Nonlinear Connections and Nearly Autoparallel Maps in General Relativity

TL;DR

This paper develops a framework using nonlinear connections and nearly autoparallel maps to model local anisotropy in general relativity, leading to new solutions and insights into anisotropic gravitational fields.

Contribution

It introduces the theory of nearly autoparallel maps and tensorial na-integration, enabling redefinition of Einstein gravity on na-backgrounds with invariant conditions.

Findings

Constructed new solutions of Einstein equations with local anisotropy.

Developed the theory of na-maps and tensorial na-integration.

Generated vacuum Einstein fields using Finsler-like metrics.

Abstract

We apply the method of moving anholonomic frames, with associated nonlinear connections, in (pseudo) Riemannian spaces and examine the conditions when various types of locally anisotropic (la) structures (Lagrange, Finsler like and more general ones) could be modeled in general relativity. New classes of solutions of the Einstein equations with generic local anisotropy are constructed. We formulate the theory of nearly autoparallel (na) maps and introduce the tensorial na-integration as the inverse operation to both covariant derivation and deformation of connections by na-maps. The problem of redefinition of the Einstein gravity theory on na-backgrounds, provided with a set of na-map invariant conditions and local conservation laws, is analyzed. There are illustrated some examples of generation of vacuum Einstein fields by Finsler like metrics and chains of na-maps.