  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.
