The Volume Inside a Black Hole

TL;DR

This paper explores the concept of volume inside a black hole, showing it varies with the choice of time coordinate and providing examples for different definitions, thus clarifying a subtle aspect of black hole geometry.

Contribution

It offers a detailed analysis of how the interior volume of a black hole depends on the chosen time coordinate, providing explicit examples and clarifications.

Findings

Black hole horizon surface is invariant across observers.

Interior volume varies with time coordinate choice.

Examples demonstrate volume can be zero or time-dependent.

Abstract

The horizon (the surface) of a black hole is a null surface, defined by those hypothetical "outgoing" light rays that just hover under the influence of the strong gravity at the surface. Because the light rays are orthogonal to the spatial 2-dimensional surface at one instant of time, the surface of the black hole is the same for all observers (i.e. the same for all coordinate definitions of "instant of time"). This value is 4*(pi)* (2Gm/c^2)^2 for nonspinning black holes, with G= Newton's constant, c= speed of light, and m= mass of the black hole. The 3-dimensional spatial volume inside a black hole, in contrast, depends explicitly on the definition of time, and can even be time dependent, or zero. We give examples of the volume found inside a standard, nonspinning spherical black hole, for several different standard time-coordinate definitions. Elucidating these results for the volume provides a new pedagogical resource of facts already known in principle to the relativity community, but rarely worked out.