Revista Matemática Iberoamericana

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

Erasmo Caponio , Miguel Ángel Javaloyes , and Miguel Sánchez

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Abstract

We obtain some results in both Lorentz and Finsler geometries, by using a correspondence between the conformal structure (Causality) of standard stationary spacetimes on $M = \mathbb{R} \times S$ and Randers metrics on $S$. In particular: (1) For stationary spacetimes: we give a simple characterization of when $\mathbb{R} \times S$ 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 metric $\tilde R$ 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.

Article information

Source
Rev. Mat. Iberoamericana, Volume 27, Number 3 (2011), 919-952.

Dates
First available in Project Euclid: 9 August 2011

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1312906783

Mathematical Reviews number (MathSciNet)
MR2895339

Zentralblatt MATH identifier
1229.53070

Subjects
Primary: 53C22: Geodesics [See also 58E10] 53C50: Lorentz manifolds, manifolds with indefinite metrics 53C60: Finsler spaces and generalizations (areal metrics) [See also 58B20] 58B20: Riemannian, Finsler and other geometric structures [See also 53C20, 53C60]

Keywords
Finsler and Randers metrics geodesics stationary spacetimes causality in Lorentzian manifolds Cauchy horizons

Citation

Caponio, Erasmo; Javaloyes, Miguel Ángel; Sánchez, Miguel. On the interplay between Lorentzian Causality and Finsler metrics of Randers type. Rev. Mat. Iberoamericana 27 (2011), no. 3, 919--952. https://projecteuclid.org/euclid.rmi/1312906783


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