einstein

63. Specifically, the issue was whether the Entwurf 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 H 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 H, 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 general 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 H.
When it is properly generalized, it permits an arbitrary H. 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 Entwurf ’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, Jürgen 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 Entwurf 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.
73. Overbye, 290.
74. Einstein, “On the General Theory of Relativity (Addendum),” Nov. 11, 1915, CPAE 6: 22; Renn and Sauer 2006, 276; Pais 1982, 252.
75. Einstein to David Hilbert, Nov. 12, 1915.
76. Einstein to Hans Albert Einstein, Nov. 15, 1915; Einstein to Mileva Mari , Nov. 15, 1915; Einstein to Heinrich Zangger, Nov. 15, 1915
(released in 2006 and printed in supplement to vol. 10).
77. Einstein to David Hilbert, Nov. 15, 1915.
78. Einstein, “Explanation of the Perihelion Motion of Mercury from the General Theory of Relativity,” Nov. 18, 1915, CPAE 6: 24.
79. Pais 1982, 253; Einstein to Paul Ehrenfest, Jan. 17, 1916; Einstein to Arnold Sommerfeld, Dec. 9, 1915.
80. Einstein to David Hilbert, Nov. 18, 1915.
81. David Hilbert to Einstein, Nov. 19, 1915.
82. The equation has been expressed in many ways. The one I use follows the formulation Einstein used in his 1921 Princeton lectures. The
entire left-hand side of the equation can be expressed more compactly as what is now known as the Einstein tensor: G μν..
83. Overbye, 293; Aczel 1999, 117; archive.ncsa.uiuc.edu/Cyberia/NumRel/Ein steinEquations.html#intro. A variation of Wheeler’s quote is on
p. 5 of the book he coauthored with Charles Misner and Kip Thorne, Gravitation.
84. Greene 2004, 74.
85. Einstein, “The Foundations of the General Theory of Relativity,”Annalen der Physik (Mar. 20, 1916), CPAE 6: 30.
86. Einstein to Heinrich Zangger, Nov. 26, 1915; Einstein to Michele Besso, Nov. 30, 1915.
87. Thorne, 119.
88. For an analysis of Hilbert’s contribution, see Sauer 1999, 529–575; Sauer 2005, 577–590. Papers describing Hilbert’s revisions include
Corry, Renn, and Stachel; Sauer 2005. For a flavor of the controversy, see also John Earman and Clark Glymour, “Einstein and
Hilbert:Two Months in the History of General Relativity,”Archive for History of Exact Sciences (1978): 291; A. A. Logunov, M. A.
Mestvirishvili, and V. A. Petrov, “How Were the Hilbert-Einstein Equations Discovered?,” Uspekhi Fizicheskikh Nauk 174, no. 6 (June
2004): 663–678; Christopher Jon Bjerknes, Albert Einstein:The Incorrigible Plagiarist , available at
home.comcast.net/~xtxinc/AEIPBook.htm; John Stachel, “Anti-Einstein Sentiment Surfaces Again,”Physics World , Apr. 2003,
physicsweb.org/articles/review/16/4/2/1; Christopher Jon Bjerknes, “The Author of Albert Einstein: The Incorrigible Plagiarist Responds
to John Stachel’s Personal Attack,” home.comcast.net/~xtxinc/Response.htm; Friedwardt Winterberg, “On ‘Belated Decision in the
Hilbert-Einstein Priority Dispute,’ ”Zeitschrift fuer Naturforschung A, (Oct. 2004): 715–719,
www.physics.unr.edu/faculty/winterberg/Hilbert-Einstein.pdf; David Rowe, “Einstein Meets Hilbert: At the Crossroads of Physics and
Mathematics,”Physics in Perspective 3, no. 4 (Nov. 2001): 379.