GRAVITATIONAL SCATTERING OF ELECTROMAGNETIC-FIELD BY SCHWARZSCHILD BLACK-HOLE

We study the electromagnetic scattering by a spherical black-hole. Maxwell's equations are written in Schwarzschild coordinates: the electromagnetic tensor is replaced with electric and magnetic fields in a three dimensional absolute space. We introduce a set of wave operators, W0+/-, W1+/-, yielding an electromagnetic field given an asymptotic behavior far from the black-hole, W0+/-, and near the Schwarzschild radius, W1+/-, as universal time t --> +/- infinity. The long range interactions are eliminated by identifying the radial coordinate in the asymptotic Minkowski space with the Regge-Wheeler parameter. After a separation of variables thanks to the generalised vector spherical harmonics of Gel'fand and Sapiro, the existence of the scattering operator is proved by using a Birman-Kato method, in particular, the asymptotic completeness of W1+/- implies the Damour-Znajeck condition: near the horizon, the fields of finite redshifted energy are described by ingoing plane waves. The Membrane Paradigm is justified: the scattering operator can be approximated by putting the impedence condition on the stretched horizon. We interpret these results on the Kruskal universe: the existence of W0-, W1- assures the characteristic Cauchy problem with data on the past horizons is well posed in the Schwarzschild submanifold, and the asymptotic completeness of W0+, W1+ allows to define the solution on the future horizons.