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

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