Epicyclic oscillations of charged particles in stationary solutions immersed in a magnetic field with application to the Kerr–Newman black hole

Abstract

We consider a stationary metric immersed in a uniform magnetic field and determine general expressions for the epicyclic frequencies of charged particles. Applications to the Kerr-Newman black hole is reach of physical consequences and reveals some new effects among which the existence of radially and vertically stable circular orbits in the region enclosed by the event horizon and the so-called "innermost" stable circular orbit in the plane of symmetry. I. WHY STUDYING AND DETERMINING THE EPICYCLIC OSCILLATIONS (ECOS)?Any small deviations from a stable, geodesic or nongeodesic, motion lead to an epicyclic motion around the stable path. If the epicyclic motion is that of a charged particle, the corresponding frequencies have direct observational effects [1]- [5].The general non-circular motion of charged particles in the geometry of a magnetized Schwarzschild black hole has been investigated [6] without, however, dealing with the epicyclic oscillations (ECOs). The easiest motion where ECOs may be determined analytically is the circular one [7], from this point of view ECOs of charged particles around circular paths of the Kerr black hole have been studied and their frequencies were determined in terms of the parameters of the black hole, the weak magnetic field in which it is immersed, the charged particle physical properties, and the fourvelocity vector of the circular geodesic motion [8,9]. One of the purposes of the present work is to determine the frequencies of the ECOs of charged particles around circular paths of a general stationary solution immersed in a magnetic field then apply them to the case of the Kerr-Newman black hole immersed in a magnetic field.In Sec. II we set the ansatz for the metric and electromagnetic field of a stationary metric immersed in a uniform magnetic field. In Sec. III we derive general expressions for the epicyclic frequencies and in Sec. IV we discuss some properties of the perturbed circular motion. In Sec. V we apply our results to the case of the Kerr-Newman black hole immersed in a magnetic field. We conclude in Sec. VI. II. THE METRIC AND THE ELECTROMAGNETIC FIELDWe split this section into two subsections dealing with the rotating charged metric, with charge Q, and the linear form of the resulting electromagnetic field due to the interaction of the charge Q with a uniform magnetic field B. A. The metricUsing the required symmetry properties of a stationary and axisymmetric spacetime that is circular [10] -a spacetime admitting the existence of two commuting 1 Killing vectors, a timelike one ξ µ t = (1, 0, 0, 0) and a spacelike one ξ 1 In a circular spacetime, there exists a family of two surfaces everywhere orthogonal to the plane defined by the two commuting Killing vectors ξ µ t and ξ µ ϕ . In such a spacetime one may choose the coordinates such that the only nonzero cross term of the metric is dtdϕ.

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