Three Quasi-local Masses

General Relativity differs from most classical field theories in that there is no welldefined notion of energy density for the gravitation field, as can be seen from Einstein’s principle of equivalence. Thus at best one can only hope to calculate the mass/energy contained within a domain, as opposed to at a point. Such a concept is referred to as quasi-local mass, that is, a functional which assigns a real number to each compact spacelike hypersurface in a spacetime. Of course there are many properties that any such functional should satisfy under appropriate conditions; most notable among these are the properties of non-negativity and rigidity (for an expanded list see Ref. 4). Although numerous definitions of quasi-local mass have been proposed, most seem to possess undesirable properties, in fact most fail the crucial test of non-negativity. However, there is one definition which appears to satisfy most of the required properties, namely the mass proposed by Bartnik. Bartnik’s idea is to localize the ADM (or total) mass in the following way. Here we restrict attention to the time symmetric case. Let (Ω, h) be a compact threemanifold with boundary, and define an admissible extension to be an asymptotically flat three-manifold (M, g) with (or without) boundary satisfying the following conditions: (M, g) has non-negative scalar curvature, the boundary (if nonempty) is