On the foundations and applications of Lorentz-Finsler Geometry

TL;DR

This paper introduces Lorentz-Finsler geometry, exploring its foundational concepts, applications in physics and classical phenomena, and recent advances such as spacetime splitting and boundary extensions, highlighting its broad utility beyond relativity.

Contribution

It provides a comprehensive introduction to Lorentz-Finsler geometry, including new results on spacetime splitting and boundary extensions, expanding its theoretical and practical scope.

Findings

Splitting of globally hyperbolic Finsler spacetimes

Analysis of extensions with timelike boundaries

Connections among Riemannian, Finsler, and Lorentz geometries

Abstract

Finslerian extensions of Special and General Relativity -- commonly referred to as Very Special and Very General Relativity -- necessitate the development of a unified Lorentz-Finsler geometry. However, the scope of this geometric framework extends well beyond relativistic physics. Indeed, it offers powerful tools for modeling wave propagation in classical mechanics, discretizing spacetimes in classical and relativistic settings, and supporting effective theories in fundamental physics. Moreover, Lorentz-Finsler geometry provides a versatile setting that facilitates the resolution of problems within Riemannian, Lorentzian, and Finslerian geometries individually. This work presents a plain introduction to the subject, reviewing foundational concepts, key applications, and future prospects. The reviewed topics include (i) basics on the setting of cones, Finsler and Lorentz-Finsler metrics and their (nonlinear, anisotropic and linear) connections, (ii) the global structure of Lorentz-Finsler manifolds and its space of null geodesics, (iii) links among Riemannian, Finsler and Lorentz geometries, (iv) applications in classical settings as wildfires and seisms propagation, and discretization in classical and relativistic settings with quantum prospects, and (v) Finslerian variational approach to Einstein equations. The new results include the splitting of globally hyperbolic Finsler spacetimes, in addition to the analysis of several extensions of the Lorentz setting, as the case of timelike boundaries.