Quasilocal energy and surface geometry of Kerr spacetime

TL;DR

This paper investigates the quasi-local energy and surface geometry of Kerr spacetime without slow rotation approximation, revealing geometric properties, critical values, and monotonic behavior of QLE in various regions.

Contribution

It provides a detailed analysis of Kerr QLE and surface geometry, including new insights into their behavior inside the ergosphere and conditions for isometric embedding.

Findings

Gaussian curvature positive for certain regions

QLE decreases monotonically from ergosphere to infinity

Critical QLE is a global minimum in some regions

Abstract

We study the quasi-local energy (QLE) and the surface geometry for Kerr spacetime in the Boyer-Lindquist coordinates without taking the slow rotation approximation. We also consider in the region $r\leq2m$, which is inside the ergosphere. For a certain region, $r>r_{k}(a)$, the Gaussian curvature of the surface with constant $t,r$ is positive, and for $r>\sqrt{3}a$ the critical value of the QLE is positive. We found that the three curves: the outer horizon $r=r_{+}(a)$, $r=r_{k}(a)$ and $r=\sqrt{3}a$ intersect at the point $a=\sqrt{3}m/2$, which is the limit for the horizon to be isometrically embedded into $\mathbb{R}^3$. The numerical result indicates that the Kerr QLE is monotonically decreasing to the ADM $m$ from the region inside the ergosphere to large $r$. Based on the second law of black hole dynamics, the QLE is increasing with respect to the irreducible mass $M_{\mathrm{ir}}$. From a results of Chen-Wang-Yau, we conclude that in a certain region, $r>r_{h}(a)$, the critical value of the Kerr QLE is a global minimum.