Maximal hypersurfaces and aspects of volume of the Kerr family of black holes

Abstract

In Schwarzschild spacetime, Reinhart (1973) has shown the hypersurface $r_R = 3M/2$ (the subscript stands for "Reinhart") to be a maximal hypersurface. This Reinhart radius $r_R$ plays a crucial role in evaluating the interior volume of a black hole. In this article, we find such a maximal hypersurface for the Kerr and Kerr-Newaman black holes. We obtain the analytical expression for the Reinhart radius as a function of the polar angle $\theta$ for a small $a/M$ limit for both the Kerr and Kerr-Newman black holes. We obtain the Reinhart radius using two independent methods: a) the vanishing trace of the extrinsic curvature, and b) the variational method. We further use the Reinhart radius to obtain an analytical expression for the interior volume of the Kerr and Kerr-Newman black hole in the small $a/M$ limit and a generic charge $Q$. We define $\mathcal{\dot{V}}$ as the rate of change of the interior volume with respect to the ingoing null coordinate $v$ and then study its behavior under various scenarios, viz., particle accretion, the Penrose process, superradiance, and Hawking radiation. We show that while under the Penrose process and superradiance, the parameter $\mathcal{\dot{V}}$ increases just as the area of a black hole but under particle accretion, $\mathcal{\dot{V}}$ can have variable signs depending on the kinematical properties of the particle. We further probe into the behavior of $\mathcal{\dot{V}}$ under Hawking radiation. These results provide important and very interesting clues toward the possible existence of laws governing the volume of black holes.