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

Abstract

A general recipe to define, via Nöther theorem, the Hamiltonian in any natural field theory is suggested. It is based on a Regge-Teitelboim-like approach applied to the variation of Nöther 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.

Conserved quantities from the equations of motion: with applications to natural and gauge natural theories of gravitation

“…We can infer this without any calculation. Indeed, it is well known that if we consider a LagrangianL = L + Div α which differs from a given Lagrangian L only for the addition of a divergence, then the reduced current remains the same, i.e.Ẽ(L, Ξ) =Ẽ(L, Ξ) while the Noether superpotential transforms as follows (see [40,48]):…”

“…When dealing with applications we shall see that the fibered morphisms U(L ′ , X) and δ X V(L, Ξ) differ indeed for a term which is nothing but the covariant Regge-Teitelboim boundary correction term; see [28,36,40,50,63].…”

“…Roughly speaking the choice of such a direction corresponds to a control mode of the fields at the boundary which is based on physical grounds (e.g. micro-canonical or grandcanonical ensembles) and it is related to the asymptotic behaviour of specific solutions and to the specific physical observable one is aimed to determine (see, e.g., [4,9,16,19,40,50,53]). We then believe that, inside a formalism which we want to be as general as possible, both physical and mathematical reasons suggest to us that only variations δQ should be reasonably considered.…”

Geometric Entropy of Self-Gravitating Systems

“…2) The quantity δ X Q B (Ξ) defined as in (13) satisfies (see [21], [32], [33]) the symplectic relation:…”

Boundary conditions, energies and gravitational heat in general relativity (a classical analysis)

“…We point out that for non-orthogonal foliations of spacetime, i. e. u µ n µ | B = 0, an extra term has to be added in (18) which corresponds to a symplectic boundary structure; see [10,15,33,42,51].…”