Mathematics – Mathematical Physics
Scientific paper
Oct 2010
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2010jmp....51j2504t&link_type=abstract
Journal of Mathematical Physics, Volume 51, Issue 10, pp. 102504-102504-57 (2010).
Mathematics
Mathematical Physics
Black Holes, Cosmology, Dirac Equation, Space-Time Configurations, Spacetime Topology, Causal Structure, Spinor Structure
Scientific paper
In this paper, we prove a complete time-dependent scattering theory for charged (massive or not) Dirac fields outside a Reissner-Nordström black hole. We shall take the point of view of observers static at infinity, well described by the Schwarzschild system of coordinates. For such observers, the exterior of a Reissner-Nordström black hole is a smooth manifold having two distinct asymptotic regions: the horizon and spacelike infinity. We first simplify the later analysis using the spherical symmetry of the Reissner-Nordström black hole and we reduce the initial 3+1 dimensional evolution equation of hyperbolic type into a 1+1 dimensional one. Then, we establish various propagation estimates for such fields in the same spirit as in the works by Dereziński and Gérard [Scattering Theory of Classical and Quantum N-Particle Systems (Springer-Verlag, Berlin, 1997)]. We construct the asymptotic velocity operators P+/- and we show that their spectra are equal to σ(P+)={-1}∪[0,1] and σ(P-)=[-1,0]∪{1}. This information points out the very distinct behaviors of Dirac fields near the two asymptotic regions of the black hole. As a consequence of this construction, we prove the existence and asymptotic completeness of (Dollard modified at infinity) wave operators.
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