Images of the lines under the MS transformations and the Concept of Velocity in the DSR theories

TL;DR

This paper explores how Maguejo-Smolin (MS) transformations affect lines in energy-momentum space within DSR theories, revealing geometric properties and velocity addition rules, and analyzing the crossing points of transformed lines.

Contribution

It provides a geometric interpretation of velocities in DSR theories and studies the effects of MS transformations on lines and their intersections in energy-momentum space.

Findings

Velocities in DSR are related to geometrical properties of lines under transformations.

MS transformations alter parallel lines, leading to crossing points at constant momentum.

The crossing point of lines is located on a specific line with constant energy-momentum ratio.

Abstract

The effect of the Maguejo-Smolin (MS) transformations on a straight line in the energy- momentum space will be studied. We will interpret the slope of this line as velocity $dE/dp$, which can leads to addition rule for the velocities in the MS doubly special relativity (DSR) case. Relation between two expressions $dE/dp$ and $p/E$ for velocity in the momentum space will be investigated more and the energy dependency of the velocities in the DSR theories is related to the geometrical properties of the lines under DSR transformations. The images of two parallel lines under the MS transformations will be studied and we will compute crossing point of these lines under the MS transformations in the energy-momentum space. The linear-fractional transformations don't keep parallelism. The crossing point is on a line in the energy-momentum space with a constant momentum $E_p/c$.