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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 [18,19]. 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 [7]. 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 which is a local minimum and that the limit of its quasi-local energy is related to the Bel-Robinson tensor. Together with earlier work [7], this completes the consistency verification of the Wang-Yau quasi-local energy with all classical limits.