  We use the techniques of Bartnik (2005) to show that the space of solutions
to the Einstein-Yang-Mills constraint equations on an asymptotically at
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. 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 where an interior boundary is present; this will be addressed in future
work.
