We use the techniques of Bartnik 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 depth throughout the 70s [1, 2, 9, 11, 13, 18, 19].
This framework allows us to prove a conjecture of Sudarsky and Wald, namely that 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 from Sudarksy and Wald, which is the case where an interior boundary is present; this will be addressed in future work.
"The phase space for the Einstein-Yang-Mills equations and the first law of black hole thermodynamics." Adv. Theor. Math. Phys. 18 (4) 799 - 825, August 2014.