Some Examples of Trapped Surfaces

Bengtsson, Ingemar

doi:10.1142/9789814425704_0009

Cited by 6 publications

(6 citation statements)

References 41 publications

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“…A key result is that the boundary of the region containing OTS in a given spacelike hypersurface turns out to be a smooth MOTS [203,7]. There are key differences between MTSs and mere MOTSs, as the latter need an outer notion and they are usually required to enclosed a piece of a spacelike hypersurface; for examples and an enlightening discussion see [23]. In the case that the foliating surfaces are truly marginally future trapped the mentioned tubes were introduced in [175] and called "future trapping horizons".…”

Section: Trapped Submanifoldsmentioning

confidence: 99%

The 1965 Penrose singularity theorem

Senovilla

Garfinkle

2015

Class. Quantum Grav.

193

226

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We review the first modern singularity theorem, published by Penrose in 1965. This is the first genuine post-Einstenian result in General Relativity, where the fundamental and fruitful concept of closed trapped surface was introduced. We include historical remarks, an appraisal of the theorem's impact, and relevant current and future work that belongs to its legacy.Singularity theorems 2 these will be discussed in sections 2 and 3-Penrose's theorem is, without a doubt, the first such theorem in its modern form containing new important ingredients and fruitful ideas that immediately led to many new developments in theoretical relativity, and to devastating physical consequences concerning the origin of the Universe and the collapse of massive stars, see sections 5 and 6. In particular, Penrose introduced geodesic incompleteness to characterize singularities (see subsection 4.1), used the notion of a Cauchy hypersurface (and thereby of global hyperbolicity, see section 4) and, more importantly, he presented the gravitational community with a precious gift in the form of a novel concept: closed trapped surfaces, see subsections 4.2 and 7.2.The fundamental, germinal and very fruitful notion of closed trapped surface is a key central idea in the physics of Black Holes, Numerical Relativity, Mathematical Relativity, Cosmology and Gravity Analogues. It has had an enormous influence as explained succinctly in section 5 and, in its refined contemporary versions -see subsections 7.2 and 7.3-, keeps generating many more advances (sections 7 and 8) of paramount importance and will probably maintain such prolific legacy with some unexpected applications in gravitational physics.As argued elsewhere [288], the singularity theorems constitute the first genuine post-Einstenian content of classical GR, not foreseen in any way by Einstein -as opposed to many other "milestones" discussed in this issue. § The global mathematical developments needed for the singularity theorems, and the ideas on incompleteness or trapping -and thus also their derived inferences -were not treated nor mentioned, neither directly nor indirectly, in any of Einstein's writings. In 1965 GR left adolescence behind, emancipated from its creator, and became a mature physical theory full of vitality and surprises. Before 1955Prior to 1965 there were many indications that the appearance of some kind of catastrophic irregularities, say "singularities", was common in GR. We give a succinct summary of some selected and instructive cases, with side historical remarks. Friedman closed models and the de Sitter solutionAs an early example, in 1922 Friedman [120] looked for solutions of (1) with the formwhere a(t) is a function of time t, F is an arbitrary function, and dΩ 2 represents the standard metric for a round sphere of unit radius. This Ansatz followed previous discussions by Einstein and de Sitter [97,87] where the space (for each t =const.) was § As a side historical remark, Einstein himself wrote a paper on "singularities" [99], later generalized to higher di...

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Section: Trapped Submanifoldsmentioning

confidence: 99%

The 1965 Penrose singularity theorem

Senovilla

Garfinkle

2015

Class. Quantum Grav.

193

226

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“…(6.12) and (6.16), respectively, in the form (6.18) where the second equation is also obtained from the first one by the replacement a → i a. Looking at these equations perturbatively (in a) one can solve exactly the condition on the left-hand-side (for W 0 (θ)) 20) and use then this solution on the right-hand-side to find the O(a 2 )-corrections according to standard procedures. We will not develop further this approach, which is beyond the scope of the present study.…”

Section: Proof Of Realizability Of Trapped Surfaces In Flrw Cosmologiesmentioning

confidence: 99%

On the local isometric embedding of trapped surfaces into three-dimensional Riemannian manifolds

Bini¹,

Esposito²

2018

Class. Quantum Grav.

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We study trapped surfaces from the point of view of local isometric embedding into three-dimensional Riemannian manifolds. When a two-surface is embedded into three-dimensional Euclidean space, the problem of finding all surfaces applicable upon it gives rise to a non-linear partial differential equation of the Monge-Ampère type, first discovered by Darboux, and later reformulated by Weingarten. Even today, this problem remains very difficult, despite some remarkable results. We find an original way of generalizing the Darboux technique, which leads to a coupled set of 6 non-linear partial differential equations. For the 3-manifolds occurring in Friedmann-(Lemaitre)-Robertson-Walker cosmologies, we show that the local isometric embedding of trapped surfaces into them can be proved by solving just one non-linear equation. Such an equation is here solved for the three kinds of Friedmann model associated with positive, zero, negative curvature of spatial sections, respectively. ‡ We will include g in the definition of inner product, i.e. < A, B >g, only when necessary, i.e. when it involves vectors of 2-and 3-dimensional sub-spaces.

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“…It has been shown [7,84] that, in a collapsing Vaidya spacetime, an event horizon forms and grows starting from the centre and an observer can cross it and be unaware of it even though his or her causal past consists entirely of a portion of Minkowski space: the event horizon cannot be detected by this observer with a physical experiment. In other words, the event horizon "knows" about events belonging to a spacetime region very far away and in its future but not causally connected to it ("clarvoyance" [5,89,90]).…”

Section: Event Horizonsmentioning

confidence: 99%

Evolving Black Hole Horizons in General Relativity and Alternative Gravity

Faraoni

2013

Galaxies

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From the microscopic point of view, realistic black holes are time-dependent and the teleological concept of the event horizon fails. At present, the apparent or trapping horizon seem to be its best replacements in various areas of black hole physics. We discuss the known phenomenology of apparent and trapping horizons for analytical solutions of General Relativity and alternative theories of gravity. These specific examples (we focus on spherically symmetric inhomogeneities in a background cosmological spacetime) are useful as toy models for research on various aspects of black hole physics

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Some Examples of Trapped Surfaces

Abstract: We present some simple pen and paper examples of trapped surfaces in order to help in visualising this key concept of the theory of gravitational collapse. We collect these examples from time-symmetric initial data, 2+1 dimensions, collapsing null shells, and the Vaidya solution.

Cited by 6 publications

References 41 publications

The 1965 Penrose singularity theorem

The 1965 Penrose singularity theorem

On the local isometric embedding of trapped surfaces into three-dimensional Riemannian manifolds

Evolving Black Hole Horizons in General Relativity and Alternative Gravity

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