Dirac Waves in the Kerr Geometry: Integral Representations, Mass Oscillation Property and the Hawking Effect
Final Report Abstract
In the framework of the first research project, we derived 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. To this end, we described the non-extreme Kerr geometry in the Newman-Penrose formalism by a regular Carter tetrad in advanced Eddington-Finkelstein-type coordinates and computed the associated spin coefficients by solving the first Maurer-Cartan equation of structure. We then determined the massive Dirac equation in a chiral Newman-Penrose dyad representation on this fixed background geometry. Applying Chandrasekhar’s separation of variables, we performed a detailed analysis of the radial asymptotics at infinity, the event horizon, and the Cauchy horizon, respectively, and quantified the decay of the corresponding errors. Furthermore, we studied the spectral properties of the angular eigenfunctions and eigenvalues. Subsequently, we rewrote the above Dirac equation in Hamiltonian form and showed the essential self-adjointness of the Hamiltonian. For the latter purpose, as the Dirac Hamiltonian fails to be elliptic at the event and the Cauchy horizon, we could not use standard elliptic methods of proof. Instead, we introduced and employed a new, general 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. In this regard and since the time evolution may not have been unitary because of Dirac particles impinging on the ring singularity, we also imposed a suitable Dirichlet-type boundary condition on a time-like inner hypersurface placed inside the Cauchy horizon, which has no effect on the particles’ dynamics outside the Cauchy horizon. We computed the resolvent of the Dirac Hamiltonian 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, we obtained an explicit integral representation of the propagator. Within the scope of the second research project, we analyzed the structure of the solution space of the Dirac equation in the exterior Schwarzschild geometry and constructed the fermionic signature operator. For this purpose, we represented the space-time inner product for families of solutions with variable mass parameter in terms of the respective scalar products and derived a so-called mass decomposition, which consists of a single mass integral involving the fermionic signature operator as well as a double integral that takes in to account the flux of Dirac currents across the event horizon. Moreover, we computed the spectrum of the fermionic signature operator and the corresponding generalized fermionic projector states. We found out that the pure quasi-free fermionic projector state obtained from the fermionic signature operator in the exterior Schwarzschild geometry coincides with the Hadamard state obtained by the frequency splitting for an observer in a rest frame at infinity.
Publications
- “Self-adjointness of the Dirac Hamiltonian for a class of non-uniformly elliptic boundary value problems,” Annals of Mathematical Sciences and Applications 1, pp. 301– 320 (2016)
F. Finster and C. Röken
(See online at https://doi.org/10.4310/AMSA.2016.v1.n2.a2) - “The fermionic projector in a time-dependent external potential: mass oscillation property and Hadamard states,” Journal of Mathematical Physics 57, id. 072303 (2016)
Finster, S. Murro, and C. Röken
(See online at https://doi.org/10.1063/1.4954806) - “The fermionic signature operator and quantum states in Rindler space-time,” Journal of Mathematical Analysis and Applications 454, pp. 385–411 (2017)
F. Finster, S. Murro, and C. Röken
(See online at https://doi.org/10.1016/j.jmaa.2017.04.044) - “The massive Dirac equation in Kerr geometry: separability in Eddington–Finkelstein-type coordinates and asymptotics,” General Relativity and Gravitation 49, id. 39 (2017)
C. Röken
(See online at https://doi.org/10.1007/s10714-017-2194-y) - The fermionic signature operator in the exterior Schwarzschild geometry
F. Finster and C. Röken
- “An integral spectral representation of the massive Dirac propagator in the Kerr geometry in Eddington–Finkelsteintype coordinates,” Advances in Theoretical and Mathematical Physics 22, pp. 47–92 (2018)
F. Finster and C. Röken
(See online at https://doi.org/10.4310/ATMP.2018.v22.n1.a3)
