Narrow Results

One way to make the variational principle based on the Einstein–Hilbert action well-defined (i.e. functionally differentiable) is to add a surface term involving the integral of the trace of the extrinsic curvature. I provide a simple derivation of this result which is constructive in the sense that it starts from the variation of Einstein–Hilbert action and obtains the correct boundary term. This is to be contrasted with the usual derivations in which one first adds this term and then shows that the unwanted parts cancel out in the variation of the total action. The approach described here also clearly identifies the variables that need to be fixed in the boundary as the three-metric, directly from the action principle.

Our covariant Hamiltonian for dynamic geometry generates the evolution of a spatial region along a vector field. It includes a boundary term which determines both the value of the Hamiltonian and the boundary conditions. The value gives the quasi-local quantities: energy-momentum, angular-momentum/center-of-mass. The boundary term depends not only on the dynamical variables but also on their reference values, the latter determine the ground state (having vanishing quasi-local quantities). For our preferred boundary term for Einstein's GR we propose using 4D isometric matching and extremizing the energy to determine the “best matched” reference metric and connection values.