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

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.