Electromagnetic Dipole Radiation Fields, Shear-Free Congruences and Complex Center of Charge World Lines

TL;DR

This paper introduces a geometric approach to identify a complex world-line representing the electromagnetic center of charge in Minkowski space, unifying electric and magnetic dipole moments and extending to general relativity.

Contribution

It establishes a unique geometric structure at null infinity to define the complex center of charge and relates it to dipole moments, extending the concept to gravitational fields.

Findings

A null direction field at null infinity uniquely determines the complex center of charge.

The complex world-line encodes both electric and magnetic dipole moments.

The approach generalizes to the gravitational case, defining a complex center of mass.

Abstract

We show that for asymptotically vanishing Maxwell fields in Minkowski space with non-vanishing total charge, one can find a unique geometric structure, a null direction field, at null infinity. From this structure a unique complex analytic world-line in complex Minkowski space that can be found and then identified as the complex center of charge. By ''sitting'' - in an imaginary sense, on this world-line both the (intrinsic) electric and magnetic dipole moments vanish. The (intrinsic) magnetic dipole moment is (in some sense) obtained from the `distance' the complex the world line is from the real space (times the charge). This point of view unifies the asymptotic treatment of the dipole moments For electromagnetic fields with vanishing magnetic dipole moments the world line is real and defines the real (ordinary center of charge). We illustrate these ideas with the Lienard-Wiechert Maxwell field. In the conclusion we discuss its generalization to general relativity where the complex center of charge world-line has its analogue in a complex center of mass allowing a definition of the spin and orbital angular momentum - the analogues of the magnetic and electric dipole moments.