Planck-scale modified dispersion relations and Finsler geometry

TL;DR

This paper demonstrates that rainbow metrics in quantum gravity phenomenology can be rigorously described using Finsler geometry, providing a new framework to analyze semiclassical quantum gravity effects and their symmetries.

Contribution

It establishes a connection between rainbow metrics and Finsler geometry, offering a novel geometric framework for quantum gravity phenomenology.

Findings

Rainbow metrics correspond to Finsler geometry.

Provides a rigorous geometric framework for semiclassical quantum gravity.

Analyzes symmetries and implications for Lorentz violation and Deformed Special Relativity.

Abstract

A common feature of all Quantum Gravity (QG) phenomenology approaches is to consider a modification of the mass shell condition of the relativistic particle to take into account quantum gravitational effects. The framework for such approaches is therefore usually set up in the cotangent bundle (phase space). However it was recently proposed that this phenomenology could be associated with an energy dependent geometry that has been coined ``rainbow metric". We show here that the latter actually corresponds to a Finsler Geometry, the natural generalization of Riemannian Geometry. We provide in this way a new and rigorous framework to study the geometrical structure possibly arising in the semiclassical regime of QG. We further investigate the symmetries in this new context and discuss their role in alternative scenarios like Lorentz violation in emergent spacetimes or Deformed Special Relativity-like models.