“I am now very satisfied with the gravitation theory. The fact that the gravitational equations are not generally covariant, which still bothered me so much some time ago, has proved to be unavoidable; it can easily be proved that a theory with generally covariant equations cannot exist if it is required that the field be mathematically completely determined by the matter...” -Albert Einstein to Ludwig Hopf, 2 November 1913
A CRITICAL MOMENT IN THE DISCOVERY OF GENERAL RELATIVITY: AUTOGRAPH LETTER SIGNED TO LUDWIG HOPF FROM 1913 WHERE EINSTEIN’S ABANDONS THE REQUIREMENT OF GENERAL COVARIANCE.
Background of the Letter - Einstein’s struggles with general covariance:
“In developing general relativity, Einstein sought to satisfy many requirements. However we shall see the his efforts were dominated by a single theme, covariance,” specifically general covariance, which was “an important concept for Einstein as he tried to generalize a theory of relativity. It meant that the relationships between their components remained the same even when there were arbitrary changes or rotations in the space and time coordinate system.” (John D. Norton, “General covariance and the foundations of general relativity: eight decades of dispute”; Walter Isaacson, Einstein: His Life and Universe).
By early 1913, Einstein had been struggling with his equations for general relativity for many years and “after many aborted attempts, Einstein eventually derived, at the end of the Zurich Notebook, a field equation that became known as the core of the Entwurf theory. It primarily satisfied the principles rooted in classical physics, namely, the correspondence principle and the conservation principle. Einstein realized that the class of coordinate systems in which the Entwurf equation takes on the same form does not satisfy the generalized principle of relativity in the way he imagined. He therefore abandoned with a heavy heart the realization of general covariance. Nevertheless, he could reassure himself that this equation was acceptable because the necessary restriction of the admissible coordinate systems could apparently be justified by the requirement to implement the conservation principle. So there seemed to be a cogent reason for the limited extent to which the generalized principle of relativity was fulfilled in the Entwurf theory... [and] Einstein assumed that it was the best that could be achieved.”
Despite the publication of the Entwurf theory, however, Einstein still could not let go of requirement for a generally covariant theory of general relativity and began to think of the lack of general covariance as an “ugly dark spot” of the theory and “had found another, more profound argument – the famous ‘hole argument’ – claiming that generally covariant theories are bound to violate causality... The hole argument and its refutation eventually became the starting point for formulating the important concept of background-independent theories, that is, of theories for which time and space are not a fixed stage for the drama of physics.
“In 1913, however, it was precisely the erroneous hole argument that motivated Einstein to further consolidate the Entwurf theory, whose main ‘ugly dark spot’ seemed to have been overcome. He concluded [in this letter to Hopf] that ‘the fact that the gravitational equations are not generally covariant, which still bothered me so much some time ago, has proved to be unavoidable; it can easily be proved that a theory with generally covariant equations cannot exist if it is required that the field be mathematically completely determined by the matter’” (Hanoch Gutfreund and Jürgen Renn, The Road to Relativity, pp. 24-26).
This oft-cited letter to Hopf documents a pivotal moment in Einstein’s thinking during the creation of general relativity. For some time after this letter, Einstein did indeed seem comfortable with abandoning the requirement for general covariance, but all that changed during the autumn of 1915 when during a flurry of creative activity he was able to discover equations for general relativity that satisfied the requirements of general covariance.
Although the generally covariant equations of general relativity have been universally acknowledged as one of the supreme achievements of human thought, the issue of covariance in general relativity has been the subject of continued concern and the subject of much modern scholarship. As noted Einstein scholar John D., Norton explains, “Einstein offered the principle of general covariance as the fundamental physical principle of his general theory of relativity and as responsible for extending the principle of relativity to accelerated motion. This view was disputed almost immediately with the counter-claim that the principle was no relativity principle and was physically vacuous. The disagreement persists today” (“General covariance and the foundations of general relativity: eight decades of dispute”, Rep. Prog. Phys. 56; 1993, p.791). See also: Kevin Hartnett, “How Einstein Lost His Bearings, and With Them, General Relativity”(Quanta Magazine, March 14, 2018) for a discussion of the current issues with general relativity and covariance.
This letter is explicitly cited in: Hanoch Gutfreund and Jürgen Renn, The Road to Relativity, pp. 24-26; Walter Isaacson, Einstein: His Life and Universe, p. 201; Galina Weinstein, Einstein’s Pathway to the Special Theory of Relativity, p. 362, and General Relativity Conflict and Rivalries, p.77; Jeroen van Dongen, Einstein’s Unification, p.22; et al.
The full text of the letter:
Zurich, 2 November 
Dear Mr. Hopf,
Thank you so much for your kind invitation, which I would have been delighted to accept. But I could not even think of it, since I must call myself happy if I can fulfill my paper-writing obligations even after having renounced all pleasure-giving extravagance. Above all, my warmest congratulations on the birth of your strapping boy. May he be as healthy and intelligent as his old man, but in addition also a little more industrious.
I am now very satisfied with the gravitation theory. The fact that the gravitational equations are not generally covariant, which still bothered me so much some time ago, has proved to be unavoidable; it can easily be proved that a theory with generally covariant equations cannot exist if it is required that the field be mathematically completely determined by the matter.
In Brussels the lecture of Bragg Sr. was extremely interesting. It is unbelievable how much this man has already found out about the lattice structure of crystals and about Röntgen rays. Now, all at once, it is possible to carry out exact determinations of the wavelengths of Röntgen rays. Some metals (e.g., radium) display very narrow emission ranges, so that it is really possible to produce monochromatic Röntgen rays. One can safely say that Laue’s figures are now totally explained. At the same time, Debije’s work on the effect of temperature is also of great importance. One hopes that Debije will soon demonstrate the incorrectness of the hypothesis of zero-point energy, the theoretical untenability of which became glaringly obvious to me soon after the publication of the paper I co-authored with Mr. Stern. It is settled that the momentum & energy fluctuations reside in the radiation.
With kindest regards to you and your wife,
English translation by Anna Beck from The Collected Papers of Albert Einstein, Volume 5, The Swiss Years: Correspondence, 1902-1914, #480.
The recipient of the letter, Ludwig Hopf (1884-1939), was a major figure in early 20th-century theoretical physics. Hopf served as an assistant to Einstein at the University of Zurich and co-authored two papers with Einstein.
One sheet, 7x9.75 inches with center fold creating four pages (with Einstein writing on three of the pages). Housed in custom presentation folder. Usual folds. In beautiful, fine condition.
LETTERS BY EINSTEIN DOCUMENTING HIS SEARCH FOR GENERAL RELATIVITY ARE PARTICULARLY RARE AND DESIRABLE.
Price: $60,000 .