The Hawking–Penrose Singularity Theorem for C 1,1-Lorentzian Metrics

Graf, Melanie; Grant, James D. E.; Kunzinger, Michael; Steinbauer, Roland

doi:10.1007/s00220-017-3047-y

Cited by 43 publications

(67 citation statements)

References 52 publications

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“…Each singularity theorem has a core which relates such causality concepts, and it is this type of result which is preserved. Our generalization is therefore of a different nature with respect to that found in [9][10][11] where the authors assume the stronger C 1,1 regularity but make sense of some other analytical objects entering the classical theorems. Also, we shall not recall the classical versions of the singularity theorems, nor shall we explain in detail why our theorems provide the causality content of such statements.…”

Section: Singularity Theoremsmentioning

confidence: 89%

“…The problem was solved in [6][7][8] where it was shown that under a C 1,1 differentiability assumption convex neighborhoods do exist and the exponential map provides a local lipeomorphism. From here most results of causality theory follow [6]; Kunzinger and collaborators explored the validity of the singularity theorems under weak differentiability assumption [9][10][11], while the author considered the non-isotropic case [12].…”

Section: Introductionmentioning

confidence: 99%

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Causality theory for closed cone structures with applications

Minguzzi

2019

Rev. Math. Phys.

214

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We develop causality theory for upper semi-continuous distributions of cones over manifolds generalizing results from mathematical relativity in two directions: non-round cones and non-regular differentiability assumptions. We prove the validity of most results of the regular Lorentzian causality theory including: causal ladder, Fermat's principle, notable singularity theorems in their causal formulation, Avez-Seifert theorem, characterizations of stable causality and global hyperbolicity by means of (smooth) time functions. For instance, we give the first proof for these structures of the equivalence between stable causality, K-causality and existence of a time function. The result implies that closed cone structures that admit continuous increasing functions also admit smooth ones. We also study proper cone structures, the fiber bundle analog of proper cones. For them we obtain most results on domains of dependence. Moreover, we prove that horismos and Cauchy horizons are generated by lightlike geodesics, the latter being defined through the achronality property. Causal geodesics and steep temporal functions are obtained with a powerful product trick. The paper also contains a study of Lorentz-Minkowski spaces under very weak regularity conditions. Finally, we introduce the concepts of stable distance and stable spacetime solving two well known problems (a) the characterization of Lorentzian manifolds embeddable in Minkowski spacetime, they turn out to be the stable spacetimes, (b) the proof that topology, order and distance (with a formula a la Connes) can be represented by the smooth steep temporal functions. The paper is self-contained, in fact we do not use any advanced result from mathematical relativity. arXiv:1709.06494v4 [gr-qc] 6 Dec 2018 Proposition 2.9. The relationJ is open, transitive and contained in J. Moreover, in a proper cone structure I ⊂J,J =J and ∂J = ∂J.One should be careful because in generalJ + (p) Int(J + (p)). Proof.It is open by definition, so let us prove its transitivity. Let (p, q) ∈J and (q, r) ∈J, then there are is a product neighborhood which satisfies (p, q) ∈ Causality theory for closed cone structures with applications 21 U × V 1 ⊂ J, and a product neighborhood which satisfies (q, r) ∈ V 2 × W ⊂ J. Since U × {q} ∪ {q} × W ⊂ J we have by composition U × W ⊂ J, thus (p, r) ∈J. For the last statement of the proposition we need only to prove J ⊂J. Let (p, q) ∈ J and let p p, q q, then since I is open and contained inJ, (p , q ) ∈J. Since p can be taken arbitrarily close to p, and analogously, q can be taken arbitrarily close to q, we have (p, q) ∈J.

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Section: Singularity Theoremsmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Causality theory for closed cone structures with applications

Minguzzi

2019

Rev. Math. Phys.

214

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“…Hence the metric (3) is of local Lipschitz regularity in u which is beyond the reach of classical smooth Lorentzian geometry. The latter reaches down to C 1,1 -at least as far as convexity and causality are concerned [16,13,14,5]. However, the Lipschitz property is decisive since it prevents the most dramatic downfalls in causality theory which are known to occur for Hölder continuous metrics [9,3,30,6,17].…”

Section: Introductionmentioning

confidence: 99%

Cut-and-paste for impulsive gravitational waves with Λ : The geometric picture

et al. 2019

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Impulsive gravitational waves in Minkowski space were introduced by Roger Penrose at the end of the 1960s, and have been widely studied over the decades. Here we focus on non-expanding waves which later have been generalised to impulses travelling in all constantcurvature backgrounds, that is also the (anti-)de Sitter universe. While Penrose's original construction was based on his vivid geometric 'scissors-and-paste' approach in a flat background, until now a comparably powerful visualisation and understanding have been missing in the Λ = 0 case. In this work we provide such a picture: The (anti-)de Sitter hyperboloid is cut along the null wave surface, and the 'halves' are then re-attached with a suitable shift of their null generators across the wave surface. This special family of global null geodesics defines an appropriate comoving coordinate system, leading to the continuous form of the metric. Moreover, it provides a complete understanding of the nature of the Penrose junction conditions and their specific form. These findings shed light on recent discussions of the memory effect in impulsive waves.

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“…In a nutshell, while some features of causality theory are rather robust and topological in nature, other results are usually proved by local arguments involving geodesically convex neighbourhoods, which do exist for C 1,1 -metrics [Min15,KSS14]. Consequently for this regularity class the bulk of causality theory [CG12,Min15,KSSV14] including the singularity theorems [KSSV15, KSV15,GGKS18] remains valid. Moreover, arguments from causality theory which neither explicitly involve the exponential map nor geodesics have been found to extend to locally Lipschitz metrics, see [CG12, Thm.…”

Section: Introductionmentioning

confidence: 99%

The future is not always open

et al. 2019

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We demonstrate the breakdown of several fundamentals of Lorentzian causality theory in low regularity. Most notably, chronological futures (defined naturally using locally Lipschitz curves) may be non-open, and may differ from the corresponding sets defined via piecewise C 1 -curves. By refining the notion of a causal bubble from [CG12], we characterize spacetimes for which such phenomena can occur, and also relate these to the possibility of deforming causal curves of positive length into timelike curves (push-up). The phenomena described here are, in particular, relevant for recent synthetic approaches to low regularity Lorentzian geometry where, in the absence of a differentiable structure, causality has to be based on locally Lipschitz curves.

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The Hawking–Penrose Singularity Theorem for C 1,1-Lorentzian Metrics

Cited by 43 publications

References 52 publications

Causality theory for closed cone structures with applications

Causality theory for closed cone structures with applications

Cut-and-paste for impulsive gravitational waves with Λ : The geometric picture

The future is not always open

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