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

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

2009-03-20

arxiv.org/abs/0903.3501v2

Rev. Mat. Iberoamericana 27:919-952, 2011

Mathematics

Differential Geometry

26 pages, AMSLaTex. Accepted for publication on Rev. Mat. Iberoamericana. v2: improved presentation of the results

Scientific paper

10.4171/RMI/658

We obtain some results in both Lorentz and Finsler geometries, by using a correspondence between the conformal structure (Causality) of standard stationary spacetimes on $M=\R\times S$ and Randers metrics on $S$. In particular, for stationary spacetimes, we give a simple characterization of when they are 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. Causality suggests that the role of completeness in many results of Riemannian Geometry (geodesic connectedness by minimizing geodesics, Bonnet-Myers, Synge theorems) is played, in Finslerian Geometry, by the compactness of symmetrized closed balls. Moreover, under this condition we show that for any Randers metric there exists another Randers metric 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.

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