Wang and Yau's Quasi-Local Energy for an Extreme Kerr Spacetime

Abstract

There exist constant radial surfaces, $\mathcal{S}$, that may not be globally embeddable in ${\mathbb{R}}^3$ for Kerr spacetimes with $a>\sqrt{3}M/2$. To compute the Brown and York (B-Y) quasi-local energy (QLE), one must isometrically embed $\mathcal{S}$ into ${\mathbb{R}}^3$. On the other hand, the Wang and Yau (W-Y) QLE embeds $\mathcal{S}$ into Minkowski space. In this paper, we examine the W-Y QLE for surfaces that may or may not be globally embeddable in ${\mathbb{R}}^3$. We show that their energy functional, $E[\tau ]$, has a critical point at $\tau =0$ for all constant radial surfaces in $t=constant$ hypersurfaces using Boyer-Lindquist coordinates. For $\tau =0$, the W-Y QLE reduces to the B-Y QLE. To examine the W-Y QLE in these cases, we write the functional explicitly in terms of $\tau$ under the assumption that $\tau$ is only a function of $\theta$. We then use a Fourier expansion of $\tau \left(\theta \right)$ to explore the values of $E[\tau \left(\theta \right)]$ in the space of coefficients. From our analysis, we discovered an open region of complex values for $E[\tau \left(\theta \right)]$. We also study the physical properties of the smallest real value of $E[\tau \left(\theta \right)]$, which lies on the boundary separating real and complex energies.

25 pages, 10 figures

References in corpus (6)

• Quasilocal mass in general relativity
• Isometric embeddings into the Minkowski space and new quasi-local mass
• Embedding of the Kerr-Newman Black Hole Surface in Euclidean Space
• Rigidity and minimizing properties of quasi-local mass
• Quasilocal energy and surface geometry of Kerr spacetime
• Quasilocal Energy in Kerr Spacetime

Cited by in corpus (6)

• Tetrad Fields, Reference Frames, and the Gravitational Energy-Momentum in the Teleparallel Equivalent of General Relativity
• The Kijowski--Liu--Yau quasi-local mass of the Kerr black hole horizon
• Quasi-Local Energy of a Rotating Object Described by Kerr Spacetime
• Energy in critical collapse
• Black Holes, Complex Curves, and Graph Theory: Revising a Conjecture by Kasner
• Strong field behavior of Wang-Yau Quasi-local energy