Minimizing Properties of Critical Points of Quasi-Local Energy

Chen, Po‐Ning; Wang, Mu-Tao; Yau, Shing–Tung

doi:10.1007/s00220-014-1909-0

Cited by 38 publications

(42 citation statements)

References 8 publications

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“…48 always vanishes and τ = 0 is always a solution. However, τ = 0 is not necessarily a local or global minimum, see [11,12] for a criterion for local minimum of a critical point in terms of a mean curvature inequality.…”

Section: The Reference Contribution To the W-y Qle In Terms Of τmentioning

confidence: 99%

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

Miller

Ray

Wang

et al. 2018

Class. Quantum Grav.

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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.

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Section: The Reference Contribution To the W-y Qle In Terms Of τmentioning

confidence: 99%

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

Miller

Ray

Wang

et al. 2018

Class. Quantum Grav.

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“…In general the optimal isometric embedding may not be unique, but as shown in [12] it is unique locally if ̺ > 0. Furthermore, nonnegativity of the energy as discussed above implies nonnegativity of the mass, and the mass is zero for surfaces in Minkowski space.…”

Section: Statement Of Main Resultsmentioning

confidence: 99%

Geometric Inequalities for Quasi-Local Masses

Alaee

Khuri

Yau

2020

Commun. Math. Phys.

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In this paper lower bounds are obtained for quasi-local masses in terms of charge, angular momentum, and horizon area. In particular we treat three quasi-local masses based on a Hamiltonian approach, namely the Brown-York, Liu-Yau, and Wang-Yau masses. The geometric inequalities are motivated by analogous results for the ADM mass. They may be interpreted as localized versions of these inequalities, and are also closely tied to the conjectured Bekenstein bounds for entropy of macroscopic bodies. In addition, we give a new proof of the positivity property for the Wang-Yau mass which is used to remove the spin condition in higher dimensions. Furthermore, we generalize a recent result of Lu and Miao to obtain a localized version of the Penrose inequality for the static Wang-Yau mass.

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“…In general, the optimal isometric embedding system is difficult to solve. Suppose (X, T ) is a solution and suppose the corresponding ρ is positive, then E(Σ, X, T ) is a local minimum [8] and the nearby optimal isometric embedding system is solvable by an inverse function theorem argument. In a perturbative configuration, when a family of surfaces limit to a surface in the Minkowski spacetime, then the optimal isometric embedding system is solvable, again subject to the positivity of the limiting mass.…”

Section: The Definition Of Wang-yau Quasilocal Massmentioning

confidence: 99%