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On the foundations and applications of Lorentz-Finsler Geometry

Miguel Sánchez

TL;DR

This work provides a comprehensive, pedagogical survey of Lorentz-Finsler geometry, unifying cone structures, anisotropic and nonlinear connections, and the Lorentz-Finsler causal framework. It develops both local and global aspects, including globally hyperbolic splittings, the space of cone geodesics, and singularity theorems, while connecting to classical geometries and practical applications in wave propagation, wildfire/seismic modeling, and discretization. A central theme is the Pfeifer–Wohlfarth variational approach to Finsler gravity, complemented by Palatini formulations, exact vacuum solutions, and cosmological models that illustrate how Finslerian corrections could influence relativity. The paper also highlights deep links between Lorentz-Finsler geometry and symplectic/ contact geometry, boundary phenomena, and c-boundaries, underscoring the framework's potential to unify geometric, physical, and numerical perspectives in spacetime theory.

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.

On the foundations and applications of Lorentz-Finsler Geometry

TL;DR

This work provides a comprehensive, pedagogical survey of Lorentz-Finsler geometry, unifying cone structures, anisotropic and nonlinear connections, and the Lorentz-Finsler causal framework. It develops both local and global aspects, including globally hyperbolic splittings, the space of cone geodesics, and singularity theorems, while connecting to classical geometries and practical applications in wave propagation, wildfire/seismic modeling, and discretization. A central theme is the Pfeifer–Wohlfarth variational approach to Finsler gravity, complemented by Palatini formulations, exact vacuum solutions, and cosmological models that illustrate how Finslerian corrections could influence relativity. The paper also highlights deep links between Lorentz-Finsler geometry and symplectic/ contact geometry, boundary phenomena, and c-boundaries, underscoring the framework's potential to unify geometric, physical, and numerical perspectives in spacetime theory.

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.

Paper Structure

This paper contains 68 sections, 17 theorems, 64 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Setting on cones, Lorentz-Finsler metrics and connections
  3. Minkowski, wind Minkowski and Lorentz-Minkowski norms
  4. Emergence of non-symmetric norms from anisotropic propagation
  5. Minkowski norms
  6. Wind norms
  7. Rheonomic emergence of cones
  8. Lorentz-Minkowski norms
  9. Cone structures and Lorentz-Finsler metrics
  10. From vector spaces to manifolds
  11. Causality and cone triples
  12. Finsler distance and exponential, Lorentz-Finsler causality
  13. Finsler distance and geodesics
  14. The exponential map, local and global causality
  15. Nonlinear, anisotropic and Finsler connections
  16. ...and 53 more sections

Key Result

Proposition 2.1

Let $S=\partial D$ be a convex closed hypersurface of $V$ such that $D$ is open and precompact and $0\in D$. For each $v\in V$, there exists a unique scalar $\parallel v\parallel \geq 0$ satisfing that $v$ belongs to the hypersurface $\lambda S$. Then, the map $\parallel \cdot \parallel: V \rightarr

Figures (18)

  • Figure 1: Hypersurface $S$ enclosing 0 and its convex hull.
  • Figure 2: Types of convexity and implications: Infinitesimal: permits flat parts in $S$. Strict: norm with strict triangle ineq., bijective $\flat$. Strong: smooth $\sharp:= \flat^{-1}$.
  • Figure 3: (Strict) conv. of $S$ equivalent to (strict) conv. of $D$.
  • Figure 4: (Strict) convexity implies that the center of the paralelogram with sides $v,w$ lies in (the interior of) $\bar{D}$, thus yielding (strict) triangle inequality. analogous picture
  • Figure 5: For $v\in S$: $T_vS=(v^{\flat})^{-1}(1)$
  • ...and 13 more figures

Theorems & Definitions (40)

  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Remark 2.9
  • ...and 30 more