München 2019 – wissenschaftliches Programm
GR 16.5: Vortrag
Freitag, 22. März 2019, 12:30–12:45, HS 4
An integral spectral representation of the massive Dirac propagator in the Kerr geometry in Eddington–Finkelstein-type coordinates — •Christian Röken — University of Granada, Faculty of Sciences, Department of Geometry and Topology, 18071 Granada, Spain
An integral spectral representation of the massive Dirac propagator in the non-extreme Kerr geometry in horizon-penetrating coordinates, which describes the dynamics of Dirac particles outside and across the event horizon, up to the Cauchy horizon, is presented. To this end, the Kerr geometry is described in the Newman–Penrose formalism by a regular Carter tetrad in advanced Eddington–Finkelstein-type coordinates and the massive Dirac equation is given in a chiral Newman–Penrose dyad representation in Hamiltonian form. The essential self-adjointness of the Hamiltonian is shown employing a new method of proof for non-uniformly elliptic mixed initial-boundary value problems on a specific class of Lorentzian manifolds that combines results from the theory of symmetric hyperbolic systems with near-boundary elliptic methods. The resolvent of this operator is computed via the projector onto a finite-dimensional, invariant spectral eigenspace of the angular operator and the radial Green’s matrix stemming from Chandrasekhar’s separation of variables. By applying Stone’s formula to the spectral measure of the Hamiltonian in the spectral decomposition of the Dirac propagator, that is, by expressing the spectral measure in terms of this resolvent, one obtains an explicit integral representation of the propagator.