Advances in Theoretical and Mathematical Physics

Volume 24 (2020)

Number 4

Small sphere limit of the quasi-local energy with anti de-Sitter space reference

Pages: 879 – 924



Po-Ning Chen (Department of Mathematics, University of California, Riverside, Calif., U.S.A.)


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 $p$.

The author is supported by NSF grant DMS-1308164 and Simons Foundation collaboration grant #584785.

Published 2 September 2020