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Journal of Gravity
Volume 2015 (2015), Article ID 901870, 6 pages
http://dx.doi.org/10.1155/2015/901870
Research Article

Field Equations and Lagrangian for the Kaluza Metric Evaluated with Tensor Algebra Software

Konfluence Research Institute, 6009 Olympic, Manitou Springs, CO 80829, USA

Received 1 August 2014; Accepted 2 December 2014

Academic Editor: Shinji Tsujikawa

Copyright © 2015 L. L. Williams. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper calculates the Kaluza field equations with the aid of a computer package for tensor algebra, xAct. The xAct file is provided with this paper. We find that Thiry’s field equations are correct, but only under limited circumstances. The full five-dimensional field equations under the cylinder condition are provided here, and we see that most of the other references miss at least some terms from them. We go on to establish the remarkable Kaluza Lagrangian, and verify that the field equations calculated from it match those calculated with xAct, thereby demonstrating self-consistency of these results. Many of these results can be found scattered throughout the literature, and we provide some pointers for historical purposes. But our intent is to provide a definitive exposition of the field equations of the classical, five-dimensional metric ansatz of Kaluza, along with the computer algebra data file to verify them, and then to recover the unique Lagrangian for the theory. In common terms, the Kaluza theory is an “” scalar field theory, but with unique electrodynamic couplings.