The phase space for the Einstein-Yang-Mills equations, black hole mechanics, and a condition for stationarity

Abstract

In this thesis, we study the phase space for the Einstein-Yang-Mills equations on an asymptotically flat manifold. The phase space is defined as a Hilbert manifold, which is modeled on weighted Sobolev spaces. We use an implicit function theorem argument to prove that the space of solutions to the constraint equations is a Hilbert submanifold of the phase space; this is equivalent to the statement that the Einstein-Yang-Mills constraints on an asymptotically flat manifold are linearisation stable. It is then shown that the energy, momentum, charge and angular momentum are smooth maps acting on the constraint submanifold. This framework allows us to prove that the first law of black hole mechanics provides a condition for initial data to be stationary, in two distinct cases: when the Cauchy surface has an interior boundary, and when it does not. Both cases are established using a Lagrange multipliers argument.