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

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 $κ$-Poincaré 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.