The Phase Space for the EinsteinYangMills Equations and the First Law of Black Hole Thermodynamics
Abstract
We use the techniques of Bartnik (2005) to show that the space of solutions to the EinsteinYangMills constraint equations on an asymptotically at manifold with one end and zero boundary components, has a Hilbert manifold structure; the EinsteinMaxwell 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 YangMills 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2013
 arXiv:
 arXiv:1302.1237
 Bibcode:
 2013arXiv1302.1237M
 Keywords:

 General Relativity and Quantum Cosmology;
 Mathematics  Analysis of PDEs;
 83C05
 EPrint:
 21 pages