Who got there first?
Timeline of relativity theory
And so, by November 1915, Einstein and Hilbert were both convinced that a generally covariant relativity theory would indeed be necessary and achievable. The following account of what happened in November is based largely on the narration Einstein and Hilbert: The Creation of General Relativity provided by Todorov (2005):
Einstein communicates to the Preussian Academy of Sciences a paper entitled Zur allgemeine Relativitätstheorie (“The Theory of General Relativity”) where he for the first time in public rejects his “Entwurf” scalar theory of gravitation of 1914, proposing in its place a new fundamental equation. Regarding his scalar theory the year before, he states:
“I lost trust in the field equations I had derived, and instead, looked for a way to limit the possibilities in a natural way. In this pursuit I arrived at the demand of general covariance, a demand from which I parted, though with a heavy heart, three years ago when i worked together with my friend Grossmann.” — Einstein, early November 1915
As Abraham Pais is later reported to have said, Einstein’s lacking ability to define his field equations in generally covariant form should indeed be quite understandable, as Einstein had been well ahead of his time exploring what he aptly called a “no man’s land”.
Einstein sends Hilbert the proofs from his November 4th paper, writing that “I recognized four weeks ago that my earlier methods of proof were deceptive”, likely responding to a communication to Sommerfeld that reports Hilbert’s objections to Einstein’s October 1914 paper. Interestingly, Einstein finishes the letter by asking Hilbert:
“I am curious whether you will be well disposed towards this solution”
Einstein’s new solution was however still not generally covariant, assuming that the determinant of the metric tensor was a constant (-1). Although Hilbert’s letter in response has been lost, it is fair to assume from their later communication and Einstein’s further revision of the theory that Hilbert was in fact not “well disposed” towards the solution.
A week later, Einstein again communicates to the Preussian Academy, under the same title as the week before, rejecting his initially proposed equation and instead putting forth a new equation, likely based on Hilbert’s feedback. Einstein now, suddenly, proposes a generally covariant equation:
However, which only coincides with his final (correct) equation if the stress-energy tensor T (and hence also R) is traceless, i.e. that the sum of the elements on the main diagonal of the matrix trace are zero), which is true for Maxwell’s electrodynamics. The very next day, Einstein sends Hilbert a second letter, announcing that with his November 11th note, he has finally found the correct formulation for his field equations, that now satisfy general covariance.
Hilbert responds to Einstein’s letter a few days later with a long note. In it, he notes that he is himself excited about his own“axiomatic solution of your grand problem”, noting that “insofar as I understand your new paper, the solution given by you is completely different from mine” from Einstein’s (Todorov, 2005 p. 8). He invites Einstein to come to Göttingen to hear him present his theory, even inviting him to spend the night in his home.
Einstein answers Hilbert’s letter asking for copies of the proofs from Hilbert’s lecture and regretfully declining Hilbert’s invitation to attend his lecture due to illness and exhaustion.
Hilbert must have responded to Einstein’s request and sent him a copy of the proofs, because by the 18th of November, Einstein again writes to Hilbert, stating “The system of equations given by you agrees — as far as I can tell, exactly with what I found in recent weeks and submitted to the Academy.” He next goes on to mention, somewhat competitively, that he and Grossman had been aware of the equation of his November 11th paper for three years, but rejected it due to its incompatibility with Newtons Law (meaning Poisson’s field equation) in the Newtonian limit. On the same day, Einstein sends his third communication to the academy for the month, entitled Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätsheorie (“Explanation of the Perihelion Motion of Mercury from the General Theory of Relativity”), providing a calculation of the precession of the perihelion of Mercury which comes out to the correct, observed 45″.
As Todorov notes, two remarks are in order based on Einstein’s paper. First, Einstein is not correct in his assertion that his equation
is equivalent to Hilbert’s equation from his November 20th paper
The two are only equivalent for one another when homogeneous, i.e. when the stress-energy tensor is equal to zero. Second, Einstein indeed does derive the correct numerical value for the precession of the perihelion of Mercury. However, he does so from the not exactly correct equation of his November 11th paper, by solving the equation for a case where the stress-energy tensor is 0 in the post-Newtonian approximation (allowing for point singularities) (Todorov, 2005. p. 9).
On the 19th, Hilbert responds to Einstein by congratulating him for having found the correct precession of the perihelion of Mercury, adding: “If I could calculate as quickly as you, then the electron would have to capitulate in the face of my equations and at the same time the hydrogen atom would have to offer its excuses for the fact that it does not radiate” (Pais 1992, p. 260).
Hilbert presents his work to Gesellschaft der Wissenschaften, the Academy in Göttingen. In the published article of his talk, he begins by acknowledging Einstein’s work, stating: “Einstein […] has brought forth profound thoughts and unique conceptions, and invented ingenious methods for dealing with them”, however also introducing ambiguity, writing “Following the axiomatic method, in fact from two simple axioms, I would like to propose a new system of the basic equations of physics. They are of ideal beauty and I believe they solve the problems of Einstein and Mie at the same time”. Hilbert then proceeds to derive the correct equations for the variational principle assuming general/reparametrization covariance and a second order equation for the metric tensor g_μν.
In Einstein’s fourth and final note to the Academy in November 1915, Die Feldgleichungen der Gravitation (“The Field Equations of Gravitation”) he rejects both of his earlier proposed fundamental equations, and submits, finally, what we now know to be the correct equations describing his general theory of relativity, albeit without a derivation of the correct stress-energy tensor his previous versions had so critically been lacking:
This time, his equation is exactly equivalent to Hilbert’s, since they both imply R+κT = 0. Einstein does not refer to Hilbert’s work in the paper.