Canonical quasilocal energy and small spheres

J. D. Brown, S. R. Lau, and J. W. York
Phys. Rev. D 59, 064028 – Published 18 February 1999
PDFExport Citation

Abstract

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 Np 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=43πr3[Tμνuμuν]|p+O(r4) and (ii) in vacuo E=190r5[Tμνλκuμuνuλuκ]|p+O(r6). Here, uμ is a unit, future-pointing, timelike vector in the tangent space at p (which defines the choice of affine radius); Tμν is the matter stress-energy-momentum tensor; Tμνλκ 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.

  • Received 2 October 1998

DOI:https://doi.org/10.1103/PhysRevD.59.064028

©1999 American Physical Society

Authors & Affiliations

J. D. Brown

  • Department of Physics, North Carolina State University, Raleigh, North Carolina 27695–8202
  • Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695–8205

S. R. Lau*

  • Department of Physics & Astronomy, University of North Carolina, CB# 3255 Phillips Hall, Chapel Hill, North Carolina 27599-3255
  • Institut für Theoretische Physik, Technische Universität Wien, Wiedner Hauptstraße 8-10, A-1040 Wien, Austria

J. W. York

  • Department of Physics, North Carolina State University, Raleigh, North Carolina 27695–8202
  • Department of Physics & Astronomy, University of North Carolina, CB# 3255 Phillips Hall, Chapel Hill, North Carolina 27599-3255

  • *Current address: Applied Mathematics Group, Department of Mathematics, University of North Carolina, CB# 3250 Phillips Hall, Chapel Hill, NC 27599-3250.

References (Subscription Required)

Click to Expand
Issue

Vol. 59, Iss. 6 — 15 March 1999

Reuse & Permissions
Access Options
Author publication services for translation and copyediting assistance advertisement

Authorization Required


×
×

Images

×

Sign up to receive regular email alerts from Physical Review D

Log In

Cancel
×

Search


Article Lookup

Paste a citation or DOI

Enter a citation
×