Separability of the Killing–Maxwell system underlying the generalized angular momentum constant in the Kerr–Newman black hole metrics

The concept of a Killing–Maxwell system may be defined by the relation Â[μ;v];ρ =(4π/3)ĵ[μgν]. In such a system the one-form Âμ is interpretable as the four-potential of an electromagnetic field F̂μv, whose source current ĵ μ is an ordinary Killing vector. Such a system determines a canonically associated duality class of source-free electromagnetic fields, its own dual being a Killing–Yano tensor, such as was found by Penrose [Ann. N.Y. Acad. Sci. 224, 125 (1973)] (with Floyd) to underlie the generalized angular momentum conservation law in the Kerr black hole metrics, the existence of the Killing–Yano tensor being also a sufficient condition for that of the Killing–Maxwell system. In the Kerr pure vacuum metric and more generally in the Kerr–Newman metrics for which a member of the associated family of source-free fields is coupled in gravitationally, it is shown that the gauge of the Killing–Maxwell one-form may be chosen so that it is expressible (in the standard Boyer–Lindquist coordinates) by 1/2 (a2 cos 2 θ−r2)dt+ 1/2 a(r2−a2)sin2 θ dφ, the corresponding source current being just (4π/3)(∂/∂t). It is found that this one-form (like that of the standard four-potential for the associated source-free field) satisfies the special requirement for separability of the corresponding coupled charged (scalar or Dirac spinor) wave equations.