Electric charge in the field of a magnetic event in three-dimensional spacetime

TL;DR

This paper studies how an electric charge moves in the field of a magnetic event in three-dimensional spacetime, revealing characteristic rotational behavior and energy transfer effects through numerical analysis and conserved quantities.

Contribution

It provides a detailed analysis of charge dynamics in magnetic fields in 3D spacetime, including conserved quantities and numerical results, highlighting the signature behavior of magnetic events.

Findings

Charge starts rotating after crossing the light cone.

The particle eventually comes to rest at the event location.

Characteristic length scales influence the particle's motion.

Abstract

We analyze the motion of an electric charge in the field of a magnetically charged event in three-dimensional spacetime. We start by exhibiting a first integral of the equations of motion in terms of the three conserved components of the spacetime angular momentum, and then proceed numerically. After crossing the light cone of the event, an electric charge initially at rest starts rotating and slowing down. There are two lengths appearing in the problem: (i) the characteristic length $\frac{q g}{2 \pi m}$, where $q$ and $m$ are the electric charge and mass of the particle, and $g$ is the magnetic charge of the event; and (ii) the spacetime impact parameter $r_0$. For $r_0 \gg \frac{q g}{2 \pi m}$, after a time of order $r_0$, the particle makes sharply a quarter of a turn and comes to rest at the same spatial position at which the event happened in the past. This jump is the main signature of the presence of the magnetic event as felt by an electric charge. A derivation of the expression for the angular momentum that uses Noether's theorem in the magnetic representation is given in the Appendix.