Maxwell fields and shear-free null geodesic congruences

Abstract

We study and report on the class of vacuum Maxwell fields in Minkowski space that possess a non-degenerate, diverging, principal null vector field (null eigenvector field of the Maxwell tensor) that is tangent to a shear-free null geodesics congruence. These congruences can be either surface forming (the tangent vectors being proportional to gradients) or not, i.e., the twisting congruences. In the non-twisting case, the associated Maxwell fields are precisely the Lienard–Wiechert fields, i.e., those Maxwell fields arising from an electric monopole moving on an arbitrary worldline. The null geodesic congruence is given by the generators of the light-cones with apex on the worldline. The twisting case is much richer, more interesting and far more complicated.In a twisting subcase, where our main interests lie, the following strange interpretation can be given. If we allow the real Minkowski space to be complexified so that the real Minkowski coordinates xa take complex values, i.e., xa → za = xa + iya with complex metric g = ηab dza dzb, the real vacuum Maxwell equations can be extended into the complex space and rewritten as This subcase of Maxwell fields can then be extended into the complex space so as to have as source, a complex analytic worldline, i.e., to now become complex Lienard–Wiechart fields. When viewed as real fields on the real Minkowski space (za = xa), they possess a real principal null vector that is shear-free but twisting and diverging. The twist is a measure of how far the complex worldline is from the real ‘slice’. Most Maxwell fields in this subcase are asymptotically flat with a time-varying set of electric and magnetic moments, all depending on the complex displacements and the complex velocities.