Solutions to the Lorentz force equation with fixed charge-to-mass ratio in globally hyperbolic spacetimes

TL;DR

This paper generalizes the Avez-Seifert theorem to include charged particles with fixed charge-to-mass ratios in curved spacetimes, establishing conditions for the existence of timelike solutions to the Lorentz force equation between two events.

Contribution

It extends classical results to charged particles, providing new existence theorems for solutions of the Lorentz force equation in globally hyperbolic spacetimes.

Findings

Existence of a range of charge-to-mass ratios with connecting solutions.

Multiple solutions exist for each ratio unless they are geodesics.

Conditions under which solutions are unique or coincide with geodesics.

Abstract

We extend the classical Avez-Seifert theorem to trajectories of charged test particles with fixed charge-to-mass ratio. In particular, given two events x_{0} and x_{1}, with x_{1} in the chronological future of x_{0}, we find an interval I=]-R,R[ such that for any q/m in I there is a timelike connecting solution of the Lorentz force equation. Moreover, under the assumption that there is no null geodesic connecting x_0 and x_1, we prove that to any value of |q/m| there correspond at least two connecting timelike solutions which coincide only if they are geodesics.