Finsler Geometries from Topological Electromagnetism

TL;DR

This paper explores Finsler geometries arising from the kinematic space of charged particles in a Rada electromagnetic field, introducing a method to compute geometric objects and analyzing duality transformations.

Contribution

It provides a systematic approach to derive Finsler geometries for particles in electromagnetic fields and examines electromagnetic duality effects within this geometric framework.

Findings

Derived a method to calculate Finsler geometric objects in electromagnetic backgrounds.

Established a duality map relating geometries of electric and magnetic particles.

Calculated dual geodesic equations illustrating electromagnetic duality effects.

Abstract

We analyse the Finsler geometries of the kinematic space of spinless and spinning electrically charged particles in an external Ra\~{n}ada field. We consider the most general actions that are invariant under the Lorentz, electromagnetic gauge and reparametrization transformations. The Finsler geometries form a set parametrized by the gauge fields in each case. We give a simple method to calculate the fundamental objects of the Finsler geometry of the kinematic space of a particle in a generic electromagnetic field. Then we apply this method to calculate the geodesic equations of the spinless and spinning particles. Also, we show that the electromagnetic duality in the Ra\~{n}ada background induces a simple dual map in the set of Finsler geometries. The duality map has a simple interpretation in terms of an electrically charged particle that interacts with the electromagnetic potential and a magnetically charged particle that interacts with the dual magnetoelectric potential. We exemplify the action of the duality map by calculating the dual geodesic equation.