7 5 8 D O C . 4 8 0 O N K A L U Z A ’ S T H E O R Y , P A R T 2 Published in Preußische Akademie der Wissenschaften (Berlin). Physikalisch-mathematische Klasse. Sitzungsberichte (1927): 26–30. Presented 17 February 1927, published 14 March 1927. Three man- uscript fragments are available: [5 065] for pp. 1–4 of the manuscript page 5 of [1 061] for the last page of the manuscript [1 068] for the addendum. [1]In what follows, as in the predecessor paper, Einstein 1927i (Doc. 459), Latin indices run from 1 to 4, whereas Greek indices run from 1 to 5 thus, represents the metric tensor of a five- dimensional spacetime, and the metric tensor of a four-dimensional spacetime. The question Einstein alludes to here is how the four-dimensional spacetime of “ordinary” general relativity is embedded in the five-dimensional spacetime. Kaluza 1921 had assumed that the four-dimensional metric of general relativity corresponds to the components to of the five-dimensional metric . However, if instead one starts by assuming the restricted group of five-dimensional coordinate transformations that had been identified in eq. (4) of Einstein 1927i (Doc. 459), one finds that is not an invariant under this symmetry group, but is. Klein, O. 1926 had identi- fied the same invariant, together with the other two invariants that Einstein gives in the first equation below. [2]Einstein introduced the sharpened cylinder condition in Einstein1927i (Doc. 459) see its note 4. [3]Klein, O. 1926, p. 896, also mentions the possibility that only the ratios of five-dimensional met- ric components, rather than their absolute value, have physical significance. He assumed this to jus- tify setting Einstein aims to justify the same on the following page (specifically that ). Einstein had considered a conformal theory, i.e., a theory in which only the ratios of metric components have physical significance, in Einstein 1921e (Vol. 7, Doc. 54) see also Vol. 12, Intro- duction, p. xlix, for references to the relevant correspondence. [4]Einstein’s eq. (8) is essentially the same as eqs. (7) and (10) of Klein, O. 1926, p. 897. The only difference is that Klein introduces two constants in front of the second term of the equation for . One of them is , which Einstein sets to 1 the other is , where κ is Einstein’s grav- itational constant. [5]When Einstein speaks of the “antisymmetric derivatives of ,” he probably means the tensor given in eq. (7) of Einstein 1927i (Doc. 459), i.e., the five-dimensional representation of the electro- magnetic field tensor. [6]The manuscript says , as it should be the typesetter made a mistake. [7]Fock 1926, sec. B, gives a more detailed discussion of the equations of motion of charged mass points as represented by five-dimensional geodesics. Kaluza 1921 had already provided a similar rep- resentation, but the idea seems due to Einstein, who had suggested it after having read a draft of Kaluza’s paper (see Einstein to Theodor Kaluza, 28 April 1919 [Vol. 9, Doc. 30]). In the published paper, Kaluza restricted the link between five-dimensional geodesics and the four-dimensional paths of charged particles subject to electromagnetic fields to small velocities and small specific charges. He noted that in the absence of this constraint the coupling of to the charge of the particle dom- inates the other terms in the equations of motion. He acknowledges Einstein for having pointed out this difficulty. Indeed, it was because of it that Einstein had retracted his offer to present (an abridged γμν gmn γ11 γ44 γμν γmn γmn γ00 -------- γ0m γ00 --------------- γ0n γ00 – gmn = γ00 const = γ00 1= gmn α γ00 = β 2κ α ------ = φm s 0 ≠ γ00