Studying conformally flat spacetimes with an elastic stress energy tensor using 1+3 formalism

TL;DR

This paper analyzes conformally flat spacetimes with elastic stress energy tensors using 1+3 formalism, deriving relations between Ricci components and pressures, and solving for non-rotating cases with ODEs.

Contribution

It introduces a detailed 1+3 formalism approach to conformally flat spacetimes with elastic stress energy, including relations and solutions for specific cases.

Findings

Derived relations between Ricci components and pressure tensors.

Formulated Einstein equations for elastic stress energy in conformally flat spacetimes.

Solved for non-rotating case using ODEs.

Abstract

Conformally flat spacetimes with an elastic stress energy tensor given by a diagonal trace-free anisotropic pressure tensor are investigated using 1+3 formalism. We show how the null tetrad Ricci components are related to the pressure components and energy density. The 1+3 Bianchi and Jacobi identities and Einstein field equations are written for this particular case. In general the commutators must be considered since they supply potentially new information on higher order derivatives of the 1+3 quantities. We solve the system for the non rotating case which consist of ODEs of a spatial coordinate.