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

TL;DR

This paper investigates the Wang and Yau quasi-local energy in Kerr spacetimes, revealing critical points, explicit functional forms, and complex energy values, thereby advancing understanding of energy definitions in rotating black hole geometries.

Contribution

It provides an explicit analysis of the Wang and Yau quasi-local energy functional for Kerr surfaces, including critical points and complex energy regions, extending previous embedding-based energy concepts.

Findings

W-Y QLE reduces to B-Y QLE at the critical point τ=0.

The energy functional has a critical point at τ=0 for all constant radial surfaces.

An open region of complex energy values is identified in the functional analysis.

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.