On General Solutions of Einstein Equations

TL;DR

This paper demonstrates a geometric method for constructing general solutions to Einstein equations with cosmological constant and matter sources, including non-Killing solutions, using anholonomic deformations and auxiliary connections.

Contribution

It introduces a novel approach to generate broad classes of exact Einstein solutions via anholonomic deformations and auxiliary connections with torsion, extending to higher dimensions.

Findings

Constructed general non-Killing solutions in 4D and 5D gravity.

Showed how to impose constraints to recover Levi-Civita connections.

Provided explicit parametrizations of off-diagonal metrics.

Abstract

We show how the Einstein equations with cosmological constant (and/or various types of matter field sources) can be integrated in a very general form following the anholonomic deformation method for constructing exact solutions in four and five dimensional gravity (S. Vacaru, IJGMMP 4 (2007) 1285). In this letter, we prove that such a geometric method can be used for constructing general non-Killing solutions. The key idea is to introduce an auxiliary linear connection which is also metric compatible and completely defined by the metric structure but contains some torsion terms induced nonholonomically by generic off-diagonal coefficients of metric. There are some classes of nonholonomic frames with respect to which the Einstein equations (for such an auxiliary connection) split into an integrable system of partial differential equations. We have to impose additional constraints on generating and integration functions in order to transform the auxiliary connection into the Levi-Civita one. This way, we extract general exact solutions (parametrized by generic off-diagonal metrics and depending on all coordinates) in Einstein gravity and five dimensional extensions.