Dynamical mechanisms for Kaluza-Klein theories

TL;DR

This paper develops variational formulations for gauge and Einstein--Yang-Mills theories within Kaluza-Klein frameworks, showing how higher-dimensional spaces can be related to four-dimensional physical space-times through topological and geometric assumptions.

Contribution

It introduces a general variational approach to Kaluza-Klein theories without assuming a fibration, linking higher-dimensional fields to four-dimensional space-time structures.

Findings

Classical solutions construct four-dimensional space-time from higher-dimensional fields.

For compact, simply connected structure groups, the higher-dimensional space admits a principal bundle structure.

The Einstein-Maxwell system can be derived when fibers are circles, indicating a fiber bundle structure in five dimensions.

Abstract

We present variational formulations of gauge theories and Einstein--Yang-Mills equations in the spirit of Kaluza-Klein theories. For gaugetheories, only a topological fibration is assumed. For gravitation coupled with gauge fields, no fibration is assumed: Fields are defined on a 'space-time' $Y$ of dimension $4 + r$ without any structure a priori, where $r$ is the dimension of the structure group. If the latter is compact and simply connected, classical solutions allow to construct a manifold $X$ of dimension 4 to be the physical space-time, in such a way that $Y$ acquires the structure of a principal bundle over $X$ and leads to solutions of the Einstein--Yang-Mills systems. The special case of the Einstein-Maxwell system is also discussed: It suffices that at least one fiber closes in on a circle to deduce that the five-dimensional space-time has afiber bundle structure.