Transformations of asymptotically AdS hyperbolic initial data and associated geometric inequalities

Sle, Ye; Khuri, Marcus

doi:10.1007/s10714-017-2323-7

Cited by 4 publications

(7 citation statements)

References 49 publications

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“…We would like to point the reader to the interesting work by Carrasco and Mars [10] giving insights into the (over-)generality of H = 0 as a condition for a horizon. In a recent paper of Cha and Khuri [12], the area A of the outermost STCMC-surface with H = 2 appears in the conjectured Penrose Inequality m ≥ √ A /16π expected to hold for an asymptotically anti-de Sitter initial data set of mass m satisfying the dominant energy condition. Because of the spacetime geometry nature of the STCMC-condition, we expect that STCMC-surfaces and STCMC-foliations will have a number of applications beyond the definition of a center of mass of an isolated system as well as beyond the setting of asymptotically Euclidean initial data sets.…”

Section: Introduction and Goalsmentioning

confidence: 99%

On center of mass and foliations by constant spacetime mean curvature surfaces for isolated systems in General Relativity

Cederbaum

Sakovich

2021

Calc. Var.

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We propose a new foliation of asymptotically Euclidean initial data sets by 2-spheres of constant spacetime mean curvature (STCMC). The leaves of the foliation have the STCMC-property regardless of the initial data set in which the foliation is constructed which asserts that there is a plethora of STCMC 2-spheres in a neighborhood of spatial infinity of any asymptotically flat spacetime. The STCMC-foliation can be understood as a equivariant relativistic generalization of the CMC-foliation suggested by Huisken and Yau (Invent Math 124:281–311, 1996). We show that a unique STCMC-foliation exists near infinity of any asymptotically Euclidean initial data set with non-vanishing energy which allows for the definition of a new notion of total center of mass for isolated systems. This STCMC-center of mass transforms equivariantly under the asymptotic Poincaré group of the ambient spacetime and in particular evolves under the Einstein evolution equations like a point particle in Special Relativity. The new definition also remedies subtle deficiencies in the CMC-approach to defining the total center of mass suggested by Huisken and Yau (Invent Math 124:281–311, 1996) which were described by Cederbaum and Nerz (Ann Henri Poincaré 16:1609–1631, 2015).

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Section: Introduction and Goalsmentioning

confidence: 99%

On center of mass and foliations by constant spacetime mean curvature surfaces for isolated systems in General Relativity

Cederbaum

Sakovich

2021

Calc. Var.

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“…Her method derives from a deformation argument of Schoen and Yau [70], which involves solving the Jang equation so that the solution admits hyperboloidal asymptotics. The induced metric on the Jang graph is then asymptotically flat, with ADM mass that agrees (up to a positive multiplicative factor) with the hyperbolic mass; further generalizations of this deformation procedure have been studied in [20,19]. The desired result then follows from the positive mass theorem in the asymptotically flat setting.…”

Section: Introductionmentioning

confidence: 92%

“…Next, notice that a consequence of (8. 19) is that ∇E is symmetric, and hence E is closed as a 1-form. Since the first cohomology H 1 (M ) is trivial, there exists a globally defined function h such that E = dh.…”

Section: A Positive Mass Theorem With Chargementioning

confidence: 99%

Spacetime Harmonic Functions and Applications to Mass

Bray¹,

Hirsch²,

Kazaras³

et al. 2021

Preprint

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In the pioneering work of Stern [73], level sets of harmonic functions have been shown to be an effective tool in the study of scalar curvature in dimension 3. Generalizations of this idea, utilizing level sets of so called spacetime harmonic functions as well as other elliptic equations, are similarly effective in treating geometric inequalities involving the ADM mass. In this paper, we survey recent results in this context, focusing on applications of spacetime harmonic functions to the asymptotically flat and asymptotically hyperbolic versions of the spacetime positive mass theorem, and additionally introduce a new concept of total mass valid in both settings which is encoded in interpolation regions between generic initial data and model geometries. Furthermore, a novel and elementary proof of the positive mass theorem with charge is presented, and the level set approach to the Penrose inequality given by Huisken and Ilmanen is related to the current developments. Lastly, we discuss several open problems.

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“…This result effectively reduces the proof of the geometrical inequality to proving existence of solutions to certain equations with appropriate fall off conditions. Cha and Khuri (2017) apply the same arguments to obtain an identical global inequality to (150) in the case that the initial data has two ends, one AdS hyperbolic and the other either asymptotically AdS hyperbolic or asymptotically cylindrical. The cosmological constant is not, however, explicitly included into the inequality.…”

Section: Non Asymptotically Flat Manifoldsmentioning

confidence: 98%

“…As we discuss in Sect. 3.2, Cha and Khuri (2017) study asymptotically AdS hyperbolic initial data and prove a global inequality where the cosmological constant does not appear explicitly. They conjecture, though, that an inequality of the form…”

Section: Cosmological Constantmentioning

confidence: 99%

Geometrical inequalities bounding angular momentum and charges in General Relativity

Dain

Gabach-Clément

2018

Living Rev Relativ

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Geometrical inequalities show how certain parameters of a physical system set restrictions on other parameters. For instance, a black hole of given mass can not rotate too fast, or an ordinary object of given size can not have too much electric charge. In this article, we are interested in bounds on the angular momentum and electromagnetic charges, in terms of total mass and size. We are mainly concerned with inequalities for black holes and ordinary objects. The former are the most studied systems in this context in General Relativity, and where most results have been found. Ordinary objects, on the other hand, present numerous challenges and many basic questions concerning geometrical estimates for them are still unanswered. We present the many results in these areas. We make emphasis in identifying the mathematical conditions that lead to such estimates, both for black holes and ordinary objects.

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Transformations of asymptotically AdS hyperbolic initial data and associated geometric inequalities

Cited by 4 publications

References 49 publications

On center of mass and foliations by constant spacetime mean curvature surfaces for isolated systems in General Relativity

On center of mass and foliations by constant spacetime mean curvature surfaces for isolated systems in General Relativity

Spacetime Harmonic Functions and Applications to Mass

Geometrical inequalities bounding angular momentum and charges in General Relativity

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