Electromagnetic dipole radiation fields, shear-free congruences and complex centre of charge world lines

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 can be found and then identified as the complex centre of charge. By ‘sitting’—in an imaginary sense—on this world line both the electric and intrinsic magnetic dipole moments vanish. The intrinsic magnetic dipole moment is (in some sense) obtained from the ‘distance’ the complex 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 centre 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 centre of charge world line has its analogue in a complex centre of mass allowing a definition of the spin and orbital angular momentum—the analogues of the magnetic and electric dipole moments.