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

TL;DR

The paper reinterprets black hole surface gravity as a local, invariant quantity derived from the growth of the horizon-bound invariant four-volume $\mathcal{V}$, showing that $\mathcal{V} \propto \ln(\lambda)$ with a horizon-constant given by the Parikh volume divided by the volume-growth rate, i.e., $\kappa = {}^3\mathcal{V}^*/\mathcal{V}^*$. By constructing regular coordinates and employing a shell that terminates on the horizon, the authors establish a depth-independent definition that extends from Schwarzschild to general static (non-degenerate) black holes and to cosmological horizons, with explicit verification in Schwarzschild and discussion for Kerr (Appendix). The key result, $\kappa = {}^3\mathcal{V}^*/\mathcal{V}^*$, offers a local invariant interpretation of surface gravity and links horizon mechanics to the rate of growth of invariant volume, with implications for black hole thermodynamics and the third law. This approach provides a new perspective on horizon structure and suggests that horizon properties are encoded in volume growth rates that are robust to the depth of integration.

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 $\mathcal{V}_{s} \varpropto \ln(λ)$ where $s$ is any shell that terminates on the horizon, $λ$ 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.

Figures (2)

• Figure 1: The collapse of a timelike boundary surface $\Sigma$ that terminates at the central singularity simultaneously with the null cone $\delta$ and produces a black hole with horizon $\mathcal{H}$. The null cone $v$ is any null cone to the future of $\delta$. The partial invariant four-volume $\mathcal{V}$ calculated here is bounded by $\delta$ and $v$, and is on and to the interior of $\mathcal{H}$. We eventually show that the depth of the integration below $\mathcal{H}$ is irrelevant as regards the surface gravity $\kappa$.
• Figure 2: $\mathcal{R}$ for the calculation of $\mathcal{V}$. The black hole is created by the collapse of some timelike boundary $\Sigma$. (The diagram to the left of $\Sigma$ is irrelevant.) The first ingoing null geodesic to hit the vacuum singularity is labeled $N$ and crosses the horizon at $v=\delta$.