The volume of the black holes — The constant curvature slicing of the spherically symmetric spacetime

Abstract

We consider the problem of determination of a volume of some bounded space-like hypersurfaces in the case of spherically symmetric spacetimes. In the case when the hypersurface is cut or bounded by a light-like hypersurface, the problem may not be well-defined. In order to define the volume of a hypersurface properly, we introduce the volume forms related to the given foliation (observer) of the considered spacetime. In the case of the constant curvature slice, the volume of the hypersurface cut by the horizon (light-like surface) becomes composed of the two, outer- and inner-horizon parts, treated differently. We compute the corresponding volumes outside and inside of the horizon of the eternal Schwarzschild black hole.