905. Others, however, disagree; a personal friend of Einstein, Maurice Solovine, later acknowledged that he and Einstein both pored for weeks over Poincaré's 1902 book, keeping them "breathless for weeks on end" [Rot06].
- Whether his wife, Mileva Marić, may have contributed to Einstein's work, although Einstein scholars say that there is no substantive evidence that she made significant contributions.
- Before 1997, "the commonly accepted view was that David Hilbert completed the general theory of relativity at least 5 days before Albert Einstein submitted his conclusive paper on this theory on 25 November 1915. Hilbert's article, bearing the date of submission 20 November 1915 but published only on 31 March 1916, presents a generally covariant theory of gravitation, including field equations essentially equivalent to those in Einstein's paper" (Corry, Renn and Stachel, 1997). Since the discovery of printer's proofs of Hilbert's paper of Nov 20, dated 6 Dec 1915, which show a number of differences from the finally published paper, this 'commonly accepted view' has been challenged.
- Whether Einstein got the correct mathematical formulation for general relativity from Hilbert, or formulated it independently. Points at issue:
- The content of Hilbert's November 16 letter/postcard to Einstein is not known. It is however, clear[according to whom?] from Einstein's response that it was an account of Hilbert's work.
- It is not known what was on the missing part of Hilbert's printer proofs. The missing portion is large enough to have contained the field equations in an explicit form. There are several competing speculations about the content of the missing piece.
- Based on the above, it is not known whether Hilbert had formulated the field equations in an explicit form before December 6 (the date of the printer's proofs) or not.
- It is known from the proofs that Hilbert introduced four non-covariant equations in order to specify the gravitational potentials and that this approach was dropped from his revised paper.
- Whether Hilbert ever tried to claim priority for the field equations - it seems clear that he regarded the theory of general relativity as Einstein's theory.
- What Hilbert thought he was referring to when he used the term "equations of my theory" about Einstein's research. Hilbert made a similar remark in a letter to Karl Schwarzschild.[B 1]
There are a large number of opinions related to these involving questions of "who should get the credit" - these are not enumerated here.[why?]
Historians of special relativity
In his History of the theories of ether and electricity from 1953, E. T. Whittaker claimed that relativity is the creation of Lorentz and Poincaré and attributed to Einstein's papers only little importance. However, most historians of science, like Gerald Holton, Arthur I. Miller, Abraham Pais, John Stachel, or Olivier Darrigol have other points of view. They admit that Lorentz and Poincaré developed the mathematics of special relativity, and many scientists originally spoke about the "Lorentz-Einstein theory". But they argue that it was Einstein who completely eliminated the classical ether and demonstrated the relativity of space and time. They also argue that Poincaré demonstrated the relativity of space and time only in his philosophical writings, but in his physical papers he maintained the ether as a privileged frame of reference that is perfectly undetectable, and continued (like Lorentz) to distinguish between "real" lengths and times measured by observers at rest within the aether, and "apparent" lengths and times measured by observers in motion within the aether.[B 2][B 3][B 4][B 5][B 6] Darrigol summarizes:
Most of the components of Einstein's paper appeared in others' anterior works on the electrodynamics of moving bodies. Poincaré and Alfred Bucherer had the relativity principle. Lorentz and Larmor had most of the Lorentz transformations, Poincaré had them all. Cohn and Bucherer rejected the ether. Poincaré, Cohn, and Abraham had a physical interpretation of Lorentz's local time. Larmor and Cohn alluded to the dilation of time. Lorentz and Poincaré had the relativistic dynamics of the electron. None of these authors, however, dared to reform the concepts of space and time. None of them imagined a new kinematics based on two postulates. None of them derived the Lorentz transformations on this basis. None of them fully understood the physical implications of these transformations. It all was Einstein's unique feat.[B 7]
Comments by Lorentz, Poincaré, and Einstein
I did not indicate the transformation which suits best. That was done by Poincaré and then by Mr. Einstein and Minkowski. [..] Because I had not thought of the direct way which led there, and because I had the idea that there is an essential difference between systems x, y, z, t and x',y',z',t'. In one we use - such was my thought - coordinate axes which have a fixed position in the aether and which we can call "true" time; in the other system, on the contrary, we would deal with simple auxiliary quantities whose introduction is only a mathematical artifice. [..] I did not establish the principle of relativity as rigorously and universally true. Poincaré, on the contrary, obtained a perfect invariance of the equations of electrodynamics, and he formulated the "postulate of relativity", terms which he was the first to employ. [..] Let us add that by correcting the imperfections of my work he never reproached me for them.
However, a 1916 reprint of his main work "The theory of electrons" contains notes (written in 1909 and 1915) in which Lorentz sketched the differences between his results and that of Einstein as follows:
[p. 230]: the chief difference [is] that Einstein simply postulates what we have deduced, with some difficulty and not altogether satisfactorily, from the fundamental equations of the electromagnetic field. [p. 321]: The chief cause of my failure was my clinging to the idea that the variable t only can be considered as the true time and that my local time t' must be regarded as no more than an auxiliary mathematical quantity. In Einstein's theory, on the contrary, t' plays the same part as t; if we want to describe phenomena in terms of x', y', z', t' we must work with these variables exactly as we could do with x, y, z, t.
