Extremal surfaces and the rigidity of null geodesic incompleteness

Silva, I P Costa e; Flores, José Luis

doi:10.1088/0264-9381/32/5/055010

Cited by 3 publications

(3 citation statements)

References 29 publications

(66 reference statements)

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“…One should also consider what happens when one or more of the hypotheses in the singularity theorems are relaxed, or suppressed altogether. This has recently been addressed in [75], trying to find theorems with milder conclusions, and considering the 'rigidity' part of the singularity theorems, see also [134]. This may open new lines worth exploring.…”

Section: Mathematical Advancesmentioning

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: Mathematical Advancesmentioning

confidence: 99%

The 1965 Penrose singularity theorem

Senovilla

Garfinkle

2015

Class. Quantum Grav.

193

226

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“…Although it might be reasonable to conjecture that a C 1;1 version of Borde and Minguzzi's theorems and the consequences obtained here could be stated based on Ref. [27], we leave the proof for the interested reader.…”

Section: On the Regularity Of The Metricmentioning

confidence: 82%

“…In general, Penrose [1], Hawking [25], and Hawking and Penrose's singularity theorems [5] hold for C 2 Lorentzian metrics. However, these results have recently been extended for metrics that are C 1;1 (metrics that are differentiable, with all derivatives locally Lipschitz) [26][27][28], which is of both mathematical and physical significance. In a case in which the regularity of the metric drops to C 1;1 , the classical theorems would predict that the curvature would become discontinuous rather than unbounded, which corresponds to a finite jump in the matter variables, and therefore such a situation would hardly be regarded as singular.…”

Section: On the Regularity Of The Metricmentioning

confidence: 99%