An Integral Representation of the Massive Dirac Propagator in the Nonextreme Kerr Geometry in Horizon-penetrating Coordinates

Abstract

The main objective of this doctoral thesis is the derivation of an integral spectral representation of the massive Dirac propagator in the nonextreme Kerr geometry in horizon-penetrating advanced Eddington– Finkelstein-type coordinates. To this end, we divide the doctoral thesis into the following three parts. In the first part, we describe the nonextreme Kerr geometry in the Newman–Penrose formalism by means of a Carter tetrad in advanced Eddington–Finkelstein-type coordinates, which are regular across the event and the Cauchy horizon, respectively, and feature a temporal function for which the level sets are partial Cauchy surfaces. On this background geometry, we define the massive Dirac equation in the Weyl representation in 2-spinor form with a Newman–Penrose dyad basis for the spinor space. We perform Chandrasekhar’s mode analysis and thus show the separability of the massive Dirac equation expressed in such horizon-penetrating coordinates into systems of radial and angular ordinary differential equations (ODEs). We compute asymptotic radial solutions at infinity, the event horizon, and the Cauchy horizon, and demonstrate that the corresponding errors have suitable decay. Furthermore, we study specific aspects of the set of eigenfunctions and the eigenvalue spectrum of the angular system. In the second part, we introduce a new method of proof for the essential self-adjointness of the Dirac Hamiltonian for a particular class of nonuniformly elliptic mixed initial-boundary value problems on smooth asymptotically flat Lorentzian manifolds, combining results from the theory of symmetric hyperbolic systems with near-boundary elliptic methods. Finally, in the third part, we present the Hamiltonian formulation of the massive Dirac equation in the nonextreme Kerr geometry in advanced Eddington–Finkelstein-type coordinates and, within this framework, derive an explicit integral spectral representation of the massive Dirac propagator, which yields the full time-dependent dynamics of massive spin-1/2 fermions outside, across, and inside the event horizon, up to the Cauchy horizon. For the construction of this propagator, we first prove that the Dirac Hamiltonian in the extended Kerr geometry is essentially self-adjoint by employing the method introduced in the second part, and then use the spectral theorem for unbounded self-adjoint operators as well as Stone’s formula, which links the spectral measure of the Dirac Hamiltonian to the associated resolvent. We determine the resolvent in a separated form in terms of the projector onto a finite-dimensional invariant spectral eigenspace of the angular operator and the radial Green’s matrix both obtained within the mode analysis of the Dirac equation presented in the first part. This propagator may be applied to study the long-time dynamics and the decay rates of massive Dirac fields in a rotating Kerr black hole spacetime. It can furthermore be used in the formulation of an algebraic quantum field theory.