4 7 4 D O C U M E N T 4 8 0 K A L U Z A S T H E O R Y : P A R T 2 480. “On Kaluza’s Theory on the Connection between Gravitation and Electricity: Part 2” [Einstein 1927j] Presented 17 February 1927 Published 14 March 1927 In: Preußische Akademie der Wissenschaften (Berlin). Physikalisch-mathematische Klasse. Sitzungsberichte (1927): 26–30. I shall present the results of further deliberations that seem to speak very much in favor of Kaluza’s ideas. The deviation from Kaluza’s considerations is purely formal. It comes from the fact that Kaluza treated the as components of the metric tensor in R4 instead of the [1] the reason is that he did not take note of the invariance properties that result from the cylinder condition.1) §1. An Interpretation of the “Sharpening” of the Cylinder Condition Even if one assumes the non-sharpened cylinder condition in the metric space R5 ,[2] one only has to require an invariance under -transformations (4) for a fixed , in addition to the invariance with respect to arbitrary substitutions of the . Thus covariance of the equations with respect to (5) must be re- quired, where must be regarded as a function of . (5) implies the invar- iance of the following quantities: But the cylinder condition requires that the Hamiltonian function contain the only in these three combinations. Now, if one assumes that not the themselves, but only the ratios of the have objective meaning, or—expressed differently—if in the space not the met- ric ( ),[3] but only the totality of the “null cones” ( ) is given, then the Hamiltonian function may depend only on the first two of the above combinations. It then does not mean any specialization of the theoretical foundations if we set we thus arrive at the “sharpened cylinder condition.” 1) To facilitate the exposition, the following will be appended to the first communication in di- rect continuation (notation, enumeration of the equations). [p. 26] γmn gmn x0 x0 x1, x2, x3, x4) ( γ00 x1, ...x4 γmn γ00 ------- - γ0m γ00 ------- ------- γ0n γ00 - ∂xn© γ0m γ00 -------· - ¹ § ∂xm© γ0n- γ00¹ ------· § γ00 γμν γμv γμv R5 dσ2 dσ2 0= [p. 27] γ00 1=
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