Regarding the fact, that in this book Lorentz only mentioned Einstein and not Poincaré in connection with a) the synchronisation by light signals, b) the reciprocity of the Lorentz transformation, and c) the relativistic transformation law for charge density, Janssen comments:[B 8]
[p.90]: My guess is that it has to do with the fact that Einstein made the physical interpretation of the Lorentz transformation the basis for a remarkably clear and simple discussion of the electrodynamics of moving bodies, whereas Poincaré's remarks on the physical interpretation of Lorentz transformed quantities may have struck Lorentz as inconsequential philosophical asides in expositions that otherwise closely followed his own. I also have a sense that Lorentz found Einstein's physically very intuitive approach more appealing than Poincaré's rather abstract but mathematically more elegant approach.
And at a conference on the Michelson–Morley experiment in 1927 at which Lorentz and Michelson were present, Michelson suggested that Lorentz was the initiator of the theory of relativity. Lorentz then replied:
I considered my time transformation only as a heuristic working hypothesis. So the theory of relativity is really solely Einstein's work. And there can be no doubt that he would have conceived it even if the work of all his predecessors in the theory of this field had not been done at all. His work is in this respect independent of the previous theories.
Poincaré attributed the development of the new mechanics almost entirely to Lorentz. He only mentioned Einstein in connection with the photoelectric effect, but not in connection with special relativity. For example, in 1912 Poincaré raises the question whether "the mechanics of Lorentz" will still exist after the development of the quantum theory. He wrote:
In all instances in which it differs from that of Newton, the mechanics of Lorentz endures. We continue to believe that no body in motion will ever be able to exceed the speed of light; that the mass of a body is not a constant, but depends on its speed and the angle formed by this speed with the force which acts upon the body; that no experiment will ever be able to determine whether a body is at rest or in absolute motion either in relation to absolute space or even in relation to the ether.
It is now known that Einstein was well aware of the scientific research of his time. The well known historian of science, Jürgen Renn, Director of the Max Planck Institute for the History of Science wrote on Einstein's contributions to the Annalen der Physik:
The Annalen also served as a source of modest additional income for Einstein, who wrote more than twenty reports for its Beiblätter - mainly on the theory of heat - thus demonstrating an impressive mastery of the contemporary literature. This activity started in 1905. and probably resulted from his earlier publications in the Annalen in this field. Going by his publications between 1900 and early 1905, one would conclude that Einstein's specialty was thermodynamics.
Einstein wrote in 1907 that one needed only to realize that an auxiliary quantity that was introduced by Lorentz and that he called "local time" can simply be defined as "time." In 1909 and 1912 Einstein explained:[B 9]
...it is impossible to base a theory of the transformation laws of space and time on the principle of relativity alone. As we know, this is connected with the relativity of the concepts of "simultaneity" and "shape of moving bodies." To fill this gap, I introduced the principle of the constancy of the velocity of light, which I borrowed from H. A. Lorentz's theory of the stationary luminiferous ether, and which, like the principle of relativity, contains a physical assumption that seemed to be justified only by the relevant experiments (experiments by Fizeau, Rowland, etc.)— Albert Einstein (1912), translated by Anna Beck (1996).
But Einstein and his supporters took the position that this "light postulate" together with the principle of relativity renders the ether superfluous and leads directly to Einstein's version of relativity. It is also known that Einstein had been reading and studying Poincaré's 1902-book Science and hypothesis well before 1905, which included:
- detailed philosophical assessments on the relativity of space, time, and simultaneity
- discussion of the reliance on conventions regarding the use of light signals for the synchronization of clocks
- the definition of the principle of relativity and the conjecture that a violation of that principle can never be detected empirically
- the possible redundancy of the ether hypothesis
- detailed remarks on the physical status of non-Euclidean geometry.
Einstein refers to Poincaré in connection with the inertia of energy in 1906 and the non-Euclidean geometry in 1921, but not in connection with the Lorentz transformation, the relativity principle or the synchronisation procedure by light signals. However, in the last years before his death Einstein acknowledged some of Poincaré's contributions (according to Darrigol, maybe because his biographer Pais in 1950 sent Einstein a copy of Poincarè's Palermo paper, which he said that he had not read before). Einstein wrote in 1953:[B 10]
There is no doubt, that the special theory of relativity, if we regard its development in retrospect, was ripe for discovery in 1905. Lorentz had already recognized that the transformations named after him are essential for the analysis of Maxwell's equations, and Poincaré deepened this insight still further. Concerning myself, I knew only Lorentz's important work of 1895 [...] but not Lorentz's later work, nor the consecutive investigations by Poincaré. In this sense my work of 1905 was independent. [..] The new feature of it was the realization of the fact that the bearing of the Lorentz transformation transcended its connection with Maxwell's equations and was concerned with the nature of space and time in general. A further new result was that the "Lorentz invariance" is a general condition for any physical theory.
Did Hilbert claim priority for parts of General Relativity?
