On the existence of maximizing curves for the charged-particle action

TL;DR

This paper extends the Avez-Seifert theorem to charged particles under Lorentz force, proving the existence of maximizing connecting curves in globally hyperbolic spacetimes, including Minkowski space.

Contribution

It generalizes the classical theorem to include Lorentz force equations for charged particles, establishing existence results for timelike connecting solutions.

Findings

Existence of timelike connecting solutions under Lorentz force in globally hyperbolic spacetimes.

Proof of at least one maximizing C^{1} connecting curve for the functional I.

Affirmative answer for Minkowski spacetime case.

Abstract

The classical Avez-Seifert theorem is generalized to the case of the Lorentz force equation for charged test particles with fixed charge-to-mass ratio. Given two events x_{0} and x_{1}, with x_{1} in the chronological future of x_{0}, and a ratio q/m, it is proved that a timelike connecting solution of the Lorentz force equation exists provided there is no null connecting geodesic and the spacetime is globally hyperbolic. As a result, the theorem answers affirmatively to the existence of timelike connecting solutions for the particular case of Minkowski spacetime. Moreover, it is proved that there is at least one C^{1} connecting curve that maximizes the functional I[\gamma]=\int_{\gamma} ds+q/(mc^2) \omega over the set of C^{1} future-directed non-spacelike connecting curves.