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

摘要

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