Anisotropic fluid in a time dependent conformally flat spacetime

TL;DR

This paper constructs a time-dependent conformally flat spacetime using special conformal transformations, revealing an anisotropic fluid with negative energy density as a source, and relates it to near-horizon Schwarzschild geometry.

Contribution

It introduces an exact solution of Einstein's equations with a novel anisotropic fluid source derived from conformal transformations.

Findings

Generated a new time-dependent conformally flat spacetime

Identified an anisotropic fluid with negative energy density as source

Connected static approximation to near-horizon Schwarzschild geometry

Abstract

The special conformal transformation (composed by inversion - translation - inversion) is used to generate a time dependent conformally flat spacetime. In order to be an exact solution of Einstein's equations, we need as a source a stress tensor corresponding to an anisotropic fluid with negative regular energy density and positive pressures. For the static approximation, the generators of the infinitesimal transformation resemble those recently obtained by Majhi and Padmanabhan for the coordinate transformation leading to the "near horizon" Schwarzschild metric in Kruskal coordinates. The static approximation corresponds to an energy momentum tensor of $\Lambda$ - form, the "cosmological constant" $\Lambda$ being proportional to the acceleration squared.