Null Geometry and the Einstein Equations

The null Penrose inequality, i.e. the Penrose inequality in terms of the Bondi energy, is studied by introducing a funtional on surfaces and studying its properties along a null hypersurface Ω extending to past null infinity. We prove a general Penrose-type inequality which involves the limit at infinity of the Hawking energy along a specific class of geodesic foliations called Geodesic Asymptotic Bondi (GAB), which are shown to always exist. Whenever, this foliation approaches large spheres, this inequality becomes the null Penrose inequality and we recover the results of Ludvigsen-Vickers and Bergqvist. By exploiting further properties of the functional along general geodesic foliations, we introduce an approach to the null Penrose inequality called Renormalized Area Method and find a set of two conditions which implies the validity of the null Penrose inequality. One of the conditions involves a limit at infinity and the other a condition on the spacetime curvature along the flow. We investigate their range of applicability in two particular but interesting cases, namely the shear-free and vacuum case, where the null Penrose inequality is known to hold from the results by Sauter, and the case of null shells propagating in the Minkowski spacetime. Finally, a general inequality bounding the area of the quasi-local black hole in terms of an asymptotic quantity intrinsic of Ω is derived.

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“…We next compute the derivative of H(λ). The extrinsic curvature K k along a null hypersurface satisfies the Ricatti equation [8] d dλ…”

Section: Propositionmentioning

confidence: 99%

On the Penrose inequality along null hypersurfaces

Mars

Soria

2016

Class. Quantum Grav.

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“…This means that any normal vector field of H consists entirely of null vectors. We will call the integral curves of these vector fields generators of H. By [14,Proposition 3.1] these generators (when given a suitable parametrization) are geodesics. By straightforward computations it holds that ⟨ , ⟩ = ⟨ ′ , ′ ⟩ and ⟨∇ , ⟩ = ⟨∇ ′ , ′ ⟩ whenever , ∈ H and − ′ = 1 and − ′ = 2 for some real numbers 1 , 2 .…”

Section: Smoothness Of Compact Cauchy Horizonsmentioning

confidence: 99%

Smoothness of Compact Horizons

Larsson

2014

Ann. Henri Poincaré

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“…An important property of null hypersurfaces is that the quotient metric γ Ω , the quotient extrinsic curvature K Ω and the ambient geometry (M, g) are related by the following equations (see e.g. [22]), which are the analog in the null case to the standard Gauss-Codazzi equations for non-degenerate submanifolds,…”

Section: Notation and Basic Definitionsmentioning

confidence: 99%

“…A crucial property of the geometry of a null hypersurface Ω is that, given any point p ∈ Ω and any embedded spacelike surface S p in Ω passing through p, the induced metric γ Sp of S p and the second fundamental form K k Sp of S p along the null normal k| p satisfy γ Sp (X, Y ) = γ Ω (X,Ȳ ) and K k Sp (X, Y ) = K Ω (X,Ȳ ), where X, Y ∈ T p S p (see e.g. [22]). In other words, the induced metric and the extrinsic geometry along k of any embedded spacelike surface in Ω depends only on p and not on the details of how S p is embedded in Ω.…”

Section: Penrose Inequality In the Minkowski Spacetime In Terms Of Thmentioning

confidence: 99%

“…It is well-known (see e.g. [22]) that given any point p ∈ Ω, an equivalence relation can be defined on T p Ω by means of X ∼ Y iff X − Y = ck with c ∈ R. The equivalence class of X ∈ T p Ω is denoted by X and the quotient space by T p Ω/k. The set T Ω/k = p∈Ω T p Ω/k, is endowed naturally with the structure of a vector bundle over Ω (with fibers of dimension n) which is called quotient bundle.…”

Section: Notation and Basic Definitionsmentioning

confidence: 99%

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On the Penrose inequality for dust null shells in the Minkowski spacetime of arbitrary dimension

Mars

Soria

2012

Class. Quantum Grav.

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A particular, yet relevant, particular case of the Penrose inequality involves null shells propagating in the Minkowski spacetime. Despite previous claims in the literature, the validity of this inequality remains open. In this paper we rewrite this inequality in terms of the geometry of the surface obtained by intersecting the past null cone of the original surface S with a constant time hyperplane and the "time height" function of S over this hyperplane. We also specialize to the case when S lies in the past null cone of a point and show the validity of the corresponding inequality in any dimension (in four dimensions this inequality was proved by Tod [1]). Exploiting properties of convex hypersurfaces in Euclidean space we write down the Penrose inequality in the Minkowski spacetime of arbitrary dimension n+2 as an inequality for two smooth functions on the sphere S n . We finally obtain a sufficient condition for the validity of the Penrose inequality in the four dimensional Minkowski spacetime and show that this condition is satisfied by a large class of surfaces.

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Null Geometry and the Einstein Equations

Cited by 17 publications

References 29 publications

On the Penrose inequality along null hypersurfaces

On the Penrose inequality along null hypersurfaces

Smoothness of Compact Horizons

On the Penrose inequality for dust null shells in the Minkowski spacetime of arbitrary dimension

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