A mathematicians' view of geometrical unification of classical physics in high-dimensional space-time

Abstract

We propose in this paper a mathematicians' view of the Kaluza-Klein idea of a five dimensional space-time unifying gravitation and electromagnetism, and extension to higher-dimensional space-time. By considering the classification of positive Einstein curvature tensors and the classical Cauchy-Choquet-Bruhat theorems in general relativity, we introduce concepts of types and rigidity. Then, abandoning the usual requirement of a Ricci-flat five dimensional space-time, we show that a unified geometrical frame can be set for gravitation and electromagnetism, giving, by projection on the classical 4-dimensional space-time, the known Einstein-Maxwell-Lorentz equations for charged fluids. Thus, although not introducing, at least at this stage, new physics, we get a very aesthetic presentation of classical physics in the spirit of general relativity. The usual physical concepts, such as mass, energy, charge, trajectory, Maxwell-Lorentz law, are shown to be only various aspects of the geometry, for example curvature, of space-time considered as a Lorentzian manifold; that is no physical objects are introduced in space-time, no laws are given, everything is only geometry. We will then extend this setting to more than 5 dimensions, giving a precise mathematical frame for possible additional physical effects, preserving gravitation and electromagnetism. Version 22 04 2013. 1

A new approach to Kaluza-Klein Theory

“…The usual physical concepts, such as mass, energy, charge, trajectory, Maxwell-Lorentz law, are shown to be only various aspects of the geometry, for example curvature, of space-time considered as a Lorentzian manifold; that is no physical objects are introduced in space-time, no laws are given, everything is only geometry.We then extend these ideas to more than 5 dimensions, by considering spacetime as a generalization of a (S 1 × W )-fiber bundle, that we named multi-fibers bundle, where S 1 is the circle and W a compact manifold. We will use this geometric structure as a possible way to model or encode deviations from standard 4-dimensional General Relativity, or "dark" effects such as dark matter or energy.This paper is a rewriting and improvement of a previous joint work, [20], with B. Vaugon, M. Dellinger, Z. Faget. 1 1. dF F F = 0, i.e F F F is closed, 2.…”

“…This paper is a rewriting and improvement of a previous joint work, [20], with B. Vaugon, M. Dellinger, Z. Faget. 1 1. dF F F = 0, i.e F F F is closed, 2.…”