In [13], a new quasi-local energy is introduced for spacetimes with a non-zero cosmological constant. In this article, we study the small sphere limit of this newly defined quasi-local energy for spacetimes with a negative cosmological constant. For such spacetimes, the anti de-Sitter space is used as the reference for the quasi-local energy. 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 quasi-local energy. The optimal embedding equation which identifies the critical points of the quasi-local energy is solved in order to evaluate the limit. Using the optimal embedding, we show that the limit recovers the stress-energy tensor of the matter field at $p$. For vacuum spacetimes, the quasi-local energy vanishes to a higher order. In this case, the limit of the quasi-local energy is related to the Bel-Robinson tensor at $p$.

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具有抗去西特空间参考的准局部能量的小球限

在[13]中，为具有非零宇宙常数的时空引入了一种新的准局域能量。在本文中，我们研究了这种新定义的具有负宇宙常数的时空准局域能量的小球面极限。对于这样的时空，反去西特空间被用作准局部能量的参考。给定时空 $N$ 中的一个点 $p$，我们考虑沿其未来零锥接近 $p$ 的典型表面族，并评估准局部能量的极限。求解识别准局部能量临界点的最优嵌入方程以评估极限。使用最佳嵌入，我们表明极限可以恢复 $p$ 处的物质场的应力-能量张量。对于真空时空，准局域能量消失到更高阶。在这种情况下，