Evaluating small sphere limit of the Wang-Yau quasi-local energy

Po-Ning Chen Columbia University Mu-Tao Wang Columbia University Shing-Tung Yau Harvard University

Differential Geometry Geometric Analysis and Geometric Topology Mathematical Physics mathscidoc:1608.10032

2015.10
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.
Wang-Yau quasi-local energy, Bel-Robinson tensor
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@inproceedings{po-ning2015evaluating,
  title={Evaluating small sphere limit of the Wang-Yau quasi-local energy},
  author={Po-Ning Chen, Mu-Tao Wang, and Shing-Tung Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160820172436721214336},
  year={2015},
}
Po-Ning Chen, Mu-Tao Wang, and Shing-Tung Yau. Evaluating small sphere limit of the Wang-Yau quasi-local energy. 2015. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160820172436721214336.
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