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September, 2011 On the interplay between Lorentzian Causality and Finsler metrics of Randers type
Erasmo Caponio , Miguel Ángel Javaloyes , Miguel Sánchez
Rev. Mat. Iberoamericana 27(3): 919-952 (September, 2011).

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.

Citation

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Erasmo Caponio . Miguel Ángel Javaloyes . Miguel Sánchez . "On the interplay between Lorentzian Causality and Finsler metrics of Randers type." Rev. Mat. Iberoamericana 27 (3) 919 - 952, September, 2011.

Information

Published: September, 2011
First available in Project Euclid: 9 August 2011

zbMATH: 1229.53070
MathSciNet: MR2895339

Subjects:
Primary: 53C22 , 53C50 , 53C60 , 58B20

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

Rights: Copyright © 2011 Departamento de Matemáticas, Universidad Autónoma de Madrid

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Vol.27 • No. 3 • September, 2011
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