The long-time dynamics of Dirac particles in the Kerr-Newman black hole geometry

Abstract

We consider the Cauchy problem for the massive Dirac equation in the non-extreme Kerr-Newman geometry outside the event horizon. We derive an integral representation for the Dirac propagator involving the solutions of the ODEs which arise in Chandrasekhar's separation of variables. It is proved that for initial data in L ∞ loc near the event horizon with L 2 decay at infinity, the probability of the Dirac particle to be in any compact region of space tends to zero as t goes to infinity. This means that the Dirac particle must either disappear in the black hole or escape to infinity. e-print archive:

The Dirac equation in Kerr–Newman–Ads black hole background

“…This turns out to be useful also for the analysis of qualitative spectral properties ofĤ. It is worth mentioning that the aforementioned problem associated with variable separation in Chandrasekhar ansatz and the occurrence of two Hilbert spaces in the analysis of the Dirac equation have been already pointed out in [9] for the case of the Dirac equation on a black hole background of the Kerr-Newman family, which has been considered in several studies [10,11,12,13,14,15] (see also [16]); the above part of our analysis can be of interest also for that case. Moreover, our analysis points out some relevant differences to be related to the AdS background considered herein, and also introduces some interesting analysis to be associated with a magnetically charged black hole: we find the Dirac charge quantization as a condition ensuring essential selfadjointness for the Hamiltonian operatorĤ on C ∞ 0 ((r + , ∞) × S 2 ) 4 .…”

“…We do it as follows. If we foliate spacetime in t = constant slices S t , the metric on any slice (considering the shift vectors) is 8) where α = 1, 2, 3 and 9) and local measure…”

An Integral Spectral Representation of the Propagator for the Wave Equation in the Kerr Geometry

“…This fact remains true even when as in [8] we consider the system in a "finite box," i.e. when the range of the radial variable r is restricted to a bounded interval r ∈ [r L , r R ] with r 1 < r L < r R < ∞.…”

“…In this paper, we turn our attention to the scalar wave equation in the Kerr geometry. Our main result is to derive an integral representation for the propagator, similar to the one obtained for the Dirac equation in [8]. In our next paper [9], we will use this integral representation to analyze the long-time dynamics and the decay of solutions in L ∞ loc .…”

The Problem of Embedded Eigenvalues for the Dirac Equation in the Schwarzschild Black Hole Metric

“…Ref. [10] inferred the completeness of this ansatz from the aforementioned integral representation. An alternative proof of the completeness of the Chandrasekhar ansatz can also be found in [11].…”