Gromov, Cauchy and causal boundaries for Riemannian, Finslerian and Lorentzian manifolds

Flores, José Luis; Herrera, J.; Sánchez, Miguel Azpitarte

doi:10.48550/arxiv.1011.1154

Cited by 10 publications

(49 citation statements)

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“…Because of ( 5), d satisfies the triangle inequality. As all the properties of a distance but symmetry are fulfilled, the pair (M, d) is referred sometimes as a generalized metric space (see [2, Section 6.2] or [19] for a detailed study). When the Finsler metric is non-reversible, d is not symmetric, because the length of a curve γ may not coincide with the length of its reverse curve γ(s) = γ(b + a − s) ∈ M .…”

Section: Finsler Metricsmentioning

confidence: 99%

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On the interplay between Lorentzian Causality and Finsler metrics of Randers type

Caponio¹,

Javaloyes²,

Sánchez³

2011

Rev. Mat. Iberoam.

147

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We obtain some results in both Lorentz and Finsler geometries, by using a correspondence between the conformal structure (Causality) of standard stationary spacetimes on M = R × S and Randers metrics on S. In particular:(1) For stationary spacetimes: we give a simple characterization of when R × S is causally continuous or globally hyperbolic (including in the latter case, when S is a Cauchy hypersurface), in terms of an associated Randers metric. Consequences for the computability of Cauchy developments are also derived.(2) For Finsler geometry: Causality suggests that the role of completeness in many results of Riemannian Geometry (geodesic connectedness by minimizing geodesics, Bonnet-Myers, Synge theorems) is played by the compactness of symmetrized closed balls in Finslerian Geometry. Moreover, under this condition we show that for any Randers metric R there exists another Randers metricR with the same pregeodesics and geodesically complete.Even more, results on the differentiability of Cauchy horizons in spacetimes yield consequences for the differentiability of the Randers distance to a subset, and vice versa.2000 Mathematics Subject Classification. 53C22, 53C50, 53C60, 58B20.

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Section: Finsler Metricsmentioning

confidence: 99%

“…(B) Even if R×S is not globally hyperbolic (or S is not Cauchy) Cauchy developments could be studied by using the Cauchy boundary associated to the Finslerian metric (see [19], for properties of this boundary).…”

Section: Cauchy Developmentsmentioning

confidence: 99%

On the interplay between Lorentzian Causality and Finsler metrics of Randers type

Caponio¹,

Javaloyes²,

Sánchez³

2011

Rev. Mat. Iberoam.

147

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“…About (ii), the conformal relation between g cl and g op is also obvious. Moreover, the future causal boundary ∂cl V can be represented by two lines T , J + with a common endpoint i + , which is the TIP equal to all V (see [18,1,6,8] for much more general computations, which include the c-boundary of all the standard static spacetimes). More precisely, the TIPs which constitute T are the chronological past of all the future-directed lightlike geodesics ρ with endpoint at x = 0.…”

Section: The Examplementioning

confidence: 99%

“…T is timelike in the sense that any two distinct TIPs P, P ∈ T satisfy either P P or P P , where the extended chronological relation can be defined here as: P P if and only if there exists some p ∈ P such that p p for all p ∈ P . It is also clear that, for the (future) chronological topology on ∂V (which here reduces to the point set convergence of the corresponding TIP's as subsets of V , see [6,7,8]) T will be homeomorphic to R. That is, in the following, T will be identified with R × {0} (each P ∈ T is identified with the endpoint in R×{0} of the lightlike geodesic whose past is equal to P ), and this identification holds at the point set, chronological and topological levels. The TIPs which constitute J + are the chronological pasts of all the future-directed lightlike ρ as above which goes to infinity (reaching arbitrarily large values of −x).…”

Section: The Examplementioning

confidence: 99%

“…Since then, a long series of redefinitions and new contributions has been carried out, and a renewed interest comes from the recent contributions by Harris [16,17,18] and Marolf and Ross [19,20] (see the review in [23] for complete references). Recently, the authors have carried out an extensive revision of both, the notion of c-boundary and the tools for its computation [5,7,8]. So, the c-boundary can be regarded now as a useful and consistent notion, which is well related to other geometric objects.…”

Section: Introductionmentioning

confidence: 99%

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Isocausal spacetimes may have different causal boundaries

Flores¹,

Herrera²,

Sánchez³

2011

Class. Quantum Grav.

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We construct an example which shows that two isocausal spacetimes, in the sense introduced recently in [11], may have c-boundaries which are not equal (more precisely, not equivalent, as no bijection between the completions can preserve all the binary relations induced by causality). This example also suggests that isocausality can be useful for the understanding and computation of the c-boundary.

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Convex Regions of Stationary Spacetimes and Randers Spaces. Applications to Lensing and Asymptotic Flatness

2015

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By using stationary-to-Randers correspondence (SRC, see [20]), a characterization of light and time-convexity of the boundary of a region of a standard stationary (n + 1)-spacetime is obtained, in terms of the convexity of the boundary of a domain in a Finsler n or (n + 1)-space of Randers type. The latter convexity is analyzed in depth and, as a consequence, the causal simplicity and the existence of causal geodesics confined in the region and connecting a point to a stationary line are characterized. Applications to asymptotically flat spacetimes include the light-convexity of hypersurfaces S n−1 (r) × R, where S n−1 (r) is a sphere of large radius in a spacelike section of an end, as well as the characterization of their time-convexity with natural physical interpretations. The lens effect of both light rays and freely falling massive particles with a finite lifetime, (i.e. the multiplicity of such connecting curves) is characterized in terms of the focalization of the geodesics in the underlying Randers manifolds.2000 Mathematics Subject Classification. 53C50, 53C60, 53C22, 58E10, 83C30.

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Gromov, Cauchy and causal boundaries for Riemannian, Finslerian and Lorentzian manifolds

Cited by 10 publications

References 0 publications

On the interplay between Lorentzian Causality and Finsler metrics of Randers type

On the interplay between Lorentzian Causality and Finsler metrics of Randers type

Isocausal spacetimes may have different causal boundaries

Convex Regions of Stationary Spacetimes and Randers Spaces. Applications to Lensing and Asymptotic Flatness

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