Dimensional Reduction of the 5D Kaluza-Klein Geodesic Deviation Equation

TL;DR

This paper extends the analysis of geodesic deviation in 5D Kaluza-Klein theory to include a scalar field, revealing how the scalar field affects charge-to-mass ratio variations and generalizing previous results.

Contribution

It generalizes prior work by incorporating the scalar field in the geodesic deviation equation, deriving an exact law for charge-to-mass ratio variation.

Findings

The scalar field modifies the geodesic deviation equation.

Charge-to-mass ratio is not conserved when the scalar field varies.

The results reduce to previous models when the scalar field is constant.

Abstract

In the work of Kerner et al. (2001) the problem of the geodesic deviation in a 5D Kaluza Klein background is faced. The 4D space-time projection of the resulting equation coincides with the usual geodesic deviation equation in the presence of the Lorenz force, provided that the fifth component of the deviation vector satisfies an extra constraint which takes into account the $q/m$ conservation along the path. The analysis was performed setting as a constant the scalar field which appears in Kaluza-Klein model. Here we focus on the extension of such a work to the model where the presence of the scalar field is considered. Our result coincides with that of Kerner et al. when the minimal case $\phi=1$ is considered, while it shows some departures in the general case. The novelty due to the presence of $\phi$ is that the variation of the $q/m$ between the two geodesic lines is not conserved during the motion; an exact law for such a behaviour has been derived.