Black hole in closed spacetime with an anisotropic fluid

TL;DR

This paper explores black hole solutions in closed spacetimes filled with anisotropic fluids, revealing conditions for their existence and classifying solutions through numerical and analytical methods within general relativity.

Contribution

It introduces a new classification of black hole solutions in closed spacetimes with anisotropic fluids, including conditions that avoid naked singularities and negative energy regions.

Findings

Static closed spaces require negative radial pressure less than density.

An exact black hole solution with a closed topology was identified.

Conditions for physically reasonable solutions involve specific inequalities on density and pressures.

Abstract

We study spherically symmetric geometries made of anisotropic perfect fluid based on general relativity. The purpose of the work is to find and classify black hole solutions in closed spacetime. In a general setting, we find that a static and closed space exists only when the radial pressure is negative but its size is smaller than the density. The Einstein equation is eventually casted into a first order autonomous equation on two-dimensional plane of scale-invariant variables, which are equivalent to the Tolman-Oppenheimer-Volkoff (TOV) equation in general relativity. Then, we display various solution curves numerically. An exact solution describing a black hole solution in a closed spacetime was known in Ref. [1], which solution bears a naked singularity and negative energy era. We find that the two deficits can be remedied when $\rho+3p_1>0$ and $\rho+p_1+2p_2< 0$, where the second violates the strong energy condition.