Black hole horizons are normally at finite spatial distance from the exterior region, but when they are degenerate (or extreme as they are usually referred to in this case) the spatial distance becomes infinite. One can still fall into the black hole in finite proper time but the crossing sphere is replaced by an "internal infinity". Near to the horizon of an extreme Kerr black hole, the scattering properties of test fields bear some similarities to what happens at an asymptotically flat infinity. This observation triggered a natural question concerning the peeling behaviour of test fields near such horizons. A geometrical tool known as the Couch-Torrence inversion is particularly well suited to studying this question. In this talk, I shall recall some essential notions on the peeling of fields at an asymptotically flat infinity and describe the Couch-Torrence inversion in the particular case of extreme Reissner-Nordström black holes, where it acts as a global conformal isometry of the spacetime. I will then show how to extend this inversion to more general spherically symmetric extreme horizons and describe what results can be obtained in terms of peeling. This is a joint ongoing project with Jack Borthwick (University of Besançon) and Eric Gourgoulhon (Paris Observatory).
