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, 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.

39

References in corpus (3)

• On a Classical, Geometric Origin of Magnetic Moments, Spin-Angular Momentum and the Dirac Gyromagnetic Ratio
• Complex Kerr Geometry and Nonstationary Kerr Solutions
• Orientifold D-String in the Source of the Kerr Spinning Particle

Cited by in corpus (19)

• Null Geodesic Congruences, Asymptotically Flat Space-Times and Their Physical Interpretation
• Classical double copy at null infinity
• The Large Footprints of H-Space on Asymptotically Flat Space-Times
• Distributional Sources for Newman's Holomorphic Field
• Electromagnetic Dipole Radiation Fields, Shear-Free Congruences and Complex Center of Charge World Lines
• Light cones in relativity: Real, complex and virtual, with applications
• On Extracting Physical Content from Asymptotically Flat Space-Time Metrics
• Algbrodynamics over complex space and phase extension of the Minkowski geometry
• CR Structures and Asymptotically Flat Space-Times
• The Universal Cut Function and Type II Metrics
• The Real Meaning of Complex Minkowski-Space World-Lines
• Maxwell, Yang-Mills, Weyl and eikonal fields defined by any null shear-free congruence
• Asymptotic twistor Theory and the Kerr Theorem
• A New Twist on the Geometry of Gravitational Plane Waves
• Nonlocal modification of the Kerr metric
• The Geometry of Regular Shear-Free Null Geodesic Congruences, CR functions and their Application to the Flat-Space Maxwell Equations
• Electromagnetic Induced Gravitational Perturbations
• The Gravitational Field of a Radiating Electromagnetic Dipole
• Newman-Janis Algorithm from Taub-NUT Instantons