1 7 6 D O C . 9 1 G E N E R A L R E L A T I V I T Y A N D M O T I O N Published in Preußische Akademie der Wissenschaften (Berlin). Physikalisch-mathematische Klasse. Sitzungsberichte (1928): 235–245. Presented 24 November 1927. A manuscript is also available [1 066]. [1] Einstein and Grommer 1927 (Vol. 15, Doc. 443). [2] Weyl had alerted Einstein to his previous work on the topic in Weyl 1923, sec. 38, after reading the page proofs of Einstein and Grommer 1927 (Vol. 15, Doc. 443) see H. Weyl to Einstein, 3 Feb- ruary 1927 and Einstein to Weyl, 26 April 1927 (Vol. 15, Docs. 473, 514). [3] In the manuscript, the reference to Lanczos 1927 is missing. [4] In §1 of Einstein and Grommer 1927 (Vol. 15, Doc. 443), the authors first reformulated the vacuum Einstein equations as a surface integral, and then argued that only if a certain equilibrium condition is satisfied for stationary fields surrounding the particle, then its equations of motion follow from the field equations (see note 31 to that paper). [5] See §3, p. 11 of Einstein and Grommer 1927 (Vol. 15, Doc. 443) for the attempt to generalize the previous derivation of geodesic motion of a singularity subject to a nonstationary external gravi- tational field. To simplify the calculation they had to make the additional hypothesis that the singu- larity is spherically symmetric. [6] In Einstein and Grommer 1927 (Vol. 15, Doc. 443), the authors only mention the connection to quantum theory in the final sentence of the paper. However, Einstein had hoped all along to produce results that would be helpful for a quantum theory of matter (see Einstein to Michele Besso, 11 August 1926 to Paul Ehrenfest, 11 January 1927 and to Hermann Weyl, 26 April 1927 [Vol. 15, Docs. 348, 450, 514] and Lehmkuhl 2019 for a detailed analysis) see also the following note. [7] This paragraph replaces the following in the manuscript: “Im folgenden wird gezeigt, dass die Feld-theorie eine Verallgemeinerung des Bewegungsgesetzes ermöglicht und nahelegt, welche es auf naheli dem Elektron eine Eigenfrequenz und ein magnetisches Eigenfeld zuschreibt. Es wird nun klar, inwieweit Annahmen über den Charakter einer in einem äusseren Felde sich bewegende Singularität möglich und richtig sind.” [8] See §3, p. 10 of Einstein and Grommer 1927 (Vol. 15, Doc. 443), for the same assumption of staticity and spherical symmetry giving the “character of the singularity” cf. note 5 above. [9] Einstein had come to question the relationship between approximate solutions and exact solu- tions as a result of his correspondence with George Yuri Rainich (see Vol. 15, Introduction, sec. II, and Einstein and Grommer 1927 [Vol. 15, Doc. 443], note 14). [10] Right before equation (4) in the manuscript, Einstein deleted “welche bekanntli Gesucht wird eine Lösun.” [11] Here, should be , as in equation (3). [12] Einstein might refer to results of Rainich, which the latter wrote to Einstein about first on 25 October 1925 (Vol. 15, Doc. 96). Rainich summarized his results as showing that the Ricci tensor in eq. (1) determines the electromagnetic field up to an integration constant (see Vol. 15, Intro- duction, sec. II, for more details and sources). [13] By eq. (1), the vanishing of the trace of the energy-momentum tensor implies the van- ishing of the Ricci scalar R. Thus eq. (1) becomes eq. (1a). [14] In the manuscript, “zentralsymmetrischen” replaces “punktsymmetrischen.” Einstein had first formulated the aim of finding static and spherically symmetric solutions in order to represent elec- trons in Einstein and Grommer 1923a, 1923b (Vol. 13, Doc. 12, note 9). The difference in the present paper is that he looks for solutions that are “infinitely close” to such an electron solution. [15] As discussed further below on p. 242, the convergence of the series is problematic because , as specified in eq. (13) diverges at the origin. [16] In the manuscript, the term in curly brackets replaces a similar replacement was made in equation (9) below. [17] The second term on the right hand side should carry an index . In the manuscript, an addi- tional term was deleted. fik ik R ik T Ti i = g ik -- 1 2 i g k x --------- - k g i x ---------- - ik g x --------- – + 1 2 ik – –