Minimizing properties of critical points of quasi-local energy

TL;DR

This paper investigates the minimizing properties of critical points of the Wang-Yau quasi-local energy in relativity, establishing conditions under which these points are local or global minima, especially in symmetric cases.

Contribution

It provides new results on when critical points of the Wang-Yau quasi-local energy are local or global minimizers under specific geometric conditions.

Findings

Critical points can be local minima under mean curvature conditions.

Certain symmetric embeddings are globally minimizing.

Conditions for minimality relate to geometric properties of the embedding.

Abstract

In relativity, the energy of a moving particle depends on the observer, and the rest mass is the minimal energy seen among all observers. The Wang-Yau quasi-local mass for a surface in spacetime introduced in [7] and [8] is defined by minimizing quasi-local energy associated with admissible isometric embeddings of the surface into the Minkowski space. A critical point of the quasi-local energy is an isometric embedding satisfying the Euler-Lagrange equation. In this article, we prove results regarding both local and global minimizing properties of critical points of the Wang-Yau quasi-local energy. In particular, under a condition on the mean curvature vector we show a critical point minimizes the quasi-local energy locally. The same condition also implies that the critical point is globally minimizing among all axially symmetric embedding provided the image of the associated isometric embedding lies in a totally geodesic Euclidean 3-space.