86. Einstein to Elsa Einstein, after Aug. 3, 1914.
87. Einstein to Mileva Mari, Sept. 15, 1914, contains the poisoning allegation. Many other letters in 1914 detail their struggle over money, furniture, and treatment of the children.
CHAPTER NINE: GENERAL RELATIVITY
1. Renn and Sauer 2006, 117.
2. The description of the equivalence principle follows the formulation that Einstein used in his yearbook article of 1907 and his comprehensive general relativity paper of 1916. Others have subsequently modified it slightly. See also Einstein, “Fundamental Ideas and Methods of Relativity Theory,” 1920, unpublished draft of a paper for Nature magazine, CPAE 7: 31.
Some of this chapter draws from a dissertation by one of the editors of the Einstein Papers Project: Jeroen van Dongen, “Einstein’s Unification: General Relativity and the Quest for Mathematical Naturalness,” 2002. He provided a copy to me along with guidance and editing for this chapter. This chapter also follows the research findings of other scholars studying Einstein’s general relativity work. I am grateful to van Dongen and others who met with me and helped me on this chapter, including Tilman Sauer, Jürgen Renn, John D. Norton, and Michel Janssen. This chapter draws on their work and also that of John Stachel, all listed in the bibliography.
3. Einstein, “The Speed of Light and the Statics of the Gravitational Field,”Annalen der Physik (Feb. 1912), CPAE 4: 3; Einstein 1922c; Janssen 2004, 9. In his 1907 and 1911 papers, Einstein refers to it as the “equivalence hypothesis,” but in this 1912 paper, he raises it to the status of an Aequivalenzprinzip.
4. Einstein, “On the Influence of Gravitation on the Propagation of Light,”Annalen der Physik (June 21): 1911, CPAE 3: 23.
5. Einstein to Erwin Freundlich, Sept. 1, 1911.
6. Stachel 1989b.
7. Record and grade transcript, CPAE 1: 25; Adolf Hurwitz to Hermann Bleuler, July 27, 1900, CPAE 1: 67; Einstein to Mileva Mari, Dec. 28, 1901.
8. Fölsing, 314; Pais 1982, 212.
9. Hartle, 13.
10. Einstein to Arnold Sommerfeld, Oct. 29, 1912.
11. Einstein, foreword to the Czech edition of his popular book Relativity, 1923; see utf.mff.cuni.cz/Relativity/Einstein.htm. In it Einstein writes, “The decisive idea of the analogy between the mathematical formulation of the theory and the Gaussian theory of surfaces came to me only in 1912 after my return to Zurich, without being aware at that time of the work of Riemann, Ricci, and Levi-Civita. This was first brought to my attention by my friend Grossmann.” Einstein 1922c: “I realized that the foundations of geometry have physical significance. My dear friend the mathematician Grossmann was there when I returned from Prague to Zurich. From him I learned for the first time about Ricci and later about Riemann.”
12. Sartori, 275.
13. Amir Aczel, “Riemann’s Metric,” in Aczel 1999, 91–101; Hoffmann 1983, 144–151.
14. I am grateful to Tilman Sauer and Craig Copi for help with this section.
15. Janssen 2002; Greene 2004, 72.
16. Calaprice, 9; Flückiger, 121.
17. The Zurich Notebook is in CPAE 4: 10. An online facsimile is available at echo.mpiwg-berlin.mpg.de/content/relativityrevolution/jnul. See also Janssen and Renn.
18. Norton 2000, 147. See also Renn and Sauer 2006, 151. I am grateful to Tilman Sauer for his editing of this section.
19. Einstein, Zurich Notebook, CPAE 4: 10 (German edition), p. 39 has the first notations of what became known as the Einstein tensor.
20. An explanation of this dilemma is in Renn and Sauer 1997, 42–43. The mystery of why Einstein in early 1913 could not find the correct gravitational tensor—and the issue of his understanding of coordinate condition options—is addressed nicely in Renn 2005b, 11–14. He builds on, and suggests some revisions to, the conclusions of Norton 1984.
