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

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.