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 $S\left(r\right),$ level in an affine radius $r,$ of the light cone $N_p$ belonging to a generic spacetime point $p.\mathrm{}$ As a power series in $r,$ we compute the energy $E$ of the gravitational and matter fields on a spacelike hypersurface Σ spanning $S\left(r\right).$ 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 $S\left(r\right)\mathrm{}$ into a genuine light cone of Minkowski spacetime. Choosing this zero point, we find the following results: (i) in the presence of matter $E=\frac{4}{3}\pi r^3\left[T_{\mu \nu }{u}^{\mu }{u}^{\nu }\right]{|}_p+\mathrm{O}{(r}^4)$ and (ii) in vacuo $E=\frac{1}{90}{r}^5\left[T_{\mu \nu \lambda \kappa }{u}^{\mu }{u}^{\nu }{u}^{\lambda }{u}^{\kappa }\right]{|}_p+\mathrm{O}{(r}^6).\mathrm{}$ Here, $u^{\mu }$ 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 stress-energy-momentum tensor; $T_{\mu \nu \lambda \kappa }$ is the Bel-Robinson gravitational super stress-energy-momentum tensor; and ${|}_p$ denotes “restriction to $p.\mathrm{}$” 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.