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

TL;DR

This paper investigates the small sphere limit of a new quasi-local energy in anti de-Sitter spacetimes, showing it recovers the stress-energy tensor or relates to the Bel-Robinson tensor depending on the spacetime's matter content.

Contribution

It provides a detailed analysis of the small sphere limit of the quasi-local energy with anti de-Sitter reference, connecting it to fundamental tensors in general relativity.

Findings

Limit recovers the stress-energy tensor at point p.

In vacuum, the quasi-local energy vanishes to higher order.

Limit relates to the Bel-Robinson tensor in vacuum spacetimes.

Abstract

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 $p$.