The interior volume calculation for an axially symmetric black hole

Abstract

Since an axially symmetric metric is much more complicated than a spherically symmetric metric, the largest hypersurface that corresponds to the interior volume of a black hole proposed by Christodoulou and Rovelli, cannot be found easily. Analogous to a Schwarzschild black hole, the particular hypersurface at constant r in a Kerr black hole has been selected directly. Using the extrinsic curvature of the hypersurface, we investigate how the hypersurface deviates from the largest hypersurface as the black hole angular momentum increases. It is shown that the hypersurface is very close to the largest hypersurface and the spinning of spacetime has pretty limited influence. Therefore, the hypersurface can be regarded as the largest hypersurface approximately in a Kerr black hole. So the volume of the hypersurface can be regarded as the interior volume of a Kerr black hole. Subsequently, we also calculate the interior volume of a Kerr–Newman black hole.