Exact plane symmetric black bounce with a perfect-fluid exterior obeying a linear equation of state

TL;DR

This paper presents an exact family of plane symmetric solutions in general relativity with a perfect fluid obeying a linear equation of state, exploring their horizons, extensions, and conditions for regular black bounce or black hole configurations.

Contribution

It identifies all regular attachments of Gamboa solutions at horizons and constructs their maximal extensions, revealing conditions for black bounce and black hole solutions in higher dimensions.

Findings

Maximal extension describes either a regular black bounce or a black hole with a singularity.

Null energy condition is violated except on the horizon.

Regularity depends on specific parameter fine-tuning and the equation of state parameter .

Abstract

We investigate an exact two-parameter family of plane symmetric solutions admitting a hypersurface-orthogonal Killing vector in general relativity with a perfect fluid obeying a linear equation of state $p=\chi\rho$ in $n(\ge 4)$ dimensions, obtained by Gamboa in 2012. The Gamboa solution is identical to the topological Schwarzschild-Tangherlini-(anti-)de~Sitter $\Lambda$-vacuum solution for $\chi=-1$ and admits a nondegenerate Killing horizon only for $\chi=-1$ and $\chi\in[-1/3,0)$. We identify all possible regular attachments of two Gamboa solutions for $\chi\in[-1/3,0)$ at the Killing horizon without a lightlike thin shell, where $\chi$ may have different values on each side of the horizon. We also present the maximal extension of the static and asymptotically topological Schwarzschild-Tangherlini Gamboa solution, realized only for $\chi\in(-(n-3)/(3n-5),0)$, under the assumption that the value of $\chi$ is unchanged in the extended dynamical region beyond the horizon. The maximally extended spacetime describes either (i) a globally regular black bounce whose Killing horizon coincides with a bounce null hypersurface or (ii) a black hole with a spacelike curvature singularity inside the horizon. The matter field inside the horizon is not a perfect fluid but rather an anisotropic fluid that can be interpreted as a spacelike (tachyonic) perfect fluid. A fine-tuning of the parameters is unnecessary for the black bounce, but the null energy condition is violated everywhere except on the horizon. In the black-bounce (black-hole) case, the metric in the regular coordinate system is $C^\infty$ only for $\chi=-1/(1+2N)$ with odd (even) $N$ satisfying $N>(n-1)/(n-3)$, and if one of the parameters in the extended region is fine-tuned.