Eur. Phys. J. C (2017) 77: 451
https://doi.org/10.1140/epjc/s10052-017-5017-0

Regular Article - Theoretical Physics

Rainbows without unicorns: metric structures in theories with modified dispersion relations

¹ Dipartimento di Fisica, Universitá “La Sapienza” and Sez. Roma1 INFN, P.le A. Moro 2, 00185, Rome, Italy
² ICRANet, Piazza della Repubblica 10, 65122, Pescara, Italy
³ CAPES Foundation, Ministry of Education of Brazil, Brasília, Brazil
⁴ Departamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008, João Pessoa, PB, CEP 58051-970, Brazil
⁵ Division of Theoretical Physics, Ruđer Bošković Institute, Bijenička c.54, 10002, Zagreb, Croatia
⁶ Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, AP 70543, 04510, Mexico, CDMX, Mexico

^* e-mail: niccolo@accatagliato.org

Received: 14 February 2017
Accepted: 25 June 2017
Published online: 7 July 2017

Abstract

Rainbow metrics are a widely used approach to the metric formalism for theories with modified dispersion relations. They have had a huge success in the quantum gravity phenomenology literature, since they allow one to introduce momentum-dependent space-time metrics into the description of systems with a 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 and a Finsler geometry approach. As the main result of this work we also present an alternative definition of a four-velocity dependent metric which allows one to handle the massless limit. This paper aims to highlight some of their properties and how to properly describe their relativistic realizations.