The Hamiltonian boundary term

The Hamiltonian for a gravitating region necessarily includes a boundary term. Its role is to determine the quasi-local values and, via the vanishing of the boundary term in the variation of the Hamiltonian (required for a well defined Hamiltonian), the boundary conditions. With this boundary variation principle, the Hamiltonian formalism tames the ambiguities inherent in the traditional approaches to the (quasi-)localization of energy-momentum. Using our covariant Hamiltonian formalism, we identified special boundary terms associated with particular physical boundary conditions. In the radiating regime the Hamiltonian fails to be "well defined". This led to new energy flux expressions. We here report on new results for homogeneous cosmologies, the small region limit, and positivity. Although there are many possibilities for the boundary term (reflecting the many choices for the boundary conditions and reference), we found that one expression is distinguished.