Accretion onto a Kerr black hole in the presence of a dipole magnetic field

We analyze the accretion of charged matter onto a rotating black hole immersed in an aligned dipolar magnetic field. We specialize to motion in the equatorial plane and calculate the ‘Keplerian’ angular momentum distribution, the marginally stable and marginally bound orbits, and the efficiency of mass-to-energy conversion as functions of the angular momentum of the black hole and of the product of the dipole moment and the charge of the infalling matter. Although the detailed results are quite different from those previously obtained in the case of an uniform magnetic field, the astrophysically relevant results are very similar; when hydrodynamical accretion is considered, these effects of the magnetic field are always very small. But for test particles the efficiency can be significantly increased for limited ranges of the parameters.     

Rotational symmetry of classical orbits, arbitrary quantization of angular momentum and the role of the gauge field in two-dimensional space

We study quantum–classical correspondence in terms of the coherent wave functions of a charged particle in two- dimensional central-scalar potentials as well as the gauge field of a magnetic flux in the sense that the probability clouds of wave functions are well localized on classical orbits. For both closed and open classical orbits, the non-integer angular-momentum quantization with the level space of angular momentum being greater or less than is determined uniquely by the same rotational symmetry of classical orbits and probability clouds of coherent wave functions, which is not necessarily 2π-periodic. The gauge potential of a magnetic flux impenetrable to the particle cannot change the quantization rule but is able to shift the spectrum of canonical angular momentum by a flux-dependent value, which results in a common topological phase for all wave functions in the given model. The well-known quantum mechanical anyon model becomes a special case of the arbitrary quantization, where the classical orbits are 2π-periodic.     

Equatorial circular orbits in Kerr–anti-de Sitter spacetimes

Equatorial circular orbits of test particles in the Kerr–anti-de Sitter black-hole and naked-singularity spacetimes are analyzed and their properties like the existence, orientation and stability are discussed. Due to the attractive cosmological constant ( $\varLambda <0$ ), all particles moving along equatorial orbits are still bound in the gravitational field of the central object. In general, there are two families of equatorial circular orbits. Particles moving along minus-family orbits possess negative angular momentum and, thus, they are counterrotating from the point of view of the locally non-rotating frames (LNRF). Particles moving along plus-family orbits possess, in most cases, positive angular momentum and belong to corotating particles from the point of view of the LNRF. Nevertheless, in stationary regions inside black holes and also near naked singularities with appropriately chosen value of the cosmological constant and rotational parameter $a<1.299$ , there are also counterrotating plus-family circular orbits. Moreover, in spacetimes with $a<1.089$ , some of these orbits are characterized by negative specific energy, indicating the bounding energy of the particle, moving along such an orbit, higher than its rest energy. In black-hole spacetimes, all such orbits are radially unstable, but in naked-singularity spacetimes, stable counterrotating orbits with negative specific energy exist.     

Geodesics around oscillatons

Oscillatons are spherically symmetric solutions to the Einstein–Klein–Gordon equations. These solutions are non-singular, asymptotically flat, and with periodic time dependency. In this paper, we investigate the geodesic motion of particles moving around of an oscillatonic field. Bound orbits are found for particular values of the particles' angular momentum L and their initial radial position r ₀. It is found that the radial coordinate of such particles oscillates in time and we are able to predict the corresponding oscillating period as well as its amplitude. We carry out this study for the quadratic V(φ) = m _Φ Φ²/2 scalar field potential. We discuss possible ways to follow in order to connect this kind of studies with astrophysical observations.     

Exact quantum mechanical description of the motion of spin-1/2 particles and of spin motion in a uniform magnetic field

The Dirac-Pauli equation is used to obtain the exact equation of spin motion for spin-1/2 particles with an anomalous magnetic moment in a constant and uniform magnetic field. Exact formulas are established for the angular velocity of the revolution of such particles along circular orbits and the rotation of the particle spin with respect to momentum. Finally, a quantum mechanical equation for the motion of the particles in a strong magnetic field is derived. Zh. éksp. Teor. Fiz. 114, 448–457 (August 1998)     

Orbits through the ergosphere of a Kerr black hole

We study the positions of orbits around a Kerr black hole with respect to its ergosphere. Ther– motions of zero-energy (E=0) null geodesies are inside truncated circular sectors, whose outer corners are on the static limit. Timelike geodesies with the same constants of motion are restricted inside a smaller area. For certain parameter values there are also orbits inside the inner horizon not reaching the center. Then we study the various types of orbits on the plane of symmetry for all the values of the angular momentum of the black hole 0aM, and of the angular momentum of the photons, or particles,L, and for all the values of the energyE. In particular we find the possible positions of the turning points with respect to the ergosphere. A restriction imposed by physical considerations is that the coordinate time increases when the proper time increases. This allows us to distinguish between positive and negative energy orbits. All negative energy orbits enter the horizon of the black hole.     

Motion of the charged particles off the equatorial plane of a Kerr black hole in an electromagnetic field

Charged particle orbits off the equatorial plane of a Kerr black hole in an external electromagnetic field is studied, both for dipole as well as uniform magnetic field. Particles are found to get trapped by the magnetic field if the initial value of the parallel velocity is small. Bending of the field lines in the vicinity of the hole and the consequent trapping of the particles in an otherwise uniform magnetic field indicates the significance of general relativistic effects in such cases.     

