First Law of Black Hole Mechanics as a Condition for Stationarity

TL;DR

This paper demonstrates that the differential relationship in the first law of black hole mechanics precisely characterizes stationarity of initial data outside a boundary surface, linking it to bifurcate Killing horizons.

Contribution

It rigorously establishes the first law's differential relationship as a condition for stationarity in Einstein-Yang-Mills spacetimes with boundary surfaces.

Findings

The first law differential relationship is necessary and sufficient for stationarity.

Stationary initial data implies the boundary is a bifurcation surface of a Killing horizon.

Evidence suggests the boundary surface corresponds to a bifurcation surface when the first law holds.

Abstract

In earlier work [arXiv:1302.1237], we provided a Hilbert manifold structure for the phase space for the Einstein-Yang-Mills equations, and used this to prove a condition for initial data to be stationary. Here we use the same phase space to consider the evolution of initial data exterior to some closed 2-surface boundary, and establish a condition for stationarity in this case. It is shown that the differential relationship given in the first law of black hole mechanics is exactly the condition required for the initial data to be stationary; this was first argued non-rigorously by Sudarsky and Wald in 1992. Furthermore, we give evidence to suggest that if this differential relationship holds then the boundary surface is the bifurcation surface of a bifurcate Killing horizon.