Modified Lorentz transformations in deformed special relativity

TL;DR

This paper develops modified Lorentz transformations within deformed special relativity, incorporating noncommutative geometry and a Planck-scale momentum cutoff, leading to finite Lorentz factors at high velocities and invariant Planck energy-momentum.

Contribution

It provides explicit forms of modified Lorentz transformations for specific deformed dispersion relations, extending special relativity to include a momentum cutoff at the Planck scale.

Findings

Lorentz factor remains finite at high velocities.

Planck energy and momentum are invariant under boosts.

The theory naturally incorporates a momentum cutoff at the Planck scale.

Abstract

We extend a recent approach to Deformed Special Relativity based on deformed dispersion laws, entailing modified Lorentz transformations and, at the same time, noncommutative geometry and intrinsically discrete spacetime. In so doing we obtain the explicit form of the modified Lorentz transformations for a special class of modified momentum-energy relations often found in literature and arising from elementary particle physics. Actually, our theory looks as a very simple and natural extension of special relativity to include a momentum cut-off at the Planck scale. In particular, the new Lorentz transformations do imply that for high boost speed ($V \sim c$) the deformed Lorentz factor does not diverge as in ordinary relativity, but results to be upperly bounded by a large finite value of the order of the ratio between the Planck mass and the particle mass. We also predict that a generic boost leaves unchanged Planck energy and momentum, which result invariant with respect to any reference frame.