2018/02/02 by Po-Ning Chen, Chen, Po-Ning
Physics and Astronomy · Mathematics · #Black Holes and Theoretical Physics #Geometric Analysis and Curvature Flows #Cosmology and Gravitation Theories
paper · pdf · doi:10.48550/arxiv.1802.00877
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.