A reference for the gravitational Hamiltonian boundary term

The Hamiltonian for physical systems and dynamic spacetime geometry generates the evolution of a spatial region along a vector field. It includes a boundary term which not only determines the value of the Hamiltonian, but also, via the boundary term in the variation of the Hamiltonian, the boundary conditions. The value of the Hamiltonian comes from its boundary term; it gives the quasi-local quantities: Energy-momentum and angularmomentum/ center-of-mass momentum. This boundary term depends not only on the dynamical variables but also on their reference values; these reference values determine the ground state the state having vanishing quasi-local quantities. Here our concern is with how to select on the quasi-local two-boundary the reference value. To determine the best matched" reference metric and connection values for our preferred boundary term for Einstein's general relativity, we propose on the boundary two-surface (i) four dimensional isometric matching, and (ii) extremizing the value of the energy.