Geodesic equation in $k$-Minkowski spacetime

TL;DR

This paper derives first-order corrections to the geodesic equation in curved spacetime due to $k$-deformation, revealing an effective drag and discussing bounds on the deformation parameter with implications for generalized uncertainty relations.

Contribution

It generalizes previous methods to include curvature effects in $k$-deformed spacetime, deriving corrected geodesic equations and uncertainty relations.

Findings

$k$-noncommutativity introduces an effective drag on particles.

Derived bounds on the deformation parameter from Newtonian limit.

Established generalized uncertainty relations in $k$-deformed spacetime.

Abstract

In this paper, we derive corrections to the geodesic equation due to the $k$-deformation of curved space-time, up to the first order in the deformation parameter a. This is done by generalizing the method from our previous paper [31], to include curvature effects. We show that the effect of $k$-noncommutativity can be interpreted as an extra drag that acts on the particle while moving in this $k$-deformed curved space. We have derived the Newtonian limit of the geodesic equation and using this, we discuss possible bounds on the deformation parameter. We also derive the generalized uncertainty relations valid in the non-relativistic limit of the $k$-space-time.