Raychaudhuri equation and singularity theorems in Finsler spacetimes

Minguzzi, E.

doi:10.1088/0264-9381/32/18/185008

Cited by 27 publications

(75 citation statements)

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“…In Let us come to Penrose's 1965 singularity theorem [93]. It was generalized to the round cone (Lorentzian) C 1,1 case [9], and to the non-round (Lorentz-Finsler) C 2 case [12]. We need a definition.…”

Section: Singularity Theoremsmentioning

confidence: 99%

“…In its first version Hawking's theorem included a global hyperbolicity assumption which was removed in [1]. A C 1,1 Lorentzian version can be found in [10] and a non-round (Lorentz-Finsler) C 2 version can be found in [12]. The argument of Hawking's theorem in the C 2 Lorentzian case looks for a contradiction in the timelike completeness assumption.…”

Section: Singularity Theoremsmentioning

confidence: 99%

“…Below we give a causal version of Hawking and Penrose's 1970 singularity theorem [93]. It has been recently generalized to C 1,1 regularity [11] and to the nonround (Lorentz-Finsler) C 2 case [12]. As for Penrose's theorem the key for the generalization to the closed cone structure case stays in the stability of compact trapped sets.…”

Section: Singularity Theoremsmentioning

confidence: 99%

“…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: 99%

Section: Singularity Theoremsmentioning

confidence: 99%

Section: Singularity Theoremsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Causality theory for closed cone structures with applications

Minguzzi

2019

Rev. Math. Phys.

214

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“…Also, the Raychaudhuri equation and singularity theorems have been recently stated in a Finsler spacetime (cf. [16]). …”

Section: Raychaudhuri Equations In a 5d Universementioning

confidence: 99%

Kinematic quantities and Raychaudhuri equations in a 5D universe

Bejancu

2015

Eur. Phys. J. C

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Based on some ideas that have emerged from the classical Kaluza-Klein theory, we present a 5D universe as a product bundle over the 4D spacetime. This enables us to introduce and study two categories of kinematic quantities (expansions, shear, vorticity) in a 5D universe. One category is related to the fourth dimension (time), and the other one comes from the assumption of the existence of the fifth dimension. The Raychaudhuri type equations that we obtain in the paper, lead us to results on the evolution of both the 4D expansion and the 5D expansion in a 5D universe.

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Affine Sphere Relativity

Minguzzi

2016

Commun. Math. Phys.

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We investigate spacetimes whose light cones could be anisotropic. We prove the equivalence of the structures: (a) Lorentz-Finsler manifold for which the mean Cartan torsion vanishes, (b) Lorentz-Finsler manifold for which the indicatrix (observer space) at each point is a convex hyperbolic affine sphere centered on the zero section, and (c) pair given by a spacetime volume and a sharp convex cone distribution. The equivalence suggests to describe (affine sphere) spacetimes with this structure, so that no algebraic-metrical concept enters the definition. As a result, this work shows how the metric features of spacetime emerge from elementary concepts such as measure and order. Non-relativistic spacetimes are obtained replacing proper spheres with improper spheres, so the distinction does not call for group theoretical elements. In physical terms, in affine sphere spacetimes the light cone distribution and the spacetime measure determine the motion of massive and massless particles (hence the dispersion relation). Furthermore, it is shown that, more generally, for Lorentz-Finsler theories non-differentiable at the cone, the lightlike geodesics and the transport of the particle momentum over them are well defined though the curve parametrization could be undefined. Causality theory is also well behaved. Several results for affine sphere spacetimes are presented. Some results in Finsler geometry, for instance in the characterization of Randers spaces, are also included.

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Raychaudhuri equation and singularity theorems in Finsler spacetimes

Cited by 27 publications

References 97 publications

Causality theory for closed cone structures with applications

Causality theory for closed cone structures with applications

Kinematic quantities and Raychaudhuri equations in a 5D universe

Affine Sphere Relativity

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