Canonical Quasilocal Energy and Small Spheres

Consider the definition E of quasilocal energy stemming from the Hamilton-Jacobi method as applied to the canonical form of the gravitational action. We examine E in the standard "small-sphere limit," first considered by Horowitz and Schmidt in their examination of Hawking's quasilocal mass. By the term "small sphere" we mean a cut S(r), level in an affine radius r, of the lightcone belonging to a generic spacetime point. As a power series in r, we compute the energy E of the gravitational and matter fields on a spacelike hypersurface spanning S(r). Much of our analysis concerns conceptual and technical issues associated with assigning the zero-point of the energy. For the small-sphere limit, we argue that the correct zero-point is obtained via a "lightcone reference," which stems from a certain isometric embedding of S(r) into a genuine lightcone of Minkowski spacetime. Choosing this zero-point, we find agreement with Hawking's quasilocal mass expression, up to and including the first non-trivial order in the affine radius. The vacuum limit relates the quasilocal energy directly to the Bel-Robinson tensor.