The Dirac Propagator in the Kerr-Newman Metric

We give an alternative proof of the completeness of the Chandrasekhar ansatz for the Dirac equation in the Kerr-Newman metric. Based on this, we derive an integral representation for smooth compactly supported functions which in turn we use to derive an integral representation for the propagator of solutions of the Cauchy problem with initial data in the above class of functions. As a by-product, we also obtain the propagator for the Dirac equation in the Minkowski space-time in oblate spheroidal coordinates. One of the most spectacular predictions of general relativity are black holes which should form when a large mass is concentrated in a sufficiently small volume. The idea of a mass-concentration which is so dense that even light would be trapped goes back to Laplace in the 18th century. Shortly after Einstein developed general relativity, Karl Schwarzschild discovered in 1916 a mathematical solution to the equations of the theory that describes such an object. It was only much later, with the work of physicists like Oppenheimer, Volkoff and Snyder in the 1930’s, that the scientific community began to think seriously about the possibility that such objects might actually exist in the Universe. It was shown that when a sufficiently massive star runs out of fuel, it is unable to support itself against its own gravitational attraction and it should collapse into a black hole. Starting with the 1960’s and the 1970’s, in the so-called Golden Era of black hole research, new interesting phenomena like the Hawking radiation and superradiance were discovered but for their rigorous mathematical description we have to wait until the 1990’s and the beginning of the new century when the rigorous analysis of the propagation and of the scattering properties of classical and quantum fields on black hole space-times started to be developed. Whenever we attempt to analyze the scattering properties of fields in the more general framework of the Kerr-Newman black hole geometry, we are faced with several difficulties which are not present in the picture of the Schwarzschild metric. First of all, the Kerr-Newman solution is only axially symmetric (cylindrical symmetry) since it possesses only two commuting Killing vector fields, namely the time coordinate vector field ∂t and the longitude coordinate vector field ∂ϕ. This implies that there is no decomposition in spin-weighted spherical harmonics. Moreover, another difficulty is due to fact that the Kerr-Newman space-time is not stationary. In