1302.1237 (Stephen McCormick)

The Phase Space for the Einstein-Yang-Mills Equations and the First Law
of Black Hole Thermodynamics    [PDF]

Stephen McCormick

We use the techniques of Bartnik (2005) to show that the space of solutions to the Einstein-Yang-Mills constraint equations on an asymptotically flat manifold with one end and zero boundary components, has a Hilbert manifold structure; the Einstein-Maxwell system can be considered as a special case. This is equivalent to the property of linearisation stability, which was studied in the 70s by Arms, Fischer, Marsden, Moncrief and others, for the case of a compact manifold without boundary. This framework allows us to prove a conjecture of Sudarsky and Wald (1992), that is, the validity of the first law of black hole thermodynamics is a suitable condition for stationarity. Since we work with a single end and no boundary conditions, this is equivalent to critical points of the ADM mass subject to variations fixing the Yang-Mills charge corresponding exactly to stationary solutions. The natural extension to this work is to prove the second conjecture of Sudarsky and Wald, which is the case when an interior boundary corresponding to a Killing horizon is present; this will be addressed in future work.

View original: http://arxiv.org/abs/1302.1237