Quasi-local energy from a Minkowski reference

The specification of energy for gravitating systems has been an unsettled issue since Einstein proposed his pseudotensor. It is now understood that energy-momentum is quasi-local (associated with a closed 2-surface). Here we consider quasi-local proposals (including pseudotensors) in the Lagrangian–Noether–Hamiltonian formulations. There are two ambiguities: (1) there are many possible expressions, (2) they depend on some non-dynamical structure, e.g., a reference frame. The Hamiltonian approach gives a handle on both problems. The Hamiltonian perspective helped us to make a remarkable discovery: with an isometric Minkowski reference a large class of expressions—namely all those that agree with the Einstein pseudotensor’s Freud superpotential to linear order—give a common quasi-local energy value. Moreover, with a best-matched reference on the boundary this is the Wang–Yau mass value.