D O C U M E N T 4 5 9 K A L U Z A’ S T H E O R Y, PA R T 1 4 5 1 459. “On Kaluza’s Theory on the Connection between Gravitation and Electricity: Part 1” [Einstein 1927i] Presented 20 January 1927 Published 14 March 1927 In: Preußische Akademie der Wissenschaften (Berlin). Physikalisch-mathematische Klasse. Sitzungsberichte (1927): 23–25. Ever since the general theory of relativity was found, theoreticians have been in- cessantly trying to bring the laws of gravitation and electricity under a unified point of view. Weyl and Eddington sought to reach this goal by a generalization of Rie- mannian geometry using a general approach for the parallel displacement of vectors.[1] Kaluza, however, proceeded fundamentally differently.1) He keeps the Riemann metric, but makes use of a continuum of five dimensions, which he, in a sense, reduces to a continuum of four dimensions by means of the “cylinder condition.”[2] Here I shall present an as of yet unconsidered aspect that is essential to Kaluza’s theory. We set out from a five-dimensional continuum of . In it there exists a Riemann metric with the line element . (1) This continuum is “cylindrical,” i.e., there exists an infinitesimal displacement vec- tor ( ), which maps the metric onto itself in the following sense:[3] If the starting point of a line element ( ) is displaced by ( ) and the end point by , then the displaced line element should have, according to (1), the same length as the non-displaced one. This means that the equations (2) can be satisfied with a suitable choice of . 1) Zum Unitätsproblem der Physik, Berl. Berichte 1921, p. 966. [p. 23] x1, x2, x3, x4, x0 dσ2 γμνdxμdxν = ξα dxμ ξμ ξμ)x ( dx + ∂xβ ∂γμν ξβ γβν--------+ ∂ξβ ∂xμ γβμ-------- ∂ξβ ∂xν + 0 = ξβ
Previous Page Next Page