Kip Thorne concludes, in remarks based on Hilbert's 1924 paper, that Hilbert regarded the General Theory of relativity as Einstein's: "Quite naturally, and in accord with Hilbert's view of things, the resulting law of warpage was quickly given the name the Einstein field equation rather than being named after Hilbert. Hilbert had carried out the last few mathematical steps to its discovery independently and almost simultaneously with Einstein, but Einstein was responsible for essentially everything that preceded those steps...".[B 11] 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."[B 11]
Arguments have been made that Hilbert claimed priority for the field equations themselves; the sources cited for this are:
- Hilbert's article (dated 20 November 1915), when it appeared in 1916, contained the text "Die so zu Stande kommenden Differentialgleichungen der Gravitation sind, wie mir scheint, mit der von Einstein in seinen späteren Abhandlungen aufgestellten großzügigen Theorie der allgemeinen Relativität in gutem Einklang." In translation, "The differential equations of gravity arrived at in this way are, I think, in good agreement with those of Einstein in his later papers in which he presented his comprehensive theory of general relativity." Hilbert refers here to the "later papers" of Einstein, obviously to distinguish them from the Entwurf theory of 1913 and the preliminary papers prior to the end of November 1915 when Einstein published the equations of general relativity in their final form.
- Wuensch[B 1] points out that Hilbert refers to the field equations of gravity as "meine Theorie" ("my theory") in his February 6, 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[B 12] and Bjerknes[B 13] 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.". These statements of course do not have any particular bearing on the matter at issue. No one disputes that Hilbert has "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 his, Einstein's, Nov 25 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, "[B 14] (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."[B 14]
- Einstein wrote to Hilbert on 20 December 1915 that there was an "ill-feeling between us" and it has been suspected that this ill feeling was the result of Einstein's bitterness over Hilbert's "nostrification" of his (Einstein's) theory. Others have suggested that Hilbert might have felt that Einstein had derived some benefit or hints from his (Hilbert's) letters, and that those had helped him to arrive at the trace term of the field equations, and if so, that Einstein should have acknowledged this in his paper. But this is pure speculation, aside from Einstein's comment that he believed others (presumably Hilbert) had tried to "nostrify" his theory.
So far, there seems to be no consensus that these statements form a clear claim by Hilbert to have published the field equations first.
Did Einstein develop the field equations independently?
For a long time, it was believed that Einstein and Hilbert found the field equations of gravity independently. While Hilbert's paper was submitted somewhat 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, a November 18, 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?[B 15]
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,[B 16] 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 November 16 talk about his theory.[B 1]
- Einstein's November 18 paper on the perihelion motion of Mercury, which still refers to the incomplete field equations of November 4 and 11. (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 November 18. 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 November 25.
- 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 November 16.
- Hilbert modified his paper in December 1915, and the November 18 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 , : the system of fundamental equations of physics". (proofs, page 7. Corry et al.). Hilbert's first theory (lecture Nov 16, lecture Nov 20, proofs Dec 6) 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.[B 14]
- One may attempt to reconstruct the way in which Einstein may have arrived at the field equations independently. This is, for instance, done in the paper of Logunov, Mestvirishvili and Petrov quoted below.[B 18] Renn and Sauer[B 19] investigate the notebook used by Einstein in 1912 and claim he was close to the correct theory at that time.
Attackers and defenders
This section cites notable publications where people have expressed a view on the issues outlined above.
Sir Edmund Whittaker (1954)
In 1954, Sir Edmund Taylor Whittaker, an English mathematician and historian of science, credited Poincaré with the equation , and he included a chapter entitled The Relativity Theory of Poincaré and Lorentz in his book A History of the Theories of Aether and Electricity.[B 20] He credited Poincaré and Lorentz, and especially alluded to Lorentz's 1904 paper (dated by Whittaker as 1903), Poincaré's St. Louis speech (The Principles of Mathematical Physics) of September 1904, and Poincaré's June 1905 paper. Whittaker attributed to Einstein's relativity paper only little importance, i.e., the formulation of the Doppler and aberration formulas. Max Born spent three years trying to dissuade Whittaker, but Whittaker insisted that everything of importance had already been said by Poincaré, and that Lorentz quite plainly had the physical interpretation.
Gerald Holton (1960)
Whittaker's claims were criticized by Gerald Holton (1960, 1973).[B 2] He argued that there are fundamental differences between the theories of Einstein on one hand, and Poincaré and Lorentz on the other hand. Einstein radically reformulated the concepts of space and time, and by that removed "absolute space" and thus the stationary luminiferous aether from physics. On the other hand, Holton argued that Poincaré and Lorentz still adhered to the stationary aether concept, and tried only to modify Newtonian dynamics, not to replace it. Holton argued, that "Poincaré's silence" (i.e., why Poincaré never mentioned Einstein's contributions to relativity) was due to their fundamentally different conceptual viewpoints. Einstein's views on space and time and the abandonment of the aether were, according to Holton, not acceptable to Poincaré, therefore the latter only referred to Lorentz as the creator of the "new mechanics". Holton also pointed out that although Poincaré's 1904 St. Louis speech was "acute and penetrating" and contained a "principle of relativity" that is confirmed by experience and needs new development, it did not "enunciate a new relativity principle". He also alluded to mistakes of Whittaker, like predating Lorentz's 1904 paper (published April 1904) to 1903.
