field equations were invariant under the non-autonomous transformation to rotating coordinates in the case of the Minkowski metric in its standard diagonal form. Janssen 2004, 29.
64
. Michele Besso memo to Einstein, Aug. 28, 1913; Janssen 2002; Norton 2000, 149; Einstein to Erwin Freundlich, Sept. 30, 1915.
65
. Einstein to Hendrik Lorentz, Oct. 12, 1915. Einstein describes his October 1915 breakthroughs in a subsequent letter to Lorentz and another one to Arnold Sommerfeld. Einstein to Hendrik Lorentz, Jan. 1, 1916: “Trying times awaited me last fall as the inaccuracy of the older gravitational field equations gradually dawned on me. I had already discovered earlier that Mercury’s perihelion motion had come out too small. In addition, I found that the equations were not covariant for substitutions corresponding to a uniform rotation of the new reference system. Finally, I found that the consideration I made last year on the determination of Lagrange’s
function for the gravitational field was thoroughly illusory, in that it could easily be modified such that no restricting conditions had to be attached to
, thus making it possible to choose
it completely freely. In this way I came to the conviction that introducing adapted systems was on the wrong track and that a more broad-reaching covariance, preferably a
covariance, must be required. Now general covariance has been achieved, whereby nothing is changed in the subsequent specialization of the frame of reference ...I had considered the current equations in essence already three years ago together with Grossmann, who had brought my attention to the Riemann tensor.” Einstein to Arnold Sommerfeld, Nov. 28, 1915: “In the last month I had one of the most stimulating and exhausting times of my life, and indeed also one of the most successful. For I realized that my existing gravitational field equations were untenable! The following indications led to this: 1) I proved that the gravitational field on a uniformly rotating system does not satisfy the field equations. 2) The motion of Mercury’s perihelion came to 18” rather than 45” per century. 3) The covariance considerations in my paper of last year do not yield the Hamiltonian function
. When it is properly generalized, it permits an arbitrary
. From this it was demonstrated that covariance with respect to ‘adapted’ coordinate systems was a flop.”
66
. Norton 2000, 152.
67
. There is a subtle divergence of opinion among the group of general relativity historians about the extent of his purported shift from the physical to the mathematical strategy in Oct.–Nov. 1915. John Norton has argued that Einstein’s “new tactic was to reverse his decision of 1913” and go back to a mathematical strategy, emphasizing a tensor analysis that would produce general covariance (Norton 2000, 151). Likewise, Jeroen van Dongen says the shift in tactics was clear: “Einstein immediately got hold of the way out of the
’s quagmire: he returned to the mathematical requirement of general covariance that he had abandoned in the Zurich notebook” (van Dongen, 25). Both scholars produce quotes from Einstein’s later years in which he claims that the big lesson he learned was to trust a mathematical strategy. On the other side, Jurgen Renn and Michel Janssen say that Norton and van Dongen (and the older Einstein in his hazy memory) make too much of this shift. Physical considerations still played a major role in finding the final theory in Nov. 1915. “In our reconstruction, however, Einstein found his way back to the generally-covariant field equations by making one important adjustment to the
theory, a theory born almost entirely out of physical considerations . . . That mathematical considerations pointed in the same direction undoubtedly inspired confidence that this was the right direction, but guiding him along this path were physical not mathematical considerations” (Janssen and Renn, 13; the quote I use in the text is on p. 10). Also, Janssen 2004, 35: “Whatever he believed, said, or wrote about it later on, Einstein only discovered the mathematical high road to the Einstein field equations after he had already found these equations at the end of a poorly paved road through physics.”
68
. Einstein to Arnold Sommerfeld, Nov. 28, 1915.
69
. Einstein, “On the General Theory of Relativity,” Nov. 4, 1915, CPAE 6: 21.
70
. Einstein to Michele Besso, Nov. 17, 1915; Einstein to Arnold Sommerfeld, Nov. 28, 1915.
71
. Einstein to Hans Albert Einstein, Nov. 4, 1915.
72
. Einstein to David Hilbert, Nov. 7, 1915.
