Hamiltonian, energy and entropy in general relativity with non-orthogonal boundaries

A general recipe to define, via the 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 the Noether-conserved quantities. The Hamiltonian for general relativity in the 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 the metric Dirichlet boundary conditions. A (conditioned) agreement with previous definitions is proved. A correspondence with the Brown–York original formulation of the first principle of black hole thermodynamics is finally established.