Dirac Waves in the Kerr Geometry: Integral Representations, Mass Oscillation Property and the Hawking Effect

• Applicant: Professor Dr. Felix Finster
• Subject Area: Mathematics
• Term: from 2014 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 262201789

In our research project, we study various aspects of the dynamics of Dirac waves in the Kerr geometry, with the goal of obtaining a novel description of the Hawking effect. We constructed a horizon-penetrating coordinate system that covers both the exterior and interior Kerr geometry and is regular across the event and Cauchy horizons. In this coordinate system, we showed that the massive Dirac equation is separable. Moreover, we gave a rigorous asymptotic analysis including decay properties. We also introduced a new method for the construction of a self-adjoint extension of the Dirac Hamiltonian for a class of non-uniformly elliptic boundary value problems. This was of paramount importance as the Dirac Hamiltonian in Kerr geometry fails to be elliptic at the horizons. On the basis of the above and a resolvent method, we worked out an integral representation for the massive Dirac propagator in Kerr geometry in horizon-penetrating coordinates, which completely describes the dynamics of Dirac waves outside, across, and inside the event horizon, up to the Cauchy horizon.In the continuation of our research project, we want to extend the notion of the ``mass oscillation property'' by introducing a general ``mass decomposition'' of the solution space of the massive Dirac equation in the Kerr geometry and give an explicit construction with the help of the above-mentioned Dirac propagator. This yields a canonical decomposition of the solution space into two subspaces. Using those structures, we intend to setup Fock spaces and study the resulting many-particle quantum states. The vacuum state constructed in the maximal extension of the non-extreme Kerr geometry should coincide with the thermal Hawking state up to corrections due to mutual couplings of spin, angular momentum and gravity, which we plan to quantify.

• DFG Programme: Research Grants