Microlocal analysis of waves across the event horizon of an extremal rotating black hole

Static black holes contain regions of spacetime, which not even light can escape from. In the center of mass frame, these blocks are separated from each other by event horizons. Unlike pointlike particles, fields can spread and interact non-causally across the horizons. The microlocal theory of this is somewhat incomplete, however. For instance, the theory of real principal type operators does not apply on the horizon. In this article, we address this issue for the extremal rotating black hole. Namely, we show that null covectors on the horizon of the extremal Kerr spacetime form an involutive double characteristic manifold and then extend the construction of parametrix across the event horizon. This provides a mathematical basis for the asymptotic oscillatory solutions in the region. Such approximations are central in quantum mechanics. In contrast to the real principal case, solutions near the horizon admit two channels for propagation of singularities. Indeed, there is additional propagation along the double characteristic variety.