Hilbert's criticism on the first version of Einstein's field equations Crossposted at HSM SE

I have read  once that Hilbert had some reservations regarding the first form of the field equations
$$ R_{\mu\nu}= k T_{\mu\nu}$$
because it was not possible to retrieve them from a variational principle. Is it true? I have not found any authoritative document that mention this and I don't know where to search and I would like some sources if they exists.
edit: I think I have found a clue in arxiv.org/pdf/physics/9811050.pdf in the footnote 85 at page 23 but I don't know where to find the source translated . Is there a collection of translated papers from hilbert?
 A: I cannot give you an expert historic account, but I would like to note a few things:

*

*The variational derivations of $T^{\mu\nu}$ for many fields did not exist in 1915. This is because most stress-energy tensors were phenomenological "mean-field" expressions for stress-energy "guessed" from the equations of motion rather than the other way around. For example, the variational principle for a perfect relativistic fluid was given by Taub only in 1954.

*Even more, I believe the link between $T^{\mu\nu}$ and the Lagrangian was not established in general. Noether's first papers including her celebrated theorem (which associates a stress-energy tensor with any Lagrangian that is invariant with respect to space-time translations) were only published in 1918, 3 years after the publication of Einstein's final equations.

As such, the right hand side of the first version of the equations was certainly not derivable from a variational principle by the time Einstein published his 1913 Entwurf. So at least in this sense the criticism of Hilbert was correct, and Einstein himself, at least implicitly, agreed, since he then stated in his November 1915 lectures for the Prussian academy that while the left hand of the final form of the Einstein equations (Einstein tensor) is made of marble, the right hand side (the stress-energy tensor) is made of wood.
But what about the left-hand side? Could one derive it variationally at least in vacuum, $T^{\mu\nu} = 0$? It turns out that yes, in fact, since the vacuum case $R_{\mu\nu} = 0$ is fully equivalent to the "final" Einstein equations $R_{\mu\nu} - R g_{\mu\nu}/2 = 0$. (You can see this by first solving for the trace of either of the equations and resubstituting.) As such, they can be derived from the same Lagrangian (albeit one has to play around with them to get the desired form).
I would assume that this fact was known both to Einstein and Hilbert before and around 1915 and they did not even care, since the vacuum theory without sources was essentially meaningless. (Recall that Hilbert and Einstein were, unlike us, not scarred by the knowledge of exact solutions with singular sources or funny boundary conditions, they knew gravity as gravity in the solar system - which is nothing without matter).
The crux, I conjecture, was really the stress-energy tensor. In his 1915 Die Grundlage der Physik, Hilbert actually coupled gravity to electromagnetism in a variational principle where the electromagnetic stress-energy tensor was the first occurence of what is now known as the Hilbert stress-energy tensor. He remarked that the tensor he derived by varying with respect to the metric actually agrees with the known stress-energy tensor of the electromagnetic field.
Hilbert knew of the theories of the electron of Abraham, Lorentz and Mie, where the mass of the electron could essentially be completely assigned to the energy of the electromagnetic field that arose due to the compression of the electron charge into a tiny radius. He hoped that, as a result, all matter-energy of ordinary matter can be composed through the matter-energy of the EM field in this manner. (Of course, this would require some averaging leading to the mean-field empirical formulas for stress-energy we had at our disposal. Our current picture is actually not that different, we just assume the stress-energy is the variational stress-energy of all the fields of the Standard model, not just EM.) So this was the ambition behind Hilbert's 1915 work and, I believe, this was exactly something that he expected from Einstein's results and did not find.
A: On arXiv you will find a paper "How Hilbert has found the Einstein Equations before Einstein and forgeries of Hilbert’s page proofs", concerning discussion between Hilbert and Einstein about proper field equation, https://arxiv.org/abs/physics/0610154. Some referenced documents there could be possibly of interest to you, too. Some other sources I have found are:
"Did Einstein "Nostrify" Hilbert's Final Form of the Field Equations for General
Relativity?", by Galina Wienstein, https://arxiv.org/abs/1412.1816, and "Einstein and Hilbert: The creation of General Relativity", by Ivan T. Todorov, https://arxiv.org/abs/physics/0504179.
