Distributional Sources for Newman's Holomorphic Field

TL;DR

This paper confirms that the holomorphic extension of the Coulomb field corresponds to a spinning charged disk in the Kerr-Newman solution, with distributions supported on the disk, clarifying the physical interpretation of Newman’s fields.

Contribution

It provides a distributional analysis of Newman’s holomorphic Coulomb field, demonstrating the source as a spinning charged disk with the rim moving at light speed.

Findings

Charge and current densities are supported on the source disk D.

The disk spins rigidly at the critical rate, with its rim moving at the speed of light.

The interpretation of the fields as arising from a spinning charged disk is validated.

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.