On Einstein's equations for spacetimes admitting a non-null Killing field

TL;DR

This paper generalizes the understanding of Einstein's equations for spacetimes with a non-null Killing field, exploring their structure with matter fields and applications to relativistic dissipative fluids.

Contribution

It extends known vacuum results to arbitrary matter fields and develops systematic methods for analyzing Einstein's equations in these spacetimes.

Findings

Basic field equations are reducible in the presence of matter.

Existence of geometrically preferred reference systems.

Application to equilibrium configurations of relativistic dissipative fluids.

Abstract

We consider the 3-dimensional formulation of Einstein's theory for spacetimes possessing a non-null Killing field $\xi^a$. It is known that for the vacuum case some of the basic field equations are deducible from the others. It will be shown here how this result can be generalized for the case of essentially arbitrary matter fields. The systematic study of the structure of the fundamental field equations is carried out. In particular, the existence of geometrically preferred reference systems is shown. Using local coordinates of this type two approaches are presented resulting resolvent systems of partial differential equations for the basic field variables. Finally, the above results are applied for perfect fluid spacetimes describing possible equilibrium configurations of relativistic dissipative fluids.