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

TL;DR

This paper explores the relationship between Lorentzian causality and Randers Finsler metrics, providing characterizations of causally well-behaved stationary spacetimes and linking geometric properties to Finslerian completeness.

Contribution

It establishes a correspondence between Lorentzian causality conditions and Randers metrics, offering new characterizations and insights into the geometry of stationary spacetimes and Finsler metrics.

Findings

Characterization of causally continuous and globally hyperbolic stationary spacetimes via Randers metrics

Equivalence of geodesic completeness in Randers metrics with compactness of symmetrized closed balls

Existence of geodesically complete Randers metrics with the same pregeodesics

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 $M=\R\times S$ 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.