Spectral analysis of radial Dirac operators in the Kerr-Newman metric and its applications to time-periodic solutions

Abstract

We investigate the existence of time-periodic solutions of the Dirac equation in the Kerr-Newman background metric. To this end, the solutions are expanded in a Fourier series with respect to the time variable t, and the Chandrasekhar separation ansatz is applied so that the question of existence of a time-periodic solution is reduced to the solvability of a certain coupled system of ordinary differential equations. First, we prove the already known result that there are no time-periodic solutions in the nonextreme case. Then, it is shown that in the extreme case for fixed black hole data there is a sequence of particle masses ͑m N ͒ NN for which a time-periodic solution of the Dirac equation does exist. The period of the solution depends only on the data of the black hole described by the Kerr-Newman metric.

The absence of normalizable time-periodic solutions for the Dirac equation in the Kerr–Newman–dS black hole background

“…In the aforementioned studies the absence of time-periodic normalizable solutions of the Dirac equation has been proved mainly in the non-extremal case. The extremal one has been shown to require further investigation, and in the Kerr-Newman case the existence of normalizable time-periodic solutions was proved in [6,5]. It is a peculiar property of the background considered herein to forbid the existence of time-periodic normalizable solutions for the Dirac equation even in the extremal case, and this can be proved in a rather straightforward way.…”

“…As it is well-known from the study of the Kerr-Newman case and of the Kerr-Newman-AdS case [16,5,6,7, 1], eigenvalues for the Hamiltonian H correspond to the solutions of the following system of coupled eigenvalue equations have to be satisfied simultaneously in L 2 ((0, π), …”

“…As in the Kerr-Newman case (cf. [5]), one needs simply to study the asymptotic behavior of the solutions of the linear system…”

“…We do not deal with this problem herein, and we limit ourselves to study the problem of the absence of time-periodic normalizable solutions of the Dirac equation. The latter topic has given rise to a number of studies in the recent literature [2,3,4,5,6,7,8,9], mostly involved in black holes of the Kerr-Newman family, or still in absence of cosmological constant. We also considered this problem in the case of Kerr-Newman-AdS black holes [1].…”

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 .…”