Distributional Sources for Newman ’ s Holomorphic Field

In [N73], Newman considered the holomorphic extension Ẽ(z) of the Coulomb field E(x) in R. By analyzing its multipole expansion, he showed that the real and imaginary parts E(x+ iy) ≡ Re Ẽ(x+ iy), H(x+ iy) ≡ Im Ẽ(x+ iy), viewed as functions of x, are the electric and magnetic fields generated by a spinning ring of charge R. This represents the EM part of the Kerr-Newman solution to the Einstein-Maxwell equations [NJ65, N65]. As already pointed out in [NJ65], 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 rigidly spinning disk of charge and mass representing the singularity of the Kerr-Newman solution. Here we confirm the above interpretation of E and H without resorting to asymptotic expansions, by computing the chargeand current densities directly as distributions in R supported in D. This shows 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 for many instructive discussions, particularly in Warsaw and during a visit to Pittsburgh. Supported by AFOSR Grant #F49620-01-1-0271. March 24, 2008