Views similar to Holton's were later (1967, 1970) expressed by his former student, Stanley Goldberg.[B 21]
G. H. Keswani (1965)
In a 1965 series of articles tracing the history of relativity,[B 22] Keswani claimed that Poincaré and Lorentz should have the main credit for special relativity - claiming that Poincaré pointedly credited Lorentz multiple times, while Lorentz credited Poincaré and Einstein, refusing to take credit for himself. He also downplayed the theory of general relativity, saying "Einstein's general theory of relativity is only a theory of gravitation and of modifications in the laws of physics in gravitational fields".[B 22] This would leave the special theory of relativity as the unique theory of relativity. Keswani cited also Vladimir Fock for this same opinion.
Dingle said, among other things, ".. the 'principle of relativity' had various meanings, and the theories associated with it were quite distinct; they were not different forms of the same theory. Each of the three protagonists.... was very well aware of the others .... but each preferred his own views"[B 23]
Keswani did not accept the criticism, and replied in two letters also published in the same journal ([B 25] and [B 26] - in his reply to Dingle, he argues that the three relativity theories were at heart the same: ".. they meant much that was common. And that much mattered the most."[B 25]
Dingle commented the year after on the history of crediting: "Until the first World War, Lorentz's and Einstein's theories were regarded as different forms of the same idea, but Lorentz, having priority and being a more established figure speaking a more familiar language, was credited with it." (Dingle 1967, Nature 216 p. 119-122).
Arthur I. Miller (1973)
Miller (1973, 1981)[B 3] agreed with the analysis of Holton and Goldberg, and further argued that although the terminology (like the principle of relativity) used by Poincaré and Einstein were very similar, their content differs sharply. According to Miller, Poincaré used this principle to complete the aether based "electromagnetic world-view" of Lorentz and Abraham. He also argued that Poincaré distinguished (in his July 1905 paper) between "ideal" and "real" systems and electrons. That is, Lorentz's and Poincaré's usage of reference frames lacks an unambiguous physical interpretation, because in many cases they are only mathematical tools, while in Einstein's theory the processes in inertial frames are not only mathematically, but also physically equivalent. Miller wrote in 1981:
- p. 172: "Although Poincaré's principle of relativity is stated in a manner similar to Einstein's, the difference in content is sharp. The critical difference is that Poincaré's principle admits the existence of the ether, and so considers the velocity of light to be exactly c only when it is measured in coordinate systems at rest in the ether. In inertial reference systems, the velocity of light is c and is independent of the emitter's motion as a result of certain compensatory effects such as the mathematical local time and the hypothesis of an unobservable contraction. Consequently, Poincaré's extension of the relativity principle of relative motion into the dynamics of the electron resided in electromagnetic theory, and not in mechanics...Poincaré came closest to rendering electrodynamics consistent, but not to a relativity theory." p. 217: "Poincaré related the imaginary system Σ' to the ether fixed system S'".
Miller (1996)[B 3] argues that Poincaré was guided by empiricism, and was willing to admit that experiments might prove relativity wrong, and so Einstein is more deserving of credit, even though he might have been substantially influenced by Poincaré's papers. Miller also argues that "Emphasis on conventionalism ... led Poincaré and Lorentz to continue to believe in the mathematical and observational equivalence of special relativity and Lorentz's electron theory. This is incorrect." [p. 96] Instead, Miller claims that the theories are mathematically equivalent but not physically equivalent. [p. 91-92]
Abraham Pais (1982)
In his Einstein biography Subtle is the Lord (1982),[B 4] Abraham Pais argued that Poincaré "comes near" to discovering special relativity (in his St. Louis lecture of September 1904, and the June 1905 paper), but eventually he failed, because in 1904 and also later in 1909, Poincaré treated length contraction as a third independent hypothesis besides the relativity principle and the constancy of the speed of light. According to Pais, Poincaré thus never understood (or at least he never accepted) special relativity, in which the whole theory including length contraction can simply be derived from two postulates. Consequently, he sharply criticized Whittaker's chapter on the "Relativity theory of Poincaré and Lorentz", saying "how well the author's lack of physical insight matches his ignorance of the literature", although Pais admitted that the first book of Whittaker's History of Aether and Electricity is a masterpiece.
He also argued that Lorentz never abandoned the stationary aether concept, either before or after 1905:
- p. 118: "Throughout the paper of 1895, the Fresnel aether is postulated explicitly"; p. 125: "Like Voigt before him, Lorentz regarded the transformation ... only as a convenient mathematical tool for proving a physical theorem ... he proposed to call t the general time and t' the local time. Although he didn't say it explicitly, it is evident that to him there was, so to speak, only one true time t."; p. 166: "8.3. Lorentz and the Aether... For example, Lorentz still opines that the contraction of the rods has a dynamic origin. There is no doubt that he had read and understood Einstein's papers by then. However, neither then nor later was he prepared to accept their conclusions as the definitive answer to the problems of the aether."
