Strong field behavior of Wang-Yau Quasi-local energy

TL;DR

This paper investigates the behavior of Wang-Yau quasi-local energy in strong gravitational fields, especially near apparent horizons, revealing conditions under which the energy diverges or remains finite.

Contribution

It provides a detailed analysis of the limit behavior of Wang-Yau quasi-local energy near horizons, distinguishing cases based on isometric embeddability into Euclidean space.

Findings

Energy blows up if horizon cannot be embedded into R^3.

Energy remains finite if horizon can be embedded into R^3.

Unique solutions to the optimal embedding equation exist at the horizon.

Abstract

We look at the strong field behavior of the Wang-Yau quasi-local energy. In particular, we examine the limit of the Wang-Yau quasi-local energy as the defining spacelike $2$-surface $\Sigma$ approaches an apparent horizon from outside. Assuming that coordinate functions of the isometric embedding are bounded in $W^{2,1}$ and mean curvature vector of the image surface remains spacelike, we find that the limit falls in two exclusive cases: 1) If the horizon cannot be isometrically embedded into $R^3$, the Wang-Yau quasi-local energy blows up as $\Sigma$ approaches the horizon while the optimal embedding equation is not solvable for $\Sigma$ near the horizon; 2) If the horizon can be isometrically embedded into $R^3$, the optimal embedding equation is solvable up to the horizon with the unique solution at the horizon corresponding to isometric embedding into $R^3$ and the Wang-Yau quasi-local mass admits a finite limit at the horizon. We discuss the implications of our results in the conclusion section.