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Time-dependent scattering theory for charged Dirac fields on a Reissner–Nordström black hole

### Abstract

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 dimensional evolution equation of hyperbolic type into a 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 and we show that their spectra are equal to and . 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.

© 2010 American Institute of Physics

Received 14 May 2010
Accepted 15 September 2010
Published online 27 October 2010

Article outline:

I. INTRODUCTION
II. REISSNER–NORDSTRÖM BLACK HOLE AND DIRAC EQUATION
III. ABSTRACT ANALYTIC FRAMEWORK AND FUNDAMENTAL PROPERTIES OF DIRAC HAMILTONIANS
A. Spin weighted harmonics
B. Symbol classes
C. Global Cauchy problem
D. Domain invariance
E. Definition of the velocity operator and Zitterbewegung
F. Absence of eigenvalues
IV. MOURRE THEORY FOR DIRAC HAMILTONIANS
A. Abstract theory
B. Locally conjugate operator for
V. WEAK PROPAGATION ESTIMATES
A. Large velocity estimates
B. Minimal velocity estimates
C. Microlocal velocity estimates
1. Microlocal velocity estimates at the horizon
2. Microlocal velocity estimates at spacelike infinity
VI. ASYMPTOTIC VELOCITY
A. Existence of
B. Spectrum of
VII. WAVE OPERATORS
A. Wave operators at the event horizon
B. Wave operators at spacelike infinity: I
C. Wave operators at spacelike infinity: II

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2010-10-27

2016-10-24

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