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 light cone N_p belonging to a generic spacetime point p. As a power series in r, we compute the energy E of the gravitational and matter fields on a spacelike hypersurface Σ 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 “light cone reference,” which stems from a certain isometric embedding of S(r) into a genuine light cone of Minkowski spacetime. Choosing this zero point, we find the following results: (i) in the presence of matter E=4/3πr³[T_μνu^μu^ν]|_p+O(r⁴) and (ii) in vacuo E=1/90r⁵[T_μνλκu^μu^νu^λu^κ]|_p+O(r⁶). Here, u^μ is a unit, future-pointing, timelike vector in the tangent space at p (which defines the choice of affine radius); T_μν is the matter stress-energy-momentum tensor; T_μνλκ is the Bel-Robinson gravitational super stress-energy-momentum tensor; and |_p denotes “restriction to p.” Hawking’s quasilocal mass expression agrees with the results (i) and (ii) up to and including the first non-trivial order in the affine radius. The non-vacuum result (i) has the expected form based on the results of Newtonian potential theory.