Elie Zahar (1983)
In several papers, Elie Zahar (1983, 2000)[B 27] argued that both Einstein (in his June paper) and Poincaré (in his July paper) independently discovered special relativity. He said that "though Whittaker was unjust towards Einstein, his positive account of Poincaré's actual achievement contains much more than a simple grain of truth". According to him, it was Poincaré's unsystematic and sometimes erroneous statements regarding his philosophical papers (often connected with conventionalism), which hindered many to give him due credit. In his opinion, Poincaré was rather a "structural realist" and from that he concludes, that Poincaré actually adhered to the relativity of time and space, while his allusions to the aether are of secondary importance. He continues, that due to his treatment of gravitation and four-dimensional space, Poincaré's 1905/6-paper was superior to Einstein's 1905-paper. Yet Zahar gives also credit to Einstein, who introduced Mass–Energy equivalence, and also transcended special relativity by taking a path leading to the development of general relativity.
John Stachel (1995)
John Stachel (1995)[B 28] argued that there is a debate over the respective contributions of Lorentz, Poincaré and Einstein to relativity. These questions depend on the definition of relativity, and Stachel argued that kinematics and the new view of space and time is the core of special relativity, and dynamical theories must be formulated in accordance with this scheme. Based on this definition, Einstein is the main originator of the modern understanding of special relativity. In his opinion, Lorentz interpreted the Lorentz transformation only as a mathematical device, while Poincaré's thinking was much nearer to the modern understanding of relativity. Yet Poincaré still believed in the dynamical effects of the aether and distinguished between observers being at rest or in motion with respect to it. Stachel wrote: "He never organized his many brilliant insights into a coherent theory that resolutely discarded the aether and the absolute time or transcended its electrodynamic origins to derive a new kinematics of space and time on a formulation of the relativity principle that makes no reference to the ether".
Peter Galison (2002)
In his book Einstein's clocks, Poincaré's maps (2002),[B 6][B 29] Peter Galison compared the approaches of both Poincaré and Einstein to reformulate the concepts of space and time. He wrote: "Did Einstein really discover relativity? Did Poincaré already have it? These old questions have grown as tedious as they are fruitless." This is because it depends on the question, which parts of relativity one considers as essential: the rejection of the aether, the Lorentz transformation, the connection with the nature of space and time, predictions of experimental results, or other parts. For Galison, it is more important to acknowledge that both thinkers were concerned with clock synchronization problems, and thus both developed the new operational meaning of simultaneity. However, while Poincaré followed a constructive approach and still adhered to the concepts of Lorentz's stationary aether and the distinction between "apparent" and "true" times, Einstein abandoned the aether and therefore all times in different inertial frames are equally valid. Galison argued that this does not mean that Poincaré was conservative, since Poincaré often alluded to the revolutionary character of the "new mechanics" of Lorentz.
Olivier Darrigol (2004)
In his 2004 article, "The Mystery of the Einstein-Poincaré Connection", Darrigol wrote:[B 7]
- "By 1905 Poincaré's and Einstein's reflections on the electrodynamics of moving bodies led them to postulate the universal validity of the relativity principle, according to which the outcome of any conceivable experiment is independent of the inertial frame of reference in which it is performed. In particular, they both assumed that the velocity of light measured in different inertial frames was the same. They further argued that the space and time measured by observers belonging to different inertial systems were related to each other through the Lorentz transformations. They both recognized that the Maxwell-Lorentz equations of electrodynamics were left invariant by these transformations. They both required that every law of physics should be invariant under these transformations. They both gave the relativistic laws of motion. They both recognized that the relativity principle and the energy principle led to paradoxes when conjointly applied to radiation processes. On several points - namely, the relativity principle, the physical interpretation of Lorentz's transformations (to first order), and the radiation paradoxes - Poincaré's relevant publications antedated Einstein's relativity paper of 1905 by at least five years, and his suggestions were radically new when they first appeared. On the remaining points, publication was nearly simultaneous."
- "I turn now to basic conceptual differences. Einstein completely eliminated the ether, required that the expression of the laws of physics should be the same in any inertial frame, and introduced a "new kinematics" in which the space and time measured in different inertial systems were all on exactly the same footing. In contrast, Poincaré maintained the ether as a privileged frame of reference in which "true" space and time were defined, while he regarded the space and time measured in other frames as only "apparent." He treated the Lorentz contraction as a hypothesis regarding the effect of the edgewise motion of a rod through the ether, whereas for Einstein it was a kinematic consequence of the difference between the space and time defined by observers in relative motion. Einstein gave the operational meaning of time dilation, whereas Poincaré never discussed it. Einstein derived the expression of the Lorentz transformation from his two postulates (the relativity principle and the constancy of the velocity of light in a given inertial system), whereas Poincaré obtained these transformations as those that leave the Maxwell-Lorentz equations invariant. Whereas Einstein, having eliminated the ether, needed a second postulate, in Poincaré's view the constancy of the velocity of light (in the ether frame) derived from the assumption of a stationary ether. Einstein obtained the dynamics of any rapidly moving particle by the direct use of Lorentz covariance, whereas Poincaré reasoned according to a specific model of the electron built up in conformity with Lorentz covariance. Einstein saw that Poincaré's radiation paradoxes could be solved only by assuming the inertia of energy, whereas Poincaré never returned to this question. Lastly, Poincaré immediately proposed a relativistic modification of Newton's law of gravitation and saw the advantages of a four-vector formalism in this context, whereas Einstein waited a couple of years to address this problem complex."
- "These differences between the two theories are sometimes regarded as implying different observable predictions even within the domain of electromagnetism and optics. In reality, there is no such disagreement, for Poincaré's ether is by assumption perfectly undetectable, and every deduction made in Einstein's theory can be translated into a deduction in Poincaré's theory ..."
