Rainbows without unicorns: Metric structures in theories with Modified Dispersion Relations

TL;DR

This paper explores various metric structures used in theories with Modified Dispersion Relations, including Rainbow metrics and Finsler geometry, highlighting their properties and relativistic descriptions in quantum gravity phenomenology.

Contribution

It introduces and compares different realizations of momentum-dependent metrics, including a novel four-velocity-dependent metric with a massless limit, clarifying their properties and relativistic formulations.

Findings

Different metric structures have unique properties in theories with Modified Dispersion Relations.

The paper clarifies how to properly describe relativistic realizations of these metrics.

A new four-velocity-dependent metric with a massless limit is proposed.

Abstract

Rainbow metrics are a widely used approach to metric formalism for theories with Modified Dispersion Relations. They have had a huge success in the Quantum Gravity Phenomenology literature, since they allow to introduce momentum-dependent spacetime metrics into the description of systems with Modified Dispersion Relation. In this paper, we introduce the reader to some realizations of this general idea: the original Rainbow metrics proposal, the momentum-space-inspired metric, the standard Finsler geometry approach and our alternative definition of a four-velocity-dependent metric with a massless limit. This paper aims to highlight some of their properties and how to properly describe their relativistic realizations.