Distributional Sources for Newman's Holomorphic Field

Abstract

In 1973, E. T. Newman considered the holomorphic extension \tilde E(x+iy) of the Coulomb field E(x) in R^3. By analyzing its multipole expansion, he showed that the real and imaginary parts of \tilde E(x+iy), viewed as functions of x for fixed y, are the electric and magnetic fields generated by a spinning ring of charge R. This represents the electromagnetic part of the Kerr-Newman solution to the Einstein-Maxwell equations. As already pointed out by Newman and Janis in 1965, this interpretation is somewhat problematic since the fields are double-valued. To make them single-valued, a branch cut must be introduced so that R is replaced by a charged disk D having R as its boundary. In the context of curved spacetime, D becomes a spinning disk of charge and mass representing the singularity of the Kerr-Newman solution. Here we confirm the above interpretation of the real and imaginary parts of \tilde E(x+iy) by computing the charge- and current densities directly as distributions in R^3 supported in the source disk D. This shows in particular that D spins rigidly at the critical rate, so that its rim R moves at the speed of light. It is a pleasure to thank Ted Newman, Andrzej Trautman and Iwo Bialinicki-Birula for many instructive discussions, particularly in Warsaw and during a visit to Pittsburgh.

15 pages, Invited paper, Workshop on Canonical and Quantum Gravity III, Polish Academy of Sciences, Warsaw

References in corpus (4)

• On a Classical, Geometric Origin of Magnetic Moments, Spin-Angular Momentum and the Dirac Gyromagnetic Ratio
• Physical wavelets and their sources: Real physics in complex spacetime
• Maxwell Fields and Shear-Free Null Geodesic Congruences
• Making electromagnetic wavelets

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