Evaluating Small Sphere Limit of the Wang–Yau Quasi-Local Energy

Evaluating Small Sphere Limit of the Wang–Yau Quasi-Local Energy In this article, we study the small sphere limit of the Wang–Yau quasi-local energy defined in Wang and Yau (Phys Rev Lett 102(2):021101, 2009, Commun Math Phys 288(3):919–942, 2009). Given a point p in a spacetime N, we consider a canonical family of surfaces approaching p along its future null cone and evaluate the limit of the Wang–Yau quasi-local energy. The evaluation relies on solving an “optimal embedding equation” whose solutions represent critical points of the quasi-local energy. For a spacetime with matter fields, the scenario is similar to that of the large sphere limit found in Chen et al. (Commun Math Phys 308(3):845–863, 2011). Namely, there is a natural solution which is a local minimum, and the limit of its quasi-local energy recovers the stress-energy tensor at p. For a vacuum spacetime, the quasi-local energy vanishes to higher order and the solution of the optimal embedding equation is more complicated. Nevertheless, we are able to show that there exists a solution that is a local minimum and that the limit of its quasi-local energy is related to the Bel–Robinson tensor. Together with earlier work (Chen et al. 2011), this completes the consistency verification of the Wang–Yau quasi-local energy with all classical limits. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Communications in Mathematical Physics Springer Journals

Evaluating Small Sphere Limit of the Wang–Yau Quasi-Local Energy

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Publisher
Springer Journals
Copyright
Copyright © 2017 by Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Physics; Theoretical, Mathematical and Computational Physics; Mathematical Physics; Quantum Physics; Complex Systems; Classical and Quantum Gravitation, Relativity Theory
ISSN
0010-3616
eISSN
1432-0916
D.O.I.
10.1007/s00220-017-3033-4
Publisher site
See Article on Publisher Site

Abstract

In this article, we study the small sphere limit of the Wang–Yau quasi-local energy defined in Wang and Yau (Phys Rev Lett 102(2):021101, 2009, Commun Math Phys 288(3):919–942, 2009). Given a point p in a spacetime N, we consider a canonical family of surfaces approaching p along its future null cone and evaluate the limit of the Wang–Yau quasi-local energy. The evaluation relies on solving an “optimal embedding equation” whose solutions represent critical points of the quasi-local energy. For a spacetime with matter fields, the scenario is similar to that of the large sphere limit found in Chen et al. (Commun Math Phys 308(3):845–863, 2011). Namely, there is a natural solution which is a local minimum, and the limit of its quasi-local energy recovers the stress-energy tensor at p. For a vacuum spacetime, the quasi-local energy vanishes to higher order and the solution of the optimal embedding equation is more complicated. Nevertheless, we are able to show that there exists a solution that is a local minimum and that the limit of its quasi-local energy is related to the Bel–Robinson tensor. Together with earlier work (Chen et al. 2011), this completes the consistency verification of the Wang–Yau quasi-local energy with all classical limits.

Journal

Communications in Mathematical PhysicsSpringer Journals

Published: Nov 18, 2017

References

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