21. Norton, Janssen, and Sauer have all suggested that Einstein’s bad experience in 1913 of abandoning a mathematical strategy for a physical one, and his subsequent belated success with a mathematical strategy, is reflected in the views he expressed in his 1933 Spencer lecture at Oxford and also his approach in the later decades of his life to finding a unified field theory.
22. Einstein, “Outline [Entwurf ] of a Generalized Theory of Relativity and of a Theory of Gravitation” (with Marcel Grossmann), before May 28, 1913, CPAE 4: 13; Janssen 2004; Janssen and Renn.
23. Einstein to Elsa Einstein, Mar. 23, 1913.
24. Einstein-Besso manuscript, CPAE 4: 14; Janssen, 2002.
25. Einstein, “On the Foundations of the General Theory of Relativity,”Annalen der Physik (Mar. 6, 1918), CPAE 7: 4. A vivid explanation of Newton’s bucket and how it connects to relativity is in Greene 2004, 23–74. Einstein is largely responsible for inferring how Mach would regard an empty universe. See Norton 1995c; Julian Barbour,“General Relativity as a Perfectly Machian Theory,” Carl Hoefer, “Einstein’s Formulation of Mach’s Principle,” and Hubert Goenner, “Mach’s Principle and Theories of Gravity,” all in Barbour and Pfister.
26. Janssen 2002, 14; Janssen 2004, 17; Janssen 2006. Janssen has done important work analyzing the Einstein-Besso collaborations of 1913. Reproductions of the Einstein-Besso manuscript and other related documents, along with an essay by Janssen on their significance, is in a 288-page catalogue from Christie’s, which auctioned the originals on Oct. 4, 2002. (The 50-page Einstein-Besso manuscript sold for $595, 000.) For an example of how Einstein dismissed Besso’s suggestion that the Minkowski metric in rotating coordinates wasn’t a valid solution to the Entwurf field equations—and how Einstein kept feeling that the Entwurf did indeed comply with Mach’s principle—see Einstein to Michele Besso, ca. Mar. 10, 1914.
27. Einstein to Ernst Mach, June 25, 1913; Misner, Thorne, and Wheeler, 544.
28. Einstein to Hendrik Lorentz, Aug. 14, 1913. But two days later, he writes Lorentz again to say that he has resigned himself to the belief that covariance is impossible: “Only now, after this ugly dark spot seems to have been eliminated, does the theory give me pleasure.” Einstein to Hendrik Lorentz, Aug. 16, 1913.
29. The hole argument basically said that a generally covariant gravitational theory would be indeterministic. Generally covariant field equations could not determine the metric field uniquely. A full specification of the metric field outside of some small region that was devoid of matter, known as “the hole,” would not be able to fix the metric field within that region. See Stachel 1989b; Norton 2005b; Janssen 2004.
30. Einstein to Ludwig Hopf, Nov. 2, 1913. See also Einstein to Paul Ehrenfest, Nov. 7, 1913: “It can be proved that generally covariant equations that determine the field completely from the matter tensor cannot exist at all. Can there be anything more beautiful than this, that the necessary specialization follows from the conservation laws? Thus, the conservation laws determine the surfaces that, from among all the surfaces, are to be privileged as coordinate surfaces. We can designate these privileged surfaces as planes, since we are left with linear substitutions as the only ones that are justified.” Einstein’s clearest explanation of the hole argument is “On the Foundations of the Generalized Theory of Relativity and the Theory of Gravitation,” Jan. 1914, CPAE 4: 25.
31. When Einstein appeared at the annual convocation of German-speaking scientists in Sept. 1913, the rival gravitation theorist Gustav Mie rose to launch a “lively” attack on him and subsequently published a violent polemic that displayed a vitriol far beyond anything explained by scientific disagreements. Einstein also engaged in a bitter debate with Max Abraham, whose own gravitational theory Einstein had attacked with great relish throughout 1912. Report on the Vienna conference, Sept. 23, 1913, CPAE 4: 17.
