The Volume of Black Holes

TL;DR

This paper introduces a new, consistent definition of volume for stationary black hole spacetimes, explores the possibility of infinite volume with finite horizon area, and discusses implications for black hole entropy.

Contribution

It provides a coordinate-independent volume definition for stationary spacetimes and shows such volumes are finite for black holes, impacting entropy interpretation.

Findings

Black hole volume in four dimensions is ${4 \\over 3} \\pi r_+^3$.

No solutions with finite horizon area and infinite volume exist in 3 or 4 dimensions.

Implications for the interpretation of Bekenstein-Hawking entropy are discussed.

Abstract

We propose a definition of volume for stationary spacetimes. The proposed volume is independent of the choice of stationary time-slicing, and applies even though the Killing vector may not be globally timelike. Moreover, it is constant in time, as well as simple: the volume of a spherical black hole in four dimensions turns out to be just ${4 \over 3} \pi r_+^3$. We then consider whether it is possible to construct spacetimes that have finite horizon area but infinite volume, by sending the radius to infinity while making discrete identifications to preserve the horizon area. We show that, in three or four dimensions, no such solutions exist that are not inconsistent in some way. We discuss the implications for the interpretation of the Bekenstein-Hawking entropy.