An equivalence of Finslerian relativistic theories

TL;DR

This paper proves that Lorentz-Finsler theories defined over cones and those over the entire tangent bundle are essentially equivalent when certain regularity conditions are met, ensuring the uniqueness of Finsler spacetime models and causality.

Contribution

It demonstrates that under regularity conditions, conic and slit tangent bundle Finsler Lagrangians coincide, establishing the essential uniqueness of Finsler spacetime theories.

Findings

Conic and slit tangent bundle Finsler Lagrangians are equivalent under regularity.

This equivalence ensures the uniqueness of Finsler causality theories.

The work clarifies the relationship between different formulations of Lorentz-Finsler geometry.

Abstract

In Lorentz-Finsler geometry it is natural to define the Finsler Lagrangian over a cone (Asanov's approach) or over the whole slit tangent bundle (Beem's approach). In the former case one might want to add differentiability conditions at the boundary of the (timelike) cone in order to retain the usual definition of lightlike geodesics. It is shown here that if this is done then the two theories coincide, namely the `conic' Finsler Lagrangian is the restriction of a slit tangent bundle Lagrangian. Since causality theory depends on curves defined through the future cone, this work establishes the essential uniqueness of (sufficiently regular) Finsler spacetime theories and Finsler causality.