Linear hyperbolic equations in a rotating black hole geometry

Linear hyperbolic equations describe the dynamics of quantum mechanical particles (Dirac equation) and of classical waves (equations for scalar, electromagnetic or gravitational waves). After a brief review of relativity and black holes, we consider the Cauchy problem for a linear hyperbolic equation in the Kerr geometry, the mathematical model of a rotating black hole. For the Dirac equation, an integral representation of the propagator is obtained, which yields pointwise decay and allows to develop the complete scattering theory. We also outline the analysis for the scalar wave equation, which is considerably harder due to the ergosphere, an annular region around the black hole where the classical energy density may be negative. We mention recent results for pointwise decay and discuss the phenomenon of superradiance. We finally give an outlook on electromagnetic and gravitational waves, and to the problem of linear stability of rotating black holes.