Evaluating Small Sphere Limit of the Wang–Yau Quasi-Local Energy
Evaluating Small Sphere Limit of the Wang–Yau Quasi-Local Energy
复制标题
评估Wang-Yau准局域能量的小球极限
DOI：
10.1007/s00220-017-3033-4
复制
发表时间：
2015
影响因子：
2.4
通讯作者：
S. Yau
中科院分区：
文献类型：
--
作者：
Po;Mu;S. Yau
在这篇文章中，我们研究了Wang和Yau（Phys Rev Lett 102（2）：021101，2009，Commun Math Phys 288（3）：919-942，2009）中定义的Wang-Yau准局域能的小球极限。给定时空N中的一个点p，我们考虑一个正则曲面族沿着它的未来零锥逼近p，并计算了Wang-Yau拟局域能量的极限。评估依赖于求解“最佳嵌入方程”，其解代表准局部能量的临界点。对于具有物质场的时空，这种情形类似于Chen et al.（Commun Math Phys 308（3）：845-863，2011）中发现的大球极限。也就是说，有一个自然的解决方案，这是一个局部最小值，其准局部能量的极限恢复的应力-能量张量在p.对于真空时空，准局部能量消失到更高的顺序和最佳嵌入方程的解决方案是更复杂的。然而，我们能够证明，存在一个解决方案，是一个局部最小值，其准局部能量的限制是有关的贝尔-罗宾逊张量。与之前的工作（Chen et al. 2011）一起，这完成了Wang-Yau准局域能量与所有经典极限的一致性验证。
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.