# Finsler spacetime geometry in Physics

## TL;DR

This review explores how Finsler geometry naturally arises in physics, especially in describing spacetime, and discusses the development of a pseudo-Finsler spacetime framework that incorporates causality, observers, and gravitational dynamics.

## Contribution

It provides a comprehensive review of Finsler geometry's role in physics and introduces a precise pseudo-Finsler spacetime framework with dynamic gravitational equations.

## Key findings

- Finsler geometry appears as a dual description of dispersion relations
- A pseudo-Finsler spacetime can be defined with causal structure and observer measurements
- A gravitational field equation for Finsler spacetime is established

## Abstract

Finsler geometry naturally appears in the description of various physical systems. In this review I divide the emergence of Finsler geometry in physics into three categories: as dual description of dispersion relations, as most general geometric clock and as geometry being compatible with the relevant Ehlers-Pirani-Schild axioms. As Finsler geometry is a straightforward generalisation of Riemannian geometry there are many attempts to use it as generalized geometry of spacetime in physics. However, this generalisation is subtle due to the existence of non-trivial null directions. I review how a pseudo-Finsler spacetime geometry can be defined such that it provides a precise notion of causal curves, observers and their measurements as well as a gravitational field equation determining the Finslerian spacetime geometry dynamically. The construction of such Finsler spacetimes lays they foundation for comparing their predictions with observations, in astrophysics as well as in laboratory experiments.