Quasilocal energy and surface geometry of Kerr spacetime

We study the quasilocal 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≤2m, which is inside the ergosphere. For a certain region, r>rk(a), the Gaussian curvature of the surface with constant t, r is positive, and for r>3a the critical value of the QLE is positive. We found that the three curves: the outer horizon r=r+(a), r=rk(a) and r=3a intersect at the point a=3m/2, which is the limit for the horizon to be isometrically embedded into R3. 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 Mir. From the results of Chen-Wang-Yau, we conclude that in a certain region, r>rh(a), the critical value of the Kerr QLE is a global minimum.