- In sum, then, Einstein could have borrowed the relativity principle, the definition of simultaneity, the physical interpretation of the Lorentz transformations, and the radiation paradoxes from Poincaré. ... The wisest attitude might be to leave the coincidence of Poincaré's and Einstein's breakthroughs unexplained, ...
Anatoly Alexeevich Logunov on special relativity (2004)
In Anatoly Logunov's book[B 18] about Poincaré's relativity theory, there is an English translation (on p. 113, using modern notations) of the part of Poincaré's 1900 article containing E=mc2. Logunov states that Poincaré's two 1905 papers are superior to Einstein's 1905 paper. According to Logunov, Poincaré was the first scientist to recognize the importance of invariance under the Poincaré group as a guideline for developing new theories in physics. In chapter 9 of this book, Logunov points out that Poincaré's second paper was the first one to formulate a complete theory of relativistic dynamics, containing the correct relativistic analogue of Newton's F=ma.
On p. 142, Logunov points out that Einstein wrote reviews for the Beiblätter Annalen der Physik, writing 21 reviews in 1905. In his view, this contradicts the claims that Einstein worked in relative isolation and with limited access to the scientific literature. Among the papers reviewed in the Beiblätter in the fourth (of 24) issue of 1905, there is a review of Lorentz' 1904-paper by Richard Gans, which contains the Lorentz transformations. In Logunov's view, this supports the view that Einstein was familiar with the Lorentz' paper containing the correct relativistic transformation in early 1905, while his June 1905 paper does not mention Lorentz in connection with this result.
Harvey R. Brown (2005)
Harvey R. Brown (2005)[B 30] (who favors a dynamical view of relativistic effects similar to Lorentz, but "without a hidden aether frame") wrote about the road to special relativity from Michelson to Einstein in section 4:
- p. 40: "The cradle of special theory of relativity was the combination of Maxwellian electromagnetism and the electron theory of Lorentz (and to a lesser extent of Larmor) based on Fresnel's notion of the stationary aether....It is well known that Einstein's special relativity was partially motivated by this failure [to find the aether wind], but in order to understand the originality of Einstein's 1905 work it is incumbent on us to review the work of the trailblazers, and in particular Michelson, FitzGerald, Lorentz, Larmor, and Poincaré. After all they were jointly responsible for the discovery of relativistic kinematics, in form if not in content, as well as a significant portion of relativistic dynamics as well."
Regarding Lorentz's work before 1905, Brown wrote about the development of Lorentz's "theorem of corresponding states" and then continued:
- p. 54: "Lorentz's interpretation of these transformations is not the one Einstein would give them and which is standardly embraced today. Indeed, until Lorentz came to terms with Einstein's 1905 work, and somehow despite Poincaré's warning, he continued to believe that the true coordinate transformations were the Galilean ones, and that the 'Lorentz' transformations ... were merely a useful formal device..." p. 56. "Lorentz consistently failed to understand the operational significance of his notions of 'local' time...He did however have an intimation of time dilation in 1899, but inevitably there are caveats...The hypotheses of Lorentz's system were starting to pile up, and the spectre of ad hocness was increasingly hard to ignore."
Then the contribution of Poincaré's to relativity:
- p. 62: "Indeed, the claim that this giant of pure and applied mathematics co-discovered special relativity is not uncommon, and it is not hard to see why. Poincaré was the first to extend the relativity principle to optics and electrodynamics exactly. Whereas Lorentz, in his theorem of corresponding states, had from 1899 effectively assumed this extension of the relativity principle up to second-order effects, Poincaré took it to hold for all orders. Poincaré was the first to show that Maxwell's equations with source terms are strictly Lorentz covariant. … Poincaré was the first to use the generalized relativity principle as a constraint on the form of the coordinate transformations. He recognized that the relativity principle implies that the transformations form a group, and in further appealing to spatial isotropy. … Poincaré was the first to see the connection between Lorentz's ‘local time’, and the issue of clock synchrony. … It is fair to say that Poincaré was the first to understand the relativity of simultaneity, and the conventionality of distant simultaneity. Poincaré anticipated Minkowski's interpretation of the Lorentz transformations as a passive, rigid rotation within a four-dimensional pseudo-Euclidean space-time. He was also aware that the the [sic] electromagnetic potentials transform in the manner of what is now called a Minkowski 4-vector. He anticipated the major results of relativistic dynamics (and in particular the relativistic relations between force, momentum and velocity), but not E=mc² in its full generality."
However, Brown continued with the reasons which speak against Poincaré's co-discovery:
- p. 63-64: "What are the grounds for denying Poincaré the title of co-discoverer of special relativity? ... Although Poincaré understood independently of Einstein how the Lorentz transformations give rise to non-Galilean transformation rules for velocities (indeed Poincaré derived the correct relativistic rules), it is not clear that he had a full appreciation of the modern operational significance attached to coordinate transformations.... he did not seem to understand the role played by the second-order terms in the transformation. Compared with the cases of Lorentz and Larmor, it is even less clear that Poincaré understood either length contraction or time dilation to be a consequence of the coordinate transformation.... What Poincaré was holding out for was no less than a new theory of ether and matter - something far more ambitions than what appeared in Einstein's 1905 relativity paper...p. 65. Like Einstein half a decade later, Poincaré wanted new physics, not a reinterpretations or reorganization of existing notions."
