D O C . 1 7 G R AV I TAT I O N A N D E L E C T R I C I T Y 6 5 Published in Preußische Akademie der Wissenschaften (Berlin). Physikalisch-mathematische Klasse. Sitzungsberichte (1925): 414–419. An ADft is also available [1 047]. [1]Einstein 1923n (Vol. 14, Doc. 52). [2]In earlier work on affine field theory Einstein had always presupposed symmetry of the affine connection in its lower indices. Dropping this assumption gives rise to the possibility of having tor- sion arising from the connection see “On the General Theory of Relativity” (Vol. 13, Doc. 417), note 4 for details. [3]In his previous work on affine field theory Einstein had always introduced the tensor density , which he will relate to the metric tensor on p. 416, independently of the affine connection , and assumed it to be symmetric. He thus introduces a so-called mixed geometry in this paper, rather than a purely affine theory (see Vizgin 1994, pp. 204–209. Goenner 2004, p. 19 and sec. 6.1, and Goenner 2005, sec. 7). He had already toyed with this possibility in early 1923: in his calculations contained in the Japan Trip Diary he considered both an asymmetric connection and an independent metric (Vol. 13, Doc. 418) and in the draft (Vol. 13, Doc. 417) of Einstein 1923e (Vol.13, Doc. 425). However, in the latter published paper he had reverted to a purely affine geometry with a symmetric connection and a (derivative) symmetric metric tensor. The idea of building a unified field theory on the basis of an asymmetric metric tensor was earlier considered by Rudolf Förster (who published under the pseudonym R. Bach) see the extensive cor- respondence between Förster and Einstein in Vol. 8, starting with Förster to Einstein, 11 November 1917 (Vol. 8A, Doc. 398). Reichenbächer 1923 also advocated an asymmetric metric tensor as the basis for a unification of gravitation and electromagnetism. He presented the approach at the 1922 GDNÄ meeting in Leipzig, which Einstein did not attend (see Einstein to Max Planck, 6 July 1922 [Vol. 13, Doc. 266] for the reasons). [4]The expression should be “16 Gleichungen.” [5]The expression should be rather than . [6]Joseph Thomas showed (see Thomas 1926) that Eq. (10a) can be interpreted as a covariant deriv- ative corresponding to the asymmetric connection . He also showed that Einstein’s setting the vector to zero further below can be seen as a result of setting the contraction of the anti- symmetric part of the connection to zero. Finally, Thomas showed how modifying the definition of covariant derivation implicit in Eq. (10a) relates the geometry employed here to Weyl’s geometry. [7]The expression should be rather than . In his previous papers, including Einstein 1923n (Vol. 14, Doc. 52), Einstein had followed Eddington in identifying the antisymmetric part of the Ricci tensor with the electromagnetic field. In the present document he instead identifies the electromag- netic field with the antisymmetric part of an asymmetric metric, a possibility that Einstein had first been alerted to by Förster (see note 3). [8]In Einstein 1924d (Vol. 14, Doc. 170), Einstein had formulated a research program according to which one should look for unified field equations that would overdetermine the field variables. He hoped that one could thereby obtain solutions that exhibit discrete features, e.g., electric charge and rest mass of only particular values. For earlier occurrences of this idea in Einstein’s thought, Vol. 14, Doc. 170, note 8. [9]Einstein will reconsider the relationship between time reversal transformations and the assign- ment of electric and magnetic field components to the components of the Faraday tensor in Einstein 1925w (Doc. 92). [10]The criterion that a satisfactory unified field theory of gravitation and electromagnetism has to give a centrally symmetric solution free of singularities that can be interpreted as representing an elec- tron was first formulated in Einstein and Grommer 1923a and 1923b (Vol. 13, Doc. 12). Einstein con- tinued to use this criterion as a litmus test in his work on affine unified field theories following Eddington see Vol. 13, Doc. 417, Einstein 1923e (Vol. 13, Doc. 425), Einstein 1923n (Vol. 14, Doc. 52), Vol. 14, Doc. 122, and Einstein 1925a (Vol. 14, Doc. 282). gμν gμν Γμαβ gμν gμν Γμαβ φα μ α = gμν gμν
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