Can we make a Finsler metric complete by a trivial projective change?

TL;DR

This paper investigates conditions under which a Finsler metric can be made both forward and backward complete through a trivial projective change, linking the problem to concepts in Lorentz geometry and global hyperbolicity.

Contribution

It characterizes when a Finsler metric can be globally completed via trivial projective changes, connecting Finsler geometry with spacetime hyperbolicity.

Findings

Conditions for trivial projective changes to achieve completeness.

Connection between Finsler metric completeness and spacetime hyperbolicity.

Insight into light-line geodesics in stationary space-times.

Abstract

A trivial projective change of a Finsler metric $F$ is the Finsler metric $F + df$. I explain when it is possible to make a given Finsler metric both forward and backward complete by a trivial projective change. The problem actually came from lorentz geometry and mathematical relativity: it was observed that it is possible to understand the light-line geodesics of a (normalized, standard) stationary 4-dimensional space-time as geodesics of a certain Finsler Randers metric on a 3-dimensional manifold. The trivial projective change of the Finsler metric corresponds to the choice of another 3-dimensional slice, and the existence of a trivial projective change that is forward and backward complete is equivalent to the global hyperbolicity of the space-time.