Canonical quasilocal energy and small spheres - art. no. 064028

Consider the definition E of quasilocal energy stemming from the Hamilton-J acobi method as applied to the canonical form of the gravitational action. We examine E in the standard "small-sphere limit," first considered by Horo witz 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 l ight cone N-p belonging to a generic spacetime point p. As a power series i n r, we compute the energy E of the gravitational and matter fields on a sp acelike hypersurface Sigma spanning S(r). Much of our analysis concerns con ceptual and technical issues associated with assigning the zero point of th e 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 isomet ric embedding of S(r) into a genuine light cone of Minkowski spacetime. Cho osing this zero point, we find the following results: (i) in the presence o f matter E=4/3 pi r(3) [T(mu nu)u(mu)u(nu)]\(p) + O(r(4)) and (ii) in vacuo E=1/90 r(5)[T(mu nu lambda kappa)u(mu)u(lambda)u(kappa)]\(p) + O(r(6)). He re, u(mu) is a unit, future-pointing, timelike vector in the tangent space at p (which defines the choice of affine radius); T-mu nu is the matter str ess-energy-momentum tensor; T mu nu lambda kappa is the Bel-Robinson gravit ational super stress-energy-momentum tensor; and \(p) denotes ''restriction to p.'' Hawking's quasilocal mass expression agrees with the results (i) a nd (ii) up to and including the first non-trivial order in the affine radiu s. The non-vacuum result (i) has the expected form based on the results of Newtonian potential theory. [S0556-2821(99)02904-5].