Hamiltonian, Energy and Entropy in General Relativity with Non-Orthogonal Boundaries

TL;DR

This paper develops a general method to define the Hamiltonian in field theories, specifically applying it to General Relativity with non-orthogonal boundaries, linking energy definitions to boundary conditions and black hole thermodynamics.

Contribution

It introduces a Noether theorem-based approach for Hamiltonian definition in field theories, applied to General Relativity with non-orthogonal boundaries, and connects it to black hole thermodynamics.

Findings

Hamiltonian defined via Noether theorem for non-orthogonal boundaries.

Energy corresponds to on-shell Hamiltonian value.

Established link with Brown-York black hole thermodynamics.

Abstract

A general recipe to define, via Noether theorem, the Hamiltonian in any natural field theory is suggested. It is based on a Regge-Teitelboim-like approach applied to the variation of Noether conserved quantities. The Hamiltonian for General Relativity in presence of non-orthogonal boundaries is analysed and the energy is defined as the on-shell value of the Hamiltonian. The role played by boundary conditions in the formalism is outlined and the quasilocal internal energy is defined by imposing metric Dirichlet boundary conditions. A (conditioned) agreement with previous definitions is proved. A correspondence with Brown-York original formulation of the first principle of black hole thermodynamics is finally established.