Near horizon limit of the Wang--Yau quasi-local mass

TL;DR

This paper analyzes the behavior of the Wang--Yau quasi-local mass as surfaces approach the apparent horizon, establishing convergence properties and the existence and uniqueness of optimal embeddings in this limit.

Contribution

It proves the continuity, existence, and uniqueness of optimal embeddings for the Wang--Yau quasi-local mass in the near horizon limit, extending previous conjectures.

Findings

The quasi-local mass converges to the total mean curvature of the horizon.

Optimal embeddings approach the isometric embedding into ^3.

The quasi-local mass is continuous in the near horizon limit.

Abstract

In this article, we compute the limit of the Wang--Yau quasi-local mass on a family of surfaces approaching the apparent horizon (the near horizon limit). Such limit is first considered in [1]. Recently, Pook-Kolb, Zhao, Andersson, Krishnan, and Yau investigated the near horizon limit of the Wang--Yau quasi-local mass in binary black hole mergers in [12] and conjectured that the optimal embeddings approach the isometric embedding of the horizon into $\R^3$. Moreover, the quasi-local mass converges to the total mean curvature of the image. The vanishing of the norm of the mean curvature vector implies special properties for the Wang--Yau quasi-local energy and the optimal embedding equation. We utilize these features to prove the existence and uniqueness of the optimal embedding and investigate the minimization of the Wang--Yau quasi-local energy. In particular, we prove the continuity of the quasi-local mass in the near horizon limit.