Canonical phase space formulation of quasi-local general relativity

We construct a Hamiltonian formulation of quasi-local 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 spacetime that interacts with its neighbourhood. This expression is found to be very similar to the Brown-York quasi-local 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 quasi-local energies are identical.