The First Law of Black Hole Mechanics

TL;DR

This work derives a strengthened first law of black hole mechanics from the Hamiltonian formulation of Einstein–Maxwell theory, showing that for any nonsingular, asymptotically flat perturbation satisfying the linearized constraints the variations of mass, charge, and angular momentum and the horizon data obey $\delta m + V \delta Q - \Omega \delta J = \frac{1}{8\pi} \kappa \delta A$ (eq. 15). The result relies on the structure of the ADM Hamiltonian and boundary terms, defining a canonical energy in terms of $m$, $Q$, $J$, and horizon data $\kappa$ and $A$. The strengthened first law is then used to close gaps in black hole uniqueness by proving that stationary, nonrotating Einstein–Maxwell black holes with an ergoregion disjoint from the horizon must be static, thereby ruling out such solutions. The argument does not straightforwardly generalize to Einstein–Yang–Mills theory, where nonstatic nonrotating configurations may exist but are expected to be unstable.

Abstract

A simple proof of a strengthened form of the first law of black hole mechanics is presented. The proof is based directly upon the Hamiltonian formulation of general relativity, and it shows that the the first law variational formula holds for arbitrary nonsingular, asymptotically flat perturbations of a stationary, axisymmetric black hole, not merely for perturbations to other stationary, axisymmetric black holes. As an application of this strengthened form of the first law, we prove that there cannot exist Einstein-Maxwell black holes whose ergoregion is disjoint from the horizon. This closes a gap in the black hole uniqueness theorems.