On the Consistency of Covariant Light-Speed Variation in Doubly Special Relativity

TL;DR

This paper examines the internal consistency of observer-independent light-speed variation in Doubly Special Relativity, revealing potential conflicts with relativistic covariance and particle motion.

Contribution

It provides a detailed analysis of LSV in DSR, highlighting structural tensions and potential issues with particle counting and motion across reference frames.

Findings

In the DSR1 model, a critical rapidity exists where the box overtakes its photon.

The overtaking rapidity is physically attainable even with macroscopic effects.

The effect challenges the resolution by relative locality alone in DSR frameworks.

Abstract

Doubly special relativity (DSR) introduces an observer-independent energy scale while preserving a deformed relativistic notion of covariance. In many realizations, this leads to an energy-dependent speed of light (light-speed variation, LSV). We investigate the consistency of such observer-independent LSV through a thought experiment involving an inertial box emitting two photons in opposite directions. We first distinguish two classes of LSV scenarios: those with the standard velocity-composition law, and those with observer-independent speed-energy relations, as in DSR. Focusing on the latter, we perform a quantitative analysis within the DSR1 model based on the $\kappa$-Poincar\'e algebra. In the subluminal case ($\ell<0$), we derive a critical rapidity above which the boosted box overtakes its own photon, and show that this rapidity is physically attainable even after taking macroscopic effects into account. Within a standard particle interpretation, this leads to tensions in particle counting and inertial motion across frames. Unlike previously discussed issues in DSR, this effect does not appear to be resolvable by relative locality alone. Our results point to a structural tension among observer-independent LSV, relativistic covariance, and standard notions of particle propagation in DSR frameworks.