Dynamics of gravitational field within a wave front and thermodynamics of black holes

Abstract

Hamiltonian dynamics of gravitational field contained in a spacetime region with boundary $S$ being a null-like hypersurface (a wave front) is discussed. Complete Hamiltonian formula for the dynamics (with no surface integrals neglected) is derived. A quasi-local proof of the first law of black holes thermodynamics is obtained as a consequence, in case when $S$ is a non-expanding horizon. The zeroth law and Penrose inequalities are discussed from this point of view.