Revista Matemática Iberoamericana

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Volume 27, Issue 3, 2011, pp. 919–952
DOI: 10.4171/RMI/658

Published online: 2011-12-06

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

Erasmo Caponio[1], Miguel Angel Javaloyes[2] and Miguel Sánchez[3]

(1) Politecnico di Bari, Italy
(2) Universidad de Granada, Spain
(3) Universidad de Granada, Spain

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.

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

Caponio Erasmo, Javaloyes Miguel Angel, Sánchez Miguel: On the interplay between Lorentzian Causality and Finsler metrics of Randers type. Rev. Mat. Iberoam. 27 (2011), 919-952. doi: 10.4171/RMI/658