Split structures in general relativity and the Kaluza-Klein theories

TL;DR

This paper develops a general geometric framework for decomposing tangent bundles of pseudo-Riemannian manifolds, unifying various decomposition methods used in general relativity and Kaluza-Klein theories, and applies it to physical models.

Contribution

It introduces an invariant, general approach to tangent bundle decomposition, encompassing all known types as special cases, and applies it to multidimensional theories and relativistic fluid configurations.

Findings

Derived invariant decomposition relations including Gauss-Codazzi-Ricci relations.

Unified various decomposition methods in a single geometric framework.

Applied the framework to Kaluza-Klein theories and relativistic fluid models.

Abstract

We construct a general approach to decomposition of the tangent bundle of pseudo-Riemannian manifolds into direct sums of subbundles, and the associated decomposition of geometric objects. An invariant structure {\cal H}^r defined as a set of r projection operators is used to induce decomposition of the geometric objects into those of the corresponding subbundles. We define the main geometric objects characterizing decomposition. Invariant non-holonomic generalizations of the Gauss-Codazzi-Ricci's relations have been obtained. All the known types of decomposition (used in the theory of frames of reference, in the Hamiltonian formulation for gravity, in the Cauchy problem, in the theory of stationary spaces, and so on) follow from the present work as special cases when fixing a basis and dimensions of subbundles, and parameterization of a basis of decomposition. Various methods of decomposition have been applied here for the Unified Multidimensional Kaluza-Klein Theory and for relativistic configurations of a perfect fluid. Discussing an invariant form of the equations of motion we have found the invariant equilibrium conditions and their 3+1 decomposed form. The formulation of the conservation law for the curl has been obtained in the invariant form.