On the Kaluza-Klein geometric theory in affine spaces

TL;DR

This paper extends Kaluza-Klein theory into a purely affine geometric framework, deriving Einstein-Maxwell equations and suggesting that the metric can emerge dynamically from affine and electromagnetic structures.

Contribution

It introduces a purely affine formulation of Kaluza-Klein theory using principal fiber bundles and Ehresmann connections, without assuming a metric.

Findings

Derivation of Einstein-Maxwell system from affine geometry

Electromagnetic field linked to non-integrability of horizontal distribution

Proposal that the metric can emerge dynamically from affine and electromagnetic fields

Abstract

In this work, we develop a generalization of Kaluza-Klein theory by considering a purely affine framework, without assuming a prior metric structure. We formulate the dimensional reduction using the geometry of principal fiber bundles and the Ehresmann connection, introducing adapted bases that allow an explicit decomposition of tensors, vectors, and connections. This formalism provides a natural geometric definition of the electromagnetic field as the difference between the horizontal space and the space generated by the observer's frame. We demonstrate that the presence of a nontrivial electromagnetic field requires the non-integrability of the horizontal distribution, and we derive a complete ansatz for decomposing the affine connection into fields defined on the reduced space. Under assumptions such as vanishing torsion, autoparallel fibers, and suitable normalization conditions, we show that the reduced theory corresponds to the Einstein-Maxwell system for purely radiative electromagnetic fields. Furthermore, we propose an interpretation where the metric emerges dynamically from the affine structure through the dynamics of the electromagnetic field.