The volume of stationary black holes and the meaning of the surface gravity

TL;DR

This paper explores the finite volume of black holes formed by gravitational collapse and proposes a new local, invariant way to define surface gravity based on the relationship between volume, affine parameter, and surface gravity.

Contribution

It introduces a novel invariant volume measure for black holes and links it to surface gravity through a logarithmic relation involving the affine generator.

Findings

Finite volume of black holes depends logarithmically on the affine generator.

Provides an alternative, local definition of surface gravity.

Connects black hole volume with horizon properties in a new way.

Abstract

The invariant four-volume $\mathcal{V}$ of a complete black hole (the volume of the spacetime at and interior to the horizon) diverges. However, if one considers the black hole set up by the gravitational collapse of an object, and integrates only a finite time to the future of the collapse, the resultant volume is well defined and finite. In this paper we examine non-degenerate stationary black holes (and cosmological horizons) and find that $\mathcal{V}_{s} \varpropto \ln(\lambda)$ where $s$ is any shell that terminates on the horizon, $\lambda$ is the affine generator of the horizon and the constant of proportionality is the Parikh volume of $s$ divided by the surface gravity. This provides an alternative local and invariant definition of the surface gravity of a stationary black hole.