On the definition and examples of cones and Finsler spacetimes

TL;DR

This paper systematically studies cone structures and Lorentz-Finsler metrics, establishing their relationships, providing explicit descriptions of Finsler spacetimes, and applying the theory to the time-dependent Zermelo navigation problem.

Contribution

It introduces cone triples linking cone structures and Finsler metrics, and offers explicit descriptions of Finsler spacetimes, including stationary and static cases, with applications to navigation problems.

Findings

Bijective correspondence between cone structures and anisotropically conformal metrics.

Explicit descriptions of stationary and static Finsler spacetimes.

Solution to the time-dependent Zermelo navigation problem.

Abstract

A systematic study of (smooth, strong) cone structures $\C$ and Lorentz-Finsler metrics $L$ is carried out. As a link between both notions, cone triples $(\Omega,T, F)$, where $\Omega$ (resp. $T$) is a 1-form (resp. vector field) with $\Omega(T)\equiv 1$ and $F$, a Finsler metric on $\ker (\Omega)$, are introduced. Explicit descriptions of all the Finsler spacetimes are given, paying special attention to stationary and static ones, as well as to issues related to differentiability. In particular, cone structures $\C$ are bijectively associated with classes of anisotropically conformal metrics $L$, and the notion of {\em cone geodesic} is introduced consistently with both structures. As a non-relativistic application, the {\em time-dependent} Zermelo navigation problem is posed rigorously, and its general solution is provided.