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

  • Erasmo Caponio

    Politecnico di Bari, Italy
  • Miguel Angel Javaloyes

    Universidad de Granada, Spain
  • Miguel Sánchez

    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=R×SM = \mathbb{R} \times S and Randers metrics on SS. In particular: (1) For stationary spacetimes: we give a simple characterization of when R×S\mathbb{R} \times S is causally continuous or globally hyperbolic (including in the latter case, when SS 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 RR there exists another Randers metric R~\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.

Cite this article

Erasmo Caponio, Miguel Angel Javaloyes, Miguel Sánchez, On the interplay between Lorentzian Causality and Finsler metrics of Randers type. Rev. Mat. Iberoam. 27 (2011), no. 3, pp. 919–952

DOI 10.4171/RMI/658