Gravitational energy is well defined

Abstract

The energy of gravitating systems has been an issue since Einstein proposed general relativity: considered to be ill defined, having no proper local density. Energy-momentum is now regarded as \emph{quasi-local} (associated with a closed 2-surface). We consider the pseudotensor and quasi-local proposals in the Lagrangian-Noether-Hamiltonian formulations. There are two ambiguities: (i) many expressions, (ii) each depends on some non-dynamical structure, e.g., a reference frame. The Hamiltonian approach gives a handle on both problems. Our remarkable discovery is that with a 4D isometric Minkowski reference a large class of expressions---those that agree with the Einstein pseudotensor's Freud superpotential to linear order---give a common quasi-local energy value. With a best-matched reference on the boundary this value is the non-negative Wang-Yau mass.