The geometry of regular shear-free null geodesic congruences, CR functions and their application to the flat-space Maxwell equations

Abstract

We describe here what appears to be a new structure, 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 worldline defined in complex Minkowski space that gives a unified Lorentz-invariant meaning to both the electric and the magnetic dipole moments. In some sense, the worldline defines a ‘complex center of charge’ around which both electric and magnetic dipole moments vanish. The question of how and where this complex worldline arises 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 spacetime. In our work, the CR structure is associated with the embedding of Penrose's real three-dimensional null infinity, , as a surface in a complex two-dimensional space, C2. 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.