Maximizing curves for the charged-particle action in globally hyperbolic spacetimes

TL;DR

This paper proves the existence of maximizing solutions to the Lorentz force equation for charged particles in globally hyperbolic spacetimes, extending known geodesic results to charged particles under weak assumptions.

Contribution

It introduces two theorems establishing the existence of charge-specific maximizing solutions to the Lorentz force equation in globally hyperbolic spacetimes.

Findings

Existence of connecting solutions for charged particles with given charge-to-mass ratio.

Solutions are shown to be maximizing in the geometrical sense.

Extends geodesic existence results to charged-particle trajectories.

Abstract

In a globally hyperbolic spacetime any pair of chronologically related events admits a connecting geodesic. We present two theorems which prove that, more generally, under weak assumptions, given a charge-to-mass ratio there is always a connecting solution of the Lorentz force equation having that ratio. A geometrical interpretation of the charged-particle action is given which shows that the constructed solutions are maximizing.