Note on cosmic censorship

Jang, Pong Soo

doi:10.1103/physrevd.20.834

Cited by 49 publications

(54 citation statements)

References 15 publications

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“…For spacetimes with electromagnetic fields the inequality M ≥ |Q|, where Q is the electric charge [11,12], has been shown (also in the presence of apparent horizons) whenever the norm of the charge 4-current is not larger than the norm of the matter 4-current. Moreover, both for the case k ij = 0 as well as for general horizons in spherically symmetric spacetimes, it is known that M ≥ A/16π + Q 2 π/A provided all charges are inside H [16,21]. Under the same requirements, the same generalized PI can be shown for general non-spherical horizons by applying the same arguments as in [21] to our final expression (11).…”

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confidence: 62%

On the Penrose Inequality for General Horizons

Mars

Simon

2002

Phys. Rev. Lett.

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For asymptotically flat initial data of Einstein's equations satisfying an energy condition, we show that the Penrose inequality holds between the ADM mass and the area of an outermost apparent horizon, if the data are restricted suitably. We prove this by generalizing Geroch's proof of monotonicity of the Hawking mass under a smooth inverse mean curvature flow, for data with non-negative Ricci scalar. Unlike Geroch we need not confine ourselves to minimal surfaces as horizons. Modulo smoothness issues we also show that our restrictions on the data can locally be fulfilled by a suitable choice of the initial surface in a given spacetime. An important issue in General Relativity is the "cosmic censorship conjecture" which reads, roughly speaking, that singularities (which necessarily develop in gravitational collapse in the future of apparent horizons) are always separated from the outside world by event horizons. Penrose gave a heuristic argument which showed that, for collapsing shells of particles with zero rest mass, cosmic censorship would imply the inequalitybetween the total mass (the ADM-mass) M of the spacetime and the area A of an apparent horizon H [1]. While for such shells inequality (1) was recently proven by Gibbons [2], there has remained the challenge of proving (independently of cosmic censorship) the Penrose inequality (PI) (1) in the "general case" characterized below.We consider a spacetime (M, 4 g µν ) and a corresponding initial data set (N , g ij , k ij ), i.e. a smooth 3-manifold N (which can be embedded in M), a positive definite metric g ij and a symmetric tensor k ij (the second fundamental form of the embedding). To be compatible with Einstein's equations on M, these quantities satisfy the constraintswhere D i is the covariant derivative and R the Ricci scalar on N , k = g ij k ij , and ρ and j i are the energy density and the matter current, respectively. We take the data to be asymptotically flat and to satisfy the dominant energy condition (which implies that ρ ≥ |j|). Furthermore, we assume that the boundary of N (if non-empty) is a "future apparent horizon" H (called a "horizon" from now on) which is a 2-surface defined by the property that all outgoing future directed null geodesics (in M) orthogonal to H have vanishing divergence, i.e. θ + = 0, and that the divergence of the outgoing past null geodesics is non-negative, i.e. θ − ≥ 0. (Past apparent horizons are defined similarly, and satisfy analogous theorems). If θ + = θ − = 0 on H, the horizon is an extremal surface of (N , g ij ) and the outermost extremal surface in an asymptotically flat space must be minimal.As an idea for proving the positive mass theorem (PMT) Geroch considered an asymptotically flat Riemannian manifold (N , g ij ) with non-negative Ricci scalar R (which naturally arises from initial data sets described above by restricting k ij suitably) and assumed that on (N , g ij ) there is a smooth "inverse mean curvature flow" (IMCF). This means that one can write the metric g ij as(where A, B = 2, 3), with smoo...

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confidence: 62%

On the Penrose Inequality for General Horizons

Mars

Simon

2002

Phys. Rev. Lett.

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“…Proof. The proof is inspired in Reiris's proof [16] and it is a simple consequence of the results presented in [11] and [10]. The crucial property of the IMCF is the Geroch monotonicity of the Hawking energy.…”

Section: Proof Of the Inequality Between Charge Energy And Size For mentioning

confidence: 94%

Bekenstein bounds and inequalities between size, charge, angular momentum, and energy for bodies

Dain

2015

Phys. Rev. D

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Bekenstein bounds for the entropy of a body imply a universal inequality between size, energy, angular momentum and charge. We prove this inequality in Electromagnetism. We also prove it, for the particular case of zero angular momentum, in General Relativity. We further discuss the relation of these inequalities with inequalities between size, angular momentum and charge recently studied in the literature.

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“…Although the weaker form of the inequality, the Bogomolny inequality m ADM ≥ | e | /G , has been proven (under assumptions on the matter content, see, e.g., [219, 508, 344, 217, 371, 213]), Gibbons’ inequality for the electric charge has been proven for special cases (for spherically-symmetric spacetimes see, e.g., [251]), and for time-symmetric initial data sets using Geroch’s inverse mean curvature flow [290]. As a consequence of the results of [278, 279] the latter has become a complete proof.…”

Section: Applications In General Relativitymentioning

confidence: 99%

Quasi-Local Energy-Momentum and Angular Momentum in General Relativity

Szabados

2009

Living Rev. Relativ.

324

390

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The present status of the quasi-local mass, energy-momentum and angular-momentum constructions in general relativity is reviewed. First, the general ideas, concepts, and strategies, as well as the necessary tools to construct and analyze the quasi-local quantities, are recalled. Then, the various specific constructions and their properties (both successes and deficiencies are discussed. Finally, some of the (actual and potential) applications of the quasi-local concepts and specific constructions are briefly mentioned.

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Note on cosmic censorship

Abstract: For initial data sets which represent charged black holes we prove some inequalities which relate the total energy, the total charge, and the size of the black hole. One of them is a necessary condition for the validity of cosmic censorship.

Cited by 49 publications

References 15 publications

On the Penrose Inequality for General Horizons

On the Penrose Inequality for General Horizons

Bekenstein bounds and inequalities between size, charge, angular momentum, and energy for bodies

Quasi-Local Energy-Momentum and Angular Momentum in General Relativity

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