Wang and Yau’s quasi-local energy for an extreme Kerr spacetime

Abstract

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

The Kijowski–Liu–Yau quasi-local mass of the Kerr black hole horizon *

“…It is therefore difficult to implement for concrete examples. This has nevertheless been attempted in [16], where the authors noted the difficulty of finding a nonzero admissible time function as in general they lead to complex energies. By examining the boundary separating the complex and real energies they have confirmed the result of Martinez [5] in the range [0, 0.4], and improved it up to j √ 3/2.…”

Tetrad Fields, Reference Frames, and the Gravitational Energy–Momentum in the Teleparallel Equivalent of General Relativity

“…Therefore we may assert that the frame given by Equation ( 57) is characterized by the following properties: (i) the frame is static, because Equation ( 56) is verified; (ii) the e (3) 𝜇 components are oriented along the symmetry axis of the black hole (the z direction). The second condition is ultimately responsible for the simple form of Equation (57).…”

“…These vectors also characterize the frame determined by Equation (57). The fixation of a and Ω is equivalent to the fixation of six components of the tetrad fields.…”

“…It is not the purpose of this review to address or critique alternative approaches that deal with the same investigation of this section, but we may just compare our result with the one obtained by other means, notably the quasi-local expression for the gravitational energy. A recent proposal for the quasi-local gravitational energy was given by Wang and Yau, [55,56] and applied to the Kerr space-time [57] to address precisely the irreducible mass of the black hole. In the context of the latter references, it was made an analysis similar to the one above that led to Figure 1.…”

“…The frame described by the latter equation yields a field of observers that are dragged in rotational motion by the gravitational field of the black hole inside the ergosphere, which is a well known feature of the Kerr space-time. On the other hand, tetrads defined by Equation (57) are not defined in the interior of the ergosphere, and for this reason they play no role in the evaluation of the irreducible mass.…”