Distributional sources for Newman's holomorphic Coulomb field

Newman (1973 J. Math. Phys. 14 102-3) considered the holomorphic extension \skew3\tilde{\bm E}({\bm z}) of the Coulomb field E(x) in {\bb R}^3 . From an analysis of its multipole expansion, he concluded that the real and imaginary parts \hspace*{-2.3pc}

{\bi E}({\bi x}+ i{\bi y}) \equiv Re\, \skew3\tilde{\bm E}({\bm x}+ i{\bm y}), \qquad

{\bm H}({\bm x}+ i{\bm y}) \equiv {Im}\, \skew3\tilde{\bm E}({\bm x}+ i{\bm y}),

viewed as functions of x, are the electric and magnetic fields generated by a spinning ring of charge {\cal R} . This represents the EM part of the Kerr-Newman solution to the Einstein-Maxwell equations (Newman E T and Janis A I 1965 J. Math. Phys. 6 915-7 Newman E T et al 1965 J. Math. Phys. 6 918-9). As already pointed out in Newman and Janis (1965 J. Math. Phys. 6 915-7), this interpretation is somewhat problematic since the fields are double-valued. To make them single-valued, a branch cut must be introduced so that {\cal R} is replaced by a charged disc {\cal D} having {\cal R} as its boundary. In the context of curved spacetime, {\cal D} becomes a spinning disc 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 charge and current densities directly as distributions in {\bb R}^3 supported in {\cal D} . This will show that {\cal D} spins rigidly at the critical rate so that its rim {\cal R} moves at the speed of light.