32. Einstein to Heinrich Zangger, ca. Jan. 20, 1914.
33. Einstein to Heinrich Zangger, Mar. 10, 1914. Jürgen Renn has pointed out that the 1913–1915 period of defending and refining the Entwurf, even though it did not save that theory, did help Einstein to better understand the difficulties that seemed to bedevil t
he tensors he had explored in the mathematical strategy. “Practically all of the technical problems Einstein had encountered in the Zurich notebook with candidates derived from the Riemann tensor were actually resolved in this period in the course of his examination of problems associated with the Entwurf theory.” Renn 2005b, 16.
34. Einstein to Erwin Freundlich, Jan. 8, 1912, mid-Aug. 1913; Einstein to George Hale, Oct. 14, 1913; George Hale to Einstein, Nov. 8, 1913.
35. Clark, 207.
36. Einstein to Erwin Freundlich, Dec. 7, 1913.
37. Einstein to Erwin Freundlich, Jan. 20, 1914.
38. Fölsing, 356–357.
39. Einstein to Paul Ehrenfest, Aug. 19, 1914.
40. Ibid.
41. Einstein to Paolo Straneo, Jan. 7, 1915.
42. For a good description from which this is drawn, see Levenson, especially 60–65.
43. Elon, 277, 303–304.
44. Fölsing, 344.
45. Einstein to Hans Albert Einstein, Jan. 25, 1915.
46. Nathan and Norden, 4; Elon, 326. Also translated as the “Manifesto to the Civilized World.”
47. Einstein to Georg Nicolai, Feb. 20, 1915. The full text is in CPAE 6: 8, and Nathan and Norden, 5. Clark, 228, makes the case that some of the writing was Einstein’s. See also Wolf William Zuelzer, The Nicolai Case (Detroit: Wayne State University Press, 1982); Overbye, 273; Levenson, 63; Fölsing, 346–347; Elon, 328.
48. Nathan and Norden, 9; Overbye, 275–276; Fölsing, 349; Clark, 238.
49. Einstein to Romain Rolland, Sept. 15, 1915; CPAE 8a: 118 (German edition), footnote 2; Romain Rolland diary, cited in Nathan and Norden, 16; Fölsing, 366.
50. Einstein to Paul Hertz, before Oct. 8, 1915; Paul Hertz to Einstein, Oct. 8, 1915; Einstein to Paul Hertz, Oct. 9, 1915.
51. Einstein, “My Opinion on the War,” Oct. 23–Nov. 11, 1915, CPAE 6: 20.
52. Einstein to Heinrich Zangger, after Dec. 27, 1914, CPAE 8: 41a, in supplement to vol. 10.
53. Hans Albert Einstein to Einstein, two postcards, before Apr. 4, 1915, part of the family correspondence trust that was under seal until 2006. CPAE 8: 69a, 8: 69b, in supplement to vol. 10.
54. Einstein to Hans Albert Einstein, ca. Apr. 4, 1915.
55. Einstein to Heinrich Zangger, July 16, 1915.
56. Einstein to Elsa Einstein, Sept. 11, 1915; Einstein to Heinrich Zangger, Oct. 15, 1915; Einstein to Hans Albert Einstein, Nov. 4, 1915. For Einstein’s complaint that he was barely able to see his boys during the Sept. 1916 visit, see Einstein to Mileva Mari, Apr. 1, 1916: “I hope that this time you will not again withhold the boys almost entirely from me.”
57. Einstein to Heinrich Zangger, Oct. 15, 1915; Michele Besso to Einstein, ca. Oct. 30, 1915.
58. Once again, I have drawn on the works of Jürgen Renn, Tilman Sauer, John Stachel, Michel Janssen, and John D. Norton.
59. Horst Kant, “Albert Einstein and the Kaiser Wilhelm Institute for Physics in Berlin,” in Renn 2005d, 168–170.
60. Wolf-Dieter Mechler, “Einstein’s Residences in Berlin,” in Renn 2005d, 268.
61. Janssen 2004, 29.
62. Einstein to Heinrich Zangger, July 7, ca. July 24, 1915; Einstein to Arnold Sommerfeld, July 15, 1915.
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. Ein
stein 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.
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