Canonical Phase Space Formulation of Quasilocal General Relativity

TL;DR

This paper develops a Hamiltonian framework for quasilocal general relativity incorporating boundary coordinates, deriving an energy expression akin to Brown-York energy, and establishing their equivalence under certain boundary conditions.

Contribution

It introduces a Hamiltonian formulation with boundary variables for quasilocal GR, connecting and unifying it with the Brown-York energy approach.

Findings

Derived a Hamiltonian expression for quasilocal energy.

Showed the equivalence of Hamiltonian and Brown-York energies under compatible boundary conditions.

Extended phase space includes boundary coordinates as configuration variables.

Abstract

We construct a Hamiltonian formulation of quasilocal general relativity using an extended phase space that includes boundary coordinates as configuration variables. This allows us to use Hamiltonian methods to derive an expression for the energy of a non-isolated region of space-time that interacts with its neighbourhood. This expression is found to be very similar to the Brown-York quasilocal energy that was originally derived by Hamilton-Jacobi methods. We examine the connection between the two formalisms and find that when the boundary conditions for the two are harmonized, the resulting quasilocal energies are identical.