Dispersion relations in finite-boost DSR

TL;DR

This paper derives finite-boost DSR transformations at first order in Planck length, classifies them via dispersion relations, and shows a case where photon speed matches special relativity without delay, impacting quantum gravity searches.

Contribution

It introduces a classification of DSR theories based on dispersion relations and identifies a case where light speed remains unchanged, challenging previous assumptions.

Findings

Four types of DSR theories classified by dispersion relations

A special case where photon speed equals c with no delay

Implication that some quantum gravity effects may be consistent with special relativity

Abstract

We find finite-boost transformations DSR theories in first order of the Planck length $l_p$, by solving differential equations for the modified generators. We obtain corresponding dispersion relations for these transformations, which help us classify the DSR theories via four types. The final type of our classification has the same special relativistic dispersion relation but the transformations are not Lorentz. In DSR theories, the velocity of photons is generally different from the ordinary speed c and possess time delay, however in this new DSR light has the same special relativistic speed with no delay. A special case demonstrates that any search for quantum gravity effects in observations which gives a special relativistic dispersion relation is consistent with DSR.