On General Solutions for Field Equations in Einstein and Higher Dimension Gravity

TL;DR

This paper introduces a geometric method using anholonomic frame deformations to solve Einstein's equations in arbitrary dimensions, enabling the construction of exact solutions in both standard and higher-dimensional gravity theories.

Contribution

It develops a general solution framework for Einstein equations using nonholonomic frames and alternative connections, broadening the scope of exact solutions in gravity theories.

Findings

Solution method applicable to arbitrary dimensions

Exact solutions for vacuum and non-vacuum cases

Framework includes higher-dimensional gravity models

Abstract

We prove that the Einstein equations can be solved in a very general form for arbitrary spacetime dimensions and various types of vacuum and non-vacuum cases following a geometric method of anholonomic frame deformations for constructing exact solutions in gravity. The main idea of this method is to introduce on (pseudo) Riemannian manifolds an alternative (to the Levi-Civita connection) metric compatible linear connection which is also completely defined by the same metric structure. Such a canonically distinguished connection is with nontrivial torsion which is induced by some nonholonomy frame coefficients and generic off-diagonal terms of metrics. It is possible to define certain classes of adapted frames of reference when the Einstein equations for such an alternative connection transform into a system of partial differential equations which can be integrated in very general forms. Imposing nonholonomic constraints on generalized metrics and connections and adapted frames (selecting Levi-Civita configurations), we generate exact solutions in Einstein gravity and extra dimension generalizations.