Stationary Metrics and Optical Zermelo-Randers-Finsler Geometry

TL;DR

This paper explores the deep connections between Zermelo navigation, Randers Finsler geometry, and higher-dimensional spacetimes, revealing how physical and geometric perspectives intertwine in these mathematical structures.

Contribution

It establishes a triality linking Zermelo navigation, Randers Finsler geometry, and spacetime models, providing new insights into their interrelations and physical interpretations.

Findings

Finsler geometries with constant flag curvature correspond to conformally flat spacetimes.

The Finsler condition relates to causality or breaks down at ergosurfaces.

Spacetime perspective simplifies understanding of Finsler and navigation problems.

Abstract

We consider a triality between the Zermelo navigation problem, the geodesic flow on a Finslerian geometry of Randers type, and spacetimes in one dimension higher admitting a timelike conformal Killing vector field. From the latter viewpoint, the data of the Zermelo problem are encoded in a (conformally) Painleve-Gullstrand form of the spacetime metric, whereas the data of the Randers problem are encoded in a stationary generalisation of the usual optical metric. We discuss how the spacetime viewpoint gives a simple and physical perspective on various issues, including how Finsler geometries with constant flag curvature always map to conformally flat spacetimes and that the Finsler condition maps to either a causality condition or it breaks down at an ergo-surface in the spacetime picture. The gauge equivalence in this network of relations is considered as well as the connection to analogue models and the viewpoint of magnetic flows. We provide a variety of examples.