The covariant description of electric and magnetic field lines of null fields: application to Hopf-Ranada solutions

TL;DR

This paper develops a covariant geometric framework for describing electric and magnetic field lines in null fields, specifically applied to Hopf-Ranada solutions, revealing their topological invariance under Lorentz transformations.

Contribution

It introduces a covariant, geometric algebra-based method to define and analyze electric and magnetic field lines for null fields, including Hopf-Ranada solutions, with topological and optical properties.

Findings

Field lines are represented by Lorentz-invariant 2D surfaces in spacetime.

Field line surfaces transform under Lorentz transformations while preserving topology.

Relations between optical helicity, chirality, and conservation laws are established.

Abstract

The concept of electric and magnetic field lines is intrinsically non-relativistic. Nonetheless, for certain types of fields satisfying certain geometric properties, field lines can be defined covariantly. More precisely, two Lorentz-invariant 2D surfaces in spacetime can be defined such that magnetic and electric field lines are determined, for any observer, by the intersection of those surfaces with spacelike hyperplanes. An instance of this type of field is constituted by the so-called Hopf-Ranada solutions of the source-free Maxwell equations, which have been studied because of their interesting topological properties, namely, linkage of their field lines. In order to describe both geometric and topological properties in a succinct manner, we employ the tools of Geometric Algebra (aka Clifford Algebra) and use the Clebsch representation for the vector potential as well as the Euler representation for both magnetic and electric fields. This description is easily made covariant, thus allowing us to define electric and magnetic field lines covariantly in a compact geometric language. The definitions of field lines can be phrased in terms of 2D surfaces in space. We display those surfaces in different reference frames, showing how those surfaces change under Lorentz transformations while keeping their topological properties. As a byproduct we also obtain relations between optical helicity, optical chirality and generalizations thereof, and their conservation laws.