Present status of the Penrose inequality

Mars, Marc

doi:10.1088/0264-9381/26/19/193001

Cited by 149 publications

(202 citation statements)

References 167 publications

(449 reference statements)

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“…A review of the status of WCC and the Penrose inequality appears in Refs. [2,3] In this paper we study a generalization of this inequality for asymptotically anti-deSitter (AAdS) spacetimes in 4-spacetime dimensions. This requires definitions of AAdS data and quasi-local mass.…”

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confidence: 99%

Penrose inequality in anti–de Sitter space

Husain

Singh

2017

Phys. Rev. D

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For asymptotically flat spacetimes the Penrose inequality gives an initial data test for the weak cosmic censorship hypothesis. We give a formulation of this inequality for asymptotically anti-deSitter (AAdS) spacetimes, and show that the inequality holds for time asymmetric data in spherical symmetry. Our analysis is motivated by the constant-negative-spatial-curvature form of the AdS black hole metric.Keywords: Cosmic Censorship, Penrose Inequality, anti-deSitter spacetime One of the most important unresolved questions in classical general relativity is Penrose's cosmic censorship hypothesis. The weak version of this hypothesis, Weak Cosmic Censorship (WCC), is the statement that a spacetime singularity arising from gravitational collapse of "reasonable" matter must be shrouded by an event horizon [1].Penrose also proposed an "initial data test" suggested by the WCC hypothesis. Consider an asymptotically flat solution of Einstein equations with matter satisfying the dominant energy condition. Then if a Cauchy slice of this solution contains an outer-trapped 2-surface S of area A(S), and if M is the Arnowitt-Deser-Misner (ADM) mass of the data on the slice, the conjectured inequality is(1)The intuition behind this inequality is that the mass m ah contained within the outermost apparent horizon cannot exceed the total mass M of the data. This is expected to be the case for an ongoing gravitational collapse since, in general, the region between this horizon and spatial infinity will contain uncollapsed (positive energy) matter. Thus the inequality may equivalently be written aswith equality in the quiescent black hole state, when all matter has fallen into the horizon. A review of the status of WCC and the Penrose inequality appears in Refs. [2,3] In this paper we study a generalization of this inequality for asymptotically anti-deSitter (AAdS) spacetimes in 4-spacetime dimensions. This requires definitions of AAdS data and quasi-local mass. The former is well known (see eg.[4]), and the latter requires a suitable generalization of one several mass definitions [5] to AAdS. The basic requirement is that any quasi-local mass should give the total mass of data in the asymptotic limit. This is guided by, and tested for, using the Schwarzschild-AdS spacetime.We will use the Hawking mass [6], which has been used to study the Penrose inequality in the asymptotically flat case [7]. This quantity is a measure of the mass contained within a closed 2-surface S in a spatial slice of spacetime. The definition iswhere A S is the area of S and θ ± are the future directed ingoing (−) and outgoing (+) null expansions of S. In terms of the ADM data (q ab , K ab ) these arewhere s a is the unit normal to S. As a prelude to introducing the class of data we will work with, let us note that the AdS black hole may be written in a form analogous to the Painleve-Gulstrand (PG) form, but rather than using the flat-slice PG form for AdS, it is more convenient to work with 3−slices of constant negative curvature. With such a choice of coord...

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confidence: 99%

Penrose inequality in anti–de Sitter space

Husain

Singh

2017

Phys. Rev. D

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“…An excellent survey of this conjecture is found in [24]. Suppose that the Cauchy data (M 3 , g, k) is complete, satisfies the nonnegative energy density condition µ ≥ |J|, and is Schwarzschild at infinity with total mass m in a chosen end.…”

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“…Penrose's heuristic argument for a future apparent horizon in this conjecture is described in more detail in [5] and [24] but roughly goes as follows: If, as is generally thought, asymptotically flat spacetimes eventually settle down to a Kerr spacetime [16], then in the distant future inequality 10 will be satisfied since explicit calculation verifies this fact for Kerr spacetimes, where A is the area of the event horizon. Given that some energy may radiate out to infinity, the total mass of these slices of Kerr may be less than the original total mass.…”

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confidence: 99%

P. D. E.'s Which Imply the Penrose Conjecture

Bray¹,

Khuri²

2011

Asian Journal of Mathematics

130

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Abstract. In this paper, we show how to reduce the Penrose conjecture to the known Riemannian Penrose inequality case whenever certain geometrically motivated systems of equations can be solved. Whether or not these special systems of equations have general existence theories is therefore an important open problem. The key tool in our method is the derivation of a new identity which we call the generalized Schoen-Yau identity, which is of independent interest. Using a generalized Jang equation, we use this identity to propose canonical embeddings of Cauchy data into corresponding static spacetimes. In addition, we discuss the Carrasco-Mars counterexample to the Penrose conjecture for generalized apparent horizons (added since the first version of this paper was posted on the arXiv) and instead conjecture the Penrose inequality for time-independent apparent horizons, which we define.

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“…This result was proved by [27] and [7]. The general case remains open, see the review article [29]. Finally, we have discussed the concept of total energy and linear momentum of an isolated system.…”

Section: Further Results and Open Problemsmentioning

confidence: 77%

“…Integrating this equation in R 3 , using for the first term in the right-hand side the Gauss theorem, the condition ψ → 1 as r → ∞ and the expression (1.23) for the energy we finally obtain 29) where dv 0 is the flat volume element. This formula proves that for metric of the form (1.17) we have E ≥ 0 if R ≥ 0 and E = 0 if and only if h ij = δ ij .…”

Section: Corollary 123 (Riemannian Positive Mass Theorem)mentioning

confidence: 99%

Positive Energy Theorems in General Relativity

Dain

2014

Springer Handbook of Spacetime

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Present status of the Penrose inequality

Cited by 149 publications

References 167 publications

Penrose inequality in anti–de Sitter space

Penrose inequality in anti–de Sitter space

P. D. E.'s Which Imply the Penrose Conjecture

Positive Energy Theorems in General Relativity

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