Metric-based Hamiltonians, null boundaries and isolated horizons

We extend the quasi-local (metric-based) Hamiltonian formulation of general relativity so that it may be used to study regions of spacetime with null boundaries. In particular, we use this generalized Brown-York formalism to study the physics of isolated horizons. We show that the first law of isolated horizon mechanics follows directly from the first variation of the Hamiltonian. This variation is not restricted to the phase space of solutions to the equations of motion but is instead through the space of all (off-shell) spacetimes that contain isolated horizons. We find 2-surface integrals evaluated on the horizons that are consistent with the Hamiltonian and which define the energy and angular momentum of these objects. These are closely related to the corresponding Komar integrals and for Kerr-Newman spacetime are equal to the corresponding ADM/Bondi quantities. Thus, the energy of an isolated horizon calculated by this method is in agreement with that recently calculated by Ashtekar and collaborators but not the same as the corresponding quasi-local energy defined by Brown and York. Isolated horizon mechanics and Brown-York thermodynamics are compared.