Brown denies the idea of other authors and historians, that the major difference between Einstein and his predecessors is Einstein's rejection of the aether, because, it is always possible to add for whatever reason the notion of a privileged frame to special relativity, as long as one accepts that it will remain unobservable, and also Poincaré argued that "some day, no doubt, the aether will thrown aside as useless". However, Brown gave some examples, what in his opinion were the new features in Einstein's work:
- p. 66: "The full meaning of relativistic kinematics was simply not properly understood before Einstein. Nor was the 'theory of relativity' as Einstein articulated it in 1905 anticipated even in its programmatic form." p. 69. "How did Albert Einstein...arrive at his special theory of relativity?...I want only to stress that it is impossible to understand Einstein's discovery (if that is the right word) of special relativity without taking on board the impacts of the quantum in physics." p. 81. "In this respect [Brown refers to the conventional nature of distant simultaneity] Einstein was doing little more than expanding on a theme that Poincaré had already introduced. Where Einstein goes well beyond the great mathematician is in his treatment of the coordinate transformations... In particular, the extraction of the phenomena of length contraction and time dilation directly from the Lorentz transformations in section 4 of the 1905 paper is completely original."
After that, Brown develops his own dynamical interpretation of special relativity as opposed to the kinematical approach of Einstein's 1905 paper (although he says that this dynamical view is already contained in Einstein's 1905-paper, "masqueraded in the language of kinematics", p. 82), and the modern understanding of space-time.
Roger Cerf (2006)
Roger Cerf (2006)[B 31] gave priority to Einstein for developing special relativity, and criticized the assertions of Leveugle and others concerning the priority of Poincaré. While Cerf agreed that Poincaré made important contributions to relativity, he argued (following Pais) that Poincaré "stopped short before the crucial step" because he handled length contraction as a "third hypothesis", therefore Poincaré lacked a complete understanding of the basic principles of relativity. "Einstein's crucial step was that he abandoned the mechanistic ether in favor of a new kinematics." He also denies the idea, that Poincaré invented E=mc² in its modern relativistic sense, because he did not realize the implications of this relationship. Cerf considers Leveugle's Hilbert-Planck-Einstein connection an implausible conspiracy theory.
Shaul Katzir (2005)
Katzir (2005)[B 32] argued that "Poincaré's work should not be seen as an attempt to formulate special relativity, but as an independent attempt to resolve questions in electrodynamics." Contrary to Miller and others, Katzir thinks that Poincaré's development of electrodynamics led him to the rejection of the pure electromagnetic world-view (due to the non-electromagnetic Poincaré-Stresses introduced in 1905), and Poincaré's theory represents a "relativistic physics" which is guided by the relativity principle. In this physics, however, "Lorentz's theory and Newton's theory remained as the fundamental bases of electrodynamics and gravitation."
Scott Walter (2005, 2007)
Walter (2005) argues that both Poincaré and Einstein put forward the theory of relativity in 1905. And in 2007 he wrote, that although Poincaré formally introduced four-dimensional spacetime in 1905/6, he was still clinging to the idea of "Galilei spacetime". That is, Poincaré preferred Lorentz covariance over Galilei covariance when it is about phenomena accessible to experimental tests; yet in terms of space and time, Poincaré preferred Galilei spacetime over Minkowski spacetime, and length contraction and time dilation "are merely apparent phenomena due to motion with respect to the ether". This is the fundamental difference in the two principal approaches to relativity theory, namely that of "Lorentz and Poincaré" on one side, and "Einstein and Minkowski" on the other side.[B 33]
Whittaker (1954)[B 20] stated that David Hilbert had derived the theory of General Relativity from an elegant variational principle almost simultaneously with Einstein's discovery of the theory.
Albrecht Fölsing on the Hilbert-Einstein interaction (1993)
From Fölsing's 1993 (English translation 1998)[B 15] Einstein biography " Hilbert, like all his other colleagues, acknowledged Einstein as the sole creator of relativity theory."
Cory/Renn/Stachel and Friedwardt Winterberg (1997/2003)
In 1997, Cory, Renn and Stachel published a 3-page article in Science entitled "Belated Decision in the Hilbert-Einstein Priority Dispute" , concluding that Hilbert had not anticipated Einstein's equations.[B 16][B 34]
Friedwardt Winterberg,[B 35] a professor of physics at the University of Nevada, Reno, disputed  these conclusions, observing that the galley proofs of Hilbert's articles had been tampered with - part of one page had been cut off. He goes on to argue that the removed part of the article contained the equations that Einstein later published, and he wrote that the cut off part of the proofs suggests a crude attempt by someone to falsify the historical record. "Science" declined to publish this; it was printed in revised form in "Zeitschrift für Naturforschung", with a dateline of June 5, 2003. Winterberg criticized Corry, Renn and Statchel for having omitted the fact that part of Hilbert's proofs were cut off. Winterberg wrote that the correct field equations are still present on the existing pages of the proofs in various equivalent forms. In this paper Winterberg asserted that Einstein sought the help of Hilbert and Klein to help him find the correct field equation, without mentioning the research of Fölsing (1997) and Sauer (1999) according to which Hilbert invited Einstein to Göttingen to give a week of lectures on general relativity in June 1915, which however does not necessarily contradict Winterberg. Hilbert at the time was looking for physics problems to solve.
