Connecting solutions of the Lorentz force equation do exist

TL;DR

This paper proves the existence of solutions to the Lorentz force equation connecting two causally related events in a globally hyperbolic spacetime with an electromagnetic field, extending previous results and providing explicit examples.

Contribution

It generalizes the maximization of the charged-particle action in globally hyperbolic spacetimes and establishes the existence of timelike solutions to the Lorentz force equation for any charge-to-mass ratio.

Findings

Existence of maximum action in causal homotopy classes.

Timelike solutions connect causally related events via Lorentz force.

Counterexample in the non-exact electromagnetic case.

Abstract

Recent results on the maximization of the charged-particle action I in a globally hyperbolic spacetime are discussed and generalized. We focus on the maximization of I over a given causal homotopy class C of curves connecting two causally related events x_0 <= x_1. Action I is proved to admit a maximum on C, and also one in the adherence of each timelike homotopy class. Moreover, the maximum on C is timelike if C contains a timelike curve (and the degree of differentiability of all the elements is at least C^2). In particular, this last result yields a complete Avez-Seifert type solution to the problem of connectedness through trajectories of charged particles in a globally hyperbolic spacetime endowed with an exact electromagnetic field: fixed any charge-to-mass ratio q/m, any two chronologically related events x_0 << x_1 can be connected by means of a timelike solution of the Lorentz force equation (LFE) corresponding to q/m. The accuracy of the approach is stressed by many examples, including an explicit counterexample (valid for all q/m) in the non-exact case. As a relevant previous step, new properties of the causal path space, causal homotopy classes and cut points on lightlike geodesics are studied.