Twisting Null Geodesic Congruences and the Einstein-Maxwell Equations

TL;DR

This paper extends previous work on asymptotically flat vacuum spacetimes to Einstein-Maxwell solutions, deriving equations of motion for a special degenerate case that resemble particle dynamics with spin and magnetic moments.

Contribution

It introduces a novel analysis of shear-free null geodesic congruences in Einstein-Maxwell fields, focusing on a degenerate case with a single complex world-line and deriving related equations of motion.

Findings

Derived equations of motion resembling particle behavior with spin and magnetic dipole moments.

Obtained classical radiation-reaction terms directly from asymptotic fields without Lorentz law.

Suggested suppression of runaway solutions via Bondi mass loss.

Abstract

The purpose of the present work is to extend the earlier results for asymptotically flat vacuum space-times to asymptotically flat solutions of the Einstein-Maxwell equations. Once again, in this case, we get a class of asymptotically shear-free null geodesic congruences depending on a complex world-line in the same four-dimensional complex space. However in this case there will be, in general, two distinct but uniquely chosen world-lines. One of which can be assigned as the complex center-of- charge while the other could be called the complex center of mass. Rather than investigating the situation where there are two distinct complex world-lines, we study instead the special degenerate case where the two world-lines coincide, i.e., where there is a single unique world-line. This mimics the case of algebraically special Einstein-Maxwell fields where the degenerate principle null vector of the Weyl tensor coincides with a Maxwell principle null vector. Again we obtain equations of motion for this world-line - but explicitly found here only in an approximation. Though there are ambiguities in assigning physical meaning to different terms it appears as if reliance on the Kerr and charged Kerr metrics and classical electromagnetic radiation theory helps considerably in this identification. In addition, the resulting equations of motion appear to have many of the properties of a particle with intrinsic spin and an intrinsic magnetic dipole moment. At first order there is even the classical radiation-reaction term 2/3{q^{2}}{c^{-3}}ddot{v}, now obtained without any use of the Lorentz force law but obtained directly from the asymptotic fields themselves. One even sees the possible suppression, via the Bondi mass loss, of the classical runaway solutions due to the radiation reaction force.