D O C U M E N T 3 8 7 C O M M E N T O N T R E F F T Z 5 9 7 Published in Preußische Akademie der Wissenschaften (Berlin). Physikalisch-mathematische Klasse. Sitzungsberichte (1922): 448–449. Submitted 23 November 1922, published 21 December 1922. [1]Erich Trefftz (1888–1937) was Professor of Mathematics at the Technical University of Aachen from 1919 until 1922, and at the Technische Hochschule in Dresden from 1922 for a short biography and comprehensive bibliography, see Stein 1997. Trefftz 1922 was received by the Mathematische Annalen (of which Einstein was a board member) on 15 September 1921. [2]As elaborated in Einstein 1919a (Vol. 7, Doc. 17), p. 351, the equations are, in fact, not equiva- lent. Validity of eq. (1a) implies validity of eq. (1) by Einstein’s argument, but eq. (1) can be rewritten as , with some constant , whereas eq. (1a) implies that the right-hand term of this equation would vanish. Nevertheless, on p. 353, Einstein argued that mul- tiplying the rewritten version of eq. (1) with ½ and substracting from eq. (1a), the cosmological con- stant λ occurring in eq. (1a) can be identified with , where is interpreted as the constant value that the curvature scalar R is supposed to take on outside of material bodies. This correspondence accounts for the fact that the metric presented by Trefftz is a solution both to eq. (1) and eq. (1a). It is in this sense that the two sets of equations are “equivalent” for the purpose at hand. [3]The solution presented in Trefftz 1922 was independently discovered by Weyl (see Weyl 1919a and Bach and Weyl 1922, sec. 4) and by Kottler (see Kottler 1922, pp. 207–210). According to Birk- hoff’s theorem, the Kottler-Trefftz-Weyl (KTW) solution is the only spherically symmetric and static solution to Einstein’s field equations with the cosmological constant but with a vanishing energy- momentum tensor (see Goenner 2001, p. 112). Mathematically, it thus plays the same role for the modified field equations from Einstein 1917b (Vol. 6, Doc. 43) and 1919a (Vol. 7, Doc. 17) that the Schwarzschild solution plays for the original Einstein field equations. In this paper, Einstein does not deny that the KTW solution is a valid solution to the field equations he challenges Trefftz’s interpre- tation of the solution, namely, interpreting it as describing two bodies that interact gravitationally and remain static. For a detailed mathematical discussion of the KTW solution involving modern tech- niques, see Geyer 1980. [4]As a result of the debate among Einstein, Willem de Sitter, Felix Klein, and Hermann Weyl, Ein- stein had come to accept that De Sitter’s solution to the cosmological field equations is matter-free, homogeneous, and fully regular, thus violating Mach’s principle (see Vol. 8, the editorial note, “The Einstein-De Sitter-Weyl-Klein Debate,” pp. 351–357). Einstein nevertheless continued to reject De Sitter’s solution as an acceptable cosmological model since it is not globally static. In Einstein 1922p (Doc. 340), Einstein also pointed out that Friedmann’s nonstatic solution looked “suspicious” (“ver- dächtig”) to him, and tried to argue that it is not a dynamic solution after all. For Friedmann’s rejection of Einstein’s argument and Einstein’s retraction, see Doc. 340, note 5. [5]“Konstante, )” should be “Konstante), .” [6]Trefftz disagreed with Einstein’s argument at this point, writing to Max von Laue: “It is not obvi- ous to me that the circumference of the mentioned sphere has to have a maximum in between the two spheres this seems to be an unwarranted inference by analogy to Euclidean geometry. Why should the field not behave in the following way, which would be consistent with my calculations (presup- posing that the field can be extended into the interior of the massive spheres): ‘The circumference of a sphere grows up to the surface of the smaller sphere, then further up to the surface of the bigger sphere, and only in the interior of the bigger sphere does it decrease back to zero’” (“[E]s ist meiner Ansicht nach nicht offenbar, daß der genannte Kugelumfang zwischen den beiden Kugeln ein Maxi- mum haben muß, das scheint mir ein unzulässiger Analogieschluß nach der Euklidischen Geometrie zu sein. Warum soll das Feld sich nicht folgendermaßen verhalten, wie es sich mit meinen Rechnun- gen [die Möglichkeit der Fortsetzung des Feldes in das innere der Massenkugeln vorausgesetzt] ver- tragen würde: ‘Der Umfang einer Kugel wächst, bis an die Oberfläche der kleineren Kugel, dann weiter bis an die Oberfläche der größeren Kugel, und nimmt erst im Inneren der größeren Kugel wie- der zu Null ab’” (Trefftz to Max von Laue, 5 June 1923, [23 050]). The cosmological interpretation of Trefftz’s solution was further discussed in Laue 1923. [7]The equation should read: . [8]Compare the discussion in Havas 1993, p. 91. Riκ 1 2 --giκR - – 1 4 --giκR0 - + 1 4 --giκ( - R R0) – –= R0 R0 4 ----- - R0 f2(x) f2(x) dw dx ------ - 1 A w --- - Bw2 + + 0 = =