Comparison of relativity theories with observer-independent scales of both velocity and length/mass

TL;DR

This paper compares two prominent relativity theories with observer-independent velocity and length/mass scales, revealing their deep connection and clarifying key concepts like mass, velocity, and energy conservation.

Contribution

It demonstrates the close relationship between two different formulations of doubly special relativity and clarifies fundamental concepts within these theories.

Findings

Both theories share the same momentum-rapidity dependence.

The connection clarifies the definitions of mass and velocity in these frameworks.

Insights into energy-momentum conservation rules in doubly special relativity.

Abstract

We consider the two most studied proposals of relativity theories with observer-independent scales of both velocity and length/mass: the one discussed by Amelino-Camelia as illustrative example for the original proposal (gr-qc/0012051) of theories with two relativistic invariants, and an alternative more recently proposed by Magueijo and Smolin (hep-th/0112090). We show that these two relativistic theories are much more closely connected than it would appear on the basis of a naive analysis of their original formulations. In particular, in spite of adopting a rather different formal description of the deformed boost generators, they end up assigning the same dependence of momentum on rapidity, which can be described as the core feature of these relativistic theories. We show that this observation can be used to clarify the concepts of particle mass, particle velocity, and energy-momentum-conservation rules in these theories with two relativistic invariants.