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

L L Williams

doi:10.1155/2015/901870

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 “ω=0” scalar field theory, but with unique electrodynamic couplings.

Highlights

In 1921, Kaluza [1] hypothesized that electrodynamics and general relativity could be unified in terms of general relativity extended to five dimensions
The complete field equations for the Kaluza metric were not discovered for another 25 years, generally attributed to Thiry [2], but recently were shown to be somewhat independently discovered by 4 research groups in the 1940s and 1950s: Jordan and colleagues in Germany, Thiry and colleagues in France, Scherrer working alone in Switzerland, and Brans and Dicke at Princeton [3]
This paper removes any ambiguity by calculating the Kaluza field equations with the aid of a computer software package for tensor algebra, xTensor in the xAct package [5]

Summary

Introduction

In 1921, Kaluza [1] hypothesized that electrodynamics and general relativity could be unified in terms of general relativity extended to five dimensions. The complete field equations for the Kaluza metric were not discovered for another 25 years, generally attributed to Thiry [2], but recently were shown to be somewhat independently discovered by 4 research groups in the 1940s and 1950s: Jordan and colleagues in Germany, Thiry and colleagues in France, Scherrer working alone in Switzerland, and Brans and Dicke at Princeton [3]. The Thiry expressions [2] for the field equations have dominated the literature subsequently for the 5D theory, and the Brans-Dicke equations have dominated the literature subsequently for general scalar-tensor theories. The components of the 5-dimensional Ricci tensor provided by Thiry do not self-consistently produce his expression for the 5D Ricci scalar, so one suspects there is a problem with them. This paper removes any ambiguity by calculating the Kaluza field equations with the aid of a computer software package for tensor algebra, xTensor in the xAct package [5]. The xTensor file is provided online [6]

The Metric Ansatz

Comparison with Previous Results