Growth conditions, Riemannian completeness and Lorentzian causality

Abstract

The stationary–Randers correspondence (SRC) provides a deep connection between the property of standard stationary spacetimes being globally hyperbolic, and the completeness of certain Finsler metrics of Randers type defined on the fibres. In order to establish further results, we investigate pointwise conformal transformations of certain Riemannian metrics on the fibres and growth conditions on the corresponding conformal factors. In general, a conformal transformation may map a complete Riemannian metric onto a complete or incomplete metric. We prove a theorem for the growth of the conformal factor such that the conformally transformed Riemannian metric is also complete. As an application, we establish novel relations between the completeness of Riemannian metrics, growth conditions on conformal factors and the Cauchy hypersurface condition on the fibres of a standard stationary spacetime. These results also imply new conditions for the completeness of Randers-type metrics by the application of the SRC.