The Geometry of Regular Shear-Free Null Geodesic Congruences, CR functions and their Application to the Flat-Space Maxwell Equations

TL;DR

This paper uncovers a hidden complex structure in Maxwell fields with charge, linking CR structures and shear-free null geodesic congruences to reveal a complex center of charge and new insights into electromagnetic dipoles.

Contribution

It introduces a novel connection between CR structures, shear-free null geodesic congruences, and Maxwell fields, revealing a complex world-line representing the charge distribution.

Findings

Extraction of a complex world-line from Maxwell fields.

Identification of a CR structure related to null infinity.

Interpretation of dipole moments via complex geometry.

Abstract

We describe here what appears to be a new structure that is hidden in all asymptotically vanishing Maxwell fields possessing a non-vanishing total charge. Though we are dealing with real Maxwell fields on real Minkowski space nevertheless, directly from the asymptotic field one can extract a complex analytic world-line defined in complex Minkowski space that gives a unified Lorentz invariant meaning to both the electric and magnetic dipole moments. In some sense the world-line defines a `complex center of charge' around which both electric and magnetic dipole moments vanish. The question of how and where does this complex world-line arise is one of the two main subjects of this work. The other subject concerns what is known in the mathematical literature as a CR structure. In GR, CR structures naturally appear in the physical context of shear-free (or asymptotically shear-free) null geodesic congruences in space-time. For us, the CR structure is associated with the embedding of Penrose's real three-dimensional null infinity, I^+, as a surface in a two complex dimensional space, C^2. It is this embedding, via a complex function, (a CR function), that is our other area of interest. Specifically we are interested in the `decomposition' of the CR function into its real and imaginary parts and the physical information contained in this decomposition.