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

TL;DR

This paper develops a Hamiltonian framework for gravitational fields at null boundaries, deriving a complete formula and providing a quasi-local proof of the black hole first law, with implications for the zeroth law and Penrose inequalities.

Contribution

It introduces a comprehensive Hamiltonian formulation for gravitational dynamics on null hypersurfaces and offers a quasi-local proof of black hole thermodynamics laws.

Findings

Derived a complete Hamiltonian formula for gravitational dynamics on null boundaries.

Provided a quasi-local proof of the first law of black hole thermodynamics.

Discussed the zeroth law and Penrose inequalities within this framework.

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.