Incorporation of Spacetime Symmetries in Einstein's Field Equations

TL;DR

This paper introduces a covariant method to incorporate spacetime symmetries into Einstein's field equations, facilitating the derivation of exact solutions and offering new insights in General Relativity.

Contribution

It develops a general, covariant formalism using double congruences to geometrize spacetime symmetries within Einstein's equations, applicable to various matter fields.

Findings

Method fully incorporates spacetime symmetries into field equations.

Application to spacelike conformal Killing vectors provides new solution approaches.

Offers new perspectives in the study of General Relativity.

Abstract

In the search for exact solutions to Einstein's field equations the main simplification tool is the introduction of spacetime symmetries. Motivated by this fact we develop a method to write the field equations for general matter in a form that fully incorporates the character of the symmetry. The method is being expressed in a covariant formalism using the framework of a double congruence. The basic notion on which it is based is that of the geometrisation of a general symmetry. As a special application of our general method we consider the case of a spacelike conformal Killing vector field on the spacetime manifold regarding special types of matter fields. New perspectives in General Relativity are discussed.