A short reply to Winterberg's article could be found at ; the original long reply can be accessed via the Internet Archive at . In this reply, Winterberg's hypothesis is called "paranoid" and "speculative". Cory et al. offer the following alternative speculation: "it is possible that Hilbert himself cropped off the top of p. 7 to include it with the three sheets he sent Klein, in order that they not end in mid-sentence."[B 36]
As of September 2006, the Max Planck Institute of Berlin has replaced the short reply with a note  saying that the Max Planck Society "distances itself from statements published on this website [...] concerning Prof. Friedwart Winterberg" and stating that "the Max Planck Society will not take a position in [this] scientific dispute".
Ivan Todorov, in a paper published on ArXiv,[B 17] says of the debate:
- Their [CRS's] attempt to support on this ground Einstein's accusation of "nostrification" goes much too far. A calm, non-confrontational reaction was soon provided by a thorough study[B 14] of Hilbert's route to the "Foundations of Physics" (see also the relatively even handed survey (Viz 01)).
In the paper recommended by Todorov as calm and non-confrontational, Tilman Sauer[B 14] concludes that the printer's proofs show conclusively that Einstein did not plagiarize Hilbert, stating
- any possibility that Einstein took the clue for the final step toward his field equations from Hilbert's note [Nov 20, 1915] is now definitely precluded.
Max Born's letters to David Hilbert, quoted in Wuensch, are quoted by Todorov as evidence that Einstein's thinking towards general covariance was influenced by the competition with Hilbert.
Todorov ends his paper by stating:
- Einstein and Hilbert had the moral strength and wisdom - after a month of intense competition, from which, in a final account, everybody (including science itself) profited - to avoid a lifelong priority dispute (something in which Leibniz and Newton failed). It would be a shame to subsequent generations of scientists and historians of science to try to undo their achievement.
Christopher Jon Bjerknes (2003)
Christopher Jon Bjerknes' book  focuses on antagonizing the general relativity priority claims in favour of Einstein expressed by Leo Corry (Cohn Institute, University of Tel-Aviv) and John Stachel (Director of the Center of Einstein Studies, Boston) regarding Hilbert-Einstein equations of gravity.
Anatoly Alexeevich Logunov on general relativity (2004)
Anatoly Logunov (a former vice president of the Soviet Academy of Sciences and currently 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.[B 37]
Wuensch and Sommer (2005)
Daniela Wuensch,[B 1] a historian of science and a Hilbert and Kaluza expert, responded to Bjerknes, Winterberg and Logunov's criticisms of the Corry/Renn/Stachel paper in a book which appeared in 2005, wherein she defends the view that the cut to Hilbert's printer proofs was made in recent times. Moreover, she presents a theory about what might have been on the missing part of the proofs, based upon her knowledge of Hilbert's papers and lectures.
She defends the view that knowledge of Hilbert's November 16, 1915 letter was crucial to Einstein's development of the field equations: Einstein arrived at the correct field equations only with Hilbert's help ("nach großer Anstrengung mit Hilfe Hilberts"), but nevertheless calls Einstein's reaction (his negative comments on Hilbert in the November 26 letter to Zangger) "understandable" ("Einsteins Reaktion ist verständlich") because Einstein had worked on the problem for a long time.
According to her publisher, Klaus Sommer, Wuensch concludes though that:
- This comprehensive study concludes with a historical interpretation. It shows that while it is true that Hilbert must be seen as the one who first discovered the field equations, the general theory of relativity is indeed Einstein's achievement, whereas Hilbert developed a unified theory of gravitation and electromagnetism. 
In 2006, Wuensch was invited to give a talk at the annual meeting of the German Physics Society (Deutsche Physikalische Gesellschaft) about her views about the priority issue for the field equations.
Wuensch's publisher, Klaus Sommer, in an article in "Physik in unserer Zeit",[B 38] supported Wuensch's view that Einstein obtained some results not independently but from the information obtained from Hilbert's November 16 letter and from the notes of Hilbert's talk. While he does not call Einstein a plagiarist, Sommer speculates that Einstein's conciliatory December 20 letter was motivated by the fear that Hilbert might comment on Einstein's behaviour in the final version of his paper. Sommer claimed that a scandal caused by Hilbert could have done more damage to Einstein than any scandal before ("Ein Skandal Hilberts hätte ihm mehr geschadet als jeder andere zuvor").
David E. Rowe (2006)
The contentions of Wuensch and Sommer have been strongly contested by the historian of mathematics and natural sciences David E. Rowe in a detailed review of Wuensch's book published in Historia Mathematica in 2006. Rowe argues that Wuensch's book offers nothing but tendentious, unsubstantiated, and in many cases highly implausible, speculations.
- History of Lorentz transformations
- History of special relativity
- History of general relativity
- Criticism of relativity theory#Accusations of plagiarism and priority discussions
- List of scientific priority disputes
- Multiple discovery
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