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

## TL;DR

This paper investigates how to define and compute the volume of bounded spacelike hypersurfaces in spherically symmetric spacetimes, especially across horizons, using volume forms related to specific foliations, with explicit calculations for Schwarzschild black holes.

## Contribution

It introduces a method to properly define the volume of hypersurfaces bounded by light-like surfaces in spherically symmetric spacetimes, including the Schwarzschild black hole case.

## Key findings

- Volumes inside and outside the horizon are explicitly calculated.
- The approach distinguishes between different parts of the hypersurface, outer and inner.
- The method clarifies the volume definition in the presence of horizons.

## 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 hypersurfaces is cut or bounded by a light-like hypersurface the problem may not be well defined. In order to define properly the volume 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 parts, outer and inner, treated differently. We compute the corresponding volumes outside and inside of the horizon of the ethernal Schwarzschild black hole.