On the conformal transformation between two anisotropic fluid spacetimes

TL;DR

This paper explores the conditions under which two anisotropic fluid spacetimes, solutions to Einstein's equations, can be related by a conformal transformation, revealing that such transformations preserve certain geometric properties.

Contribution

It establishes that conformal transformations between these spacetimes require the conformal factor to be constant on specific hypersurfaces, especially when one spacetime generalizes the Robertson-Walker solution.

Findings

Conformal factor must be constant on hypersurfaces orthogonal to fluid flow.

If one spacetime is a Robertson-Walker generalization, the other must be as well.

Evolution equation for the conformal factor derived from Ricci tensor transformation.

Abstract

We investigate the necessary conditions for the two spacetimes, which are solutions to the Einstein field equations with an anisotropic matter source, to be related to each other by means of a conformal transformation. As a result, we obtain that if one of such spacetimes is a generalization of the Robertson-Walker solution with vanishing acceleration and vorticity, then the other one has to be in this class as well, i.e. the conformal factor will be a constant function on the hypersurfaces orthogonal to the fluid flow lines. The evolution equation for this function appears naturally as a direct consequence of the conformal transformation of the Ricci tensor.