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

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.