Maxwell, Yang-Mills, Weyl and eikonal fields defined by any null shear-free congruence

TL;DR

This paper demonstrates that shear-free null geodesic congruences in Minkowski space have intrinsic algebraic structures that relate to complex fields like Maxwell and Yang-Mills, enabling explicit solutions and particle-like field configurations.

Contribution

It introduces a unified algebraic framework linking shear-free null congruences with complex gauge and field structures, extending the Kerr theorem via twistor methods.

Findings

Explicit solutions for shear-free null geodesic congruences using twistor variables

Association of these congruences with complex Maxwell and Yang-Mills fields

Particle-like field distributions with bounded singularities and collective dynamics

Abstract

We show that (specifically scaled) equations of shear-free null geodesic congruences on the Minkowski space-time possess intrinsic self-dual, restricted gauge and algebraic structures. The complex eikonal, Weyl 2-spinor, $SL(2,\mathbb C)$ Yang-Mills and complex Maxwell fields, the latter produced by integer-valued electric charges ("elementary" for the Kerr-like congruences), can all be explicitly associated with any shear-free null geodesic congruence. Using twistor variables, we derive the general solution of the equations of the shear-free null geodesic congruence (as a modification of the Kerr theorem) and analyze the corresponding "particle-like" field distributions, with bounded singularities of the associated physical fields. These can be obtained in a straightforward algebraic way and exhibit non-trivial collective dynamics simulating physical interactions