The two-centre problem in general relativity: the scattering of radiation by two extreme Reissner-Nordstrom black-holes

A special case of the Majumdar-Papapetrou solution of the Einstein-Maxwell equations is the static placement of two extreme Reissner-Nordstrom black-holes at a finite distance apart. The axisymmetric perturbations of this special solution are analysed in prolate spheroidal coordinates. In this paper, the analysis is restricted to axial perturbations. The problem is reduced to a pair of coupled partial differential equations. The equations allow the definition of an energy flux whose divergence vanishes. The asymptotic behaviours of the solutions at infinity and at the two singularities (where the horizons of the two black-holes are located) are determined. It is shown that in the vicinity of the singularities, the gravitational and the electromagnetic radiation prevails as coupled photon-graviton waves expressed in terms of a common amplitude and phase, for each specified angular momentum. The scattering of radiation by the two black-holes is a four-channel problem (in the terminology of nuclear physics). The four channels correspond to the incidence of the two sorts of radiation (gravitational and electromagnetic) from infinity and the incidence of the coupled photon-graviton waves on the horizons of the two black-holes. The flux integral, evaluated at infinity and at the two horizons, enables the formulation of the scattering problem in terms of a scattering matrix; and provides four identities among the scattering amplitudes as expressions of the conservation of energy in the scattering process. In an Appendix, the equations of geodesic motion in the space-time of the two black-holes are derived; and three families of null geodesics in the meridian plane are displayed