Strong field behavior of Wang-Yau Quasi-local energy

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.

References in corpus (17)

• Quasilocal Energy and Conserved Charges Derived from the Gravitational Action
• Spectral Methods for Numerical Relativity
• Positivity of Quasilocal Mass
• Quasilocal mass in general relativity
• Isometric embeddings into the Minkowski space and new quasi-local mass
• Concentration-compactness phenomena in the higher order Liouville's equation
• A comment on Liu and Yau's positive quasi-local mass
• Conserved quantities in general relativity: from the quasi-local level to spatial infinity
• Limit of quasilocal mass at spatial infinity
• Evaluating quasilocal energy and solving optimal embedding equation at null infinity
• Minimizing properties of critical points of quasi-local energy
• Open problems in mathematical physics
• Wang and Yau's Quasi-Local Energy for an Extreme Kerr Spacetime
• The Kijowski--Liu--Yau quasi-local mass of the Kerr black hole horizon
• Quasilocal energy and surface geometry of Kerr spacetime
• Properties of Quasi-local mass in binary black hole mergers
• Some Remarks on Wang-Yau Quasi-Local Mass