  We study and report on the class of vacuum Maxwell fields in Minkowski space
that possess a non-degenerate, diverging, principle 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 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
world-line. The twisting case is much richer, more interesting and far more
complicated. In a twisting subcase, where our main interests lie, it can be
given the following strange interpretation. If we allow the real Minkowski
space to be complexified so that the real Minkowski coordinates x^a take
complex values, i.e., x^a => z^a=x^a+iy^a with complex metric g=eta_abdz^adz^b,
the real vacuum Maxwell equations can be extended into the complex and
rewritten as curlW =iWdot, divW with W =E+iB. This subcase of Maxwell fields
can then be extended into the complex so as to have as source, a complex
analytic world-line, i.e., to now become complex Lienard-Wiechart fields. When
viewed as real fields on the real Minkowski space, z^a=x^a, they possess a real
principle null vector that is shear-free but twisting and diverging. The twist
is a measure of how far the complex world-line 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.