Dynamics of a single ion in a perturbed Penning trap. Sextupolar perturbation

In the frame work of classical mechanics, we study the nonlinear dynamics of a single ion trapped in a Penning trap perturbed by an electrostatic sextupolar perturbation. The perturbation is caused by a deformation in the configuration of the electrodes. By using a Hamiltonian formulation, we obtain that the system is governed by three parameters: the z-component of the canonical angular momentum P _φ - which is a constant of the motion because the perturbation we assume is axial-symmetric -, the parameter δ that determines the ratio between the axial and the cyclotron frequencies, and the parameter a which indicates how far from the ideal design the electrodes are. We study the case P _φ = 0. By means of surfaces of section, we show that the phase space structure is made of three fundamental families of orbits: arch, loop and box orbits. The coexistence of these kinds of orbits depends on the parameter δ. The escape is also explained on the basis of the shape of the potential energy surface as well as of the phase space structure. Received 6 September 2001 / Received in final form 19 March 2002 Published online 28 June 2002     

Orbital and epicyclic motion of charged test particles around non-rotating Einstein-Æther black holes

The study of charged test particle dynamics in the combined black hole gravitational field and magnetic field around it could provide important theoretical insight into astrophysical processes around such compact object. We have explored the orbital and epicyclic motion of charged test particles in the background of non-rotating Einstein-Æther black holes in the presence of external uniform magnetic field. We numerically integrate the equations of motion and analyze the trajectories of the charged test particles. We examined the stability of circular orbits using effective potential technique and study the characteristics of innermost stable circular orbits. We analyze the key features of quasi-harmonic oscillations of charged test particles nearby the stable circular orbits in an equatorial plane of the black hole, and investigate the radial profiles of the frequencies of latitudinal as well as radial harmonic oscillations in dependence on the strength of magnetic field, mass of the black hole and dimensionless coupling constants of the theory. We demonstrate that the magnetic field and dimensionless parameters of the theory have strong influence on charged particle motion around Einstein-Æther black holes.     

NULL GEODESICS IN STATIONARY ERNST-WILD SPACE-TIME

Null geodesics in equatorial plane of stationary Ernst-Wild space-time representing a Kerr black hole immersed in axlsymmetric magnetic field are investigated.As the case of static Ernst space-time there are two types of geodesics which depend on a dimensionless parameter β of the magnetic field.Differing from the static Ernst space-time,however,the critical β_C now depends on a dimensionless parameter q of the angular momentum of the Kerr black hole,and the relation between β_C and q is qualitatively different in two cases:L>O and L<0 where L is a dimensionless parameter of angular momentum of test particles moving along nu11 geodesics.     

The wigner-lubanski first integral in the Papapetrou-Corinaldesi equations of motion

Some years ago Papapetrou and Corinaldesi applied Papapetrou's equation's of motion of spinning particles to the case of motion in the Schwarzschild field. For the particular case of motion in the equatorial plane they found an extra integral of motion (in addition to the constants of energy and total angular momentum). We here give a group-theoretical interpretation to the origin of this constant by relating it to the Wigner-Lubanski constant known from the theory of representations of the Poincaré group.Work supported by the Israel National Academy of Sciences, Grant No. 197 (B)P.     

An effective potential for classical Yang-Mills fields as outline for bifurcation on gauge orbit space

Classical Yang-Mills (Y.M.) equations with static external sources are formulated as a Hamiltonian system with gauge symmetry in A₀ = 0 gauge. Using the concept of a “momentum mapping” (J. Marsden and A. Weinstein, Rep. Math. Phys.5 (1974), 121) on symplectic manifolds with symmetry, an analogue of centrifugal potential of a mass point in a spherically symmetric potential is derived. This gives rise to an effective potentialV_eff, whose critical points are rigorously proved to be in one-to-one correspondence with static Y.M. solutions. V_eff additionally depends on the prescribed external source ?, which is as a constant of motion analogous to angular moment of the mass point. Thus bifurcation of static solutions is caused by bifurcation of critical points of V_eff under variation of the external parameter ?. Some closing remarks on dynamics and stability on gauge orbit space are added.     

Magnetic moment of a two-dimensional degenerate electron gas in mesoscopic samples

The orbital magnetism of two-dimensional electrons in mesoscopic samples is studied in models where the interaction between electrons is neglected. Various geometries are considered as there are disc, plaquette, bracelet with hard wall confinement and also a confinement with a parabolic potential. We calculate the average magnetic moment which means an average with respect to size fluctuations and de Haas-van Alphen oscillations which arise in the case of a sharp Fermi cutoff. We see three distinct ranges in the magnetic field: (i) small field region where perturbation theory applies; (ii) moderate fields where edge currents play a prominent role; and (iii) the high field range with a Landau type susceptibility. In a quasiclassical picture, the electronic orbits are not qualitatively changed by a magnetic field in (i); skipping orbits are important in (ii); and in (iii), the cyclotron radius is smaller than the sample size. As a rule, we find an enhancement of the magnetic response which increases with k_FL, that is, with sample size divided by the Fermi wave length. Also, we have found out that the quasiclassical approximation fails in the calculation of the magnetic properties; on the other hand, we have seen no essential differences between the canonical ensemble (fixed particle number) and the grand canonical ensemble (chemical potential given). In the case of plaquettes, in particular for samples in the form of squares, we have found agreement with experimental results by Lévy, Reich, Pfeiffer and West.     

Stability of Circular Orbits around a Tidal Charged Black Hole

We study the effects of the tidal charge on the equatorial circular motion of neutral test particles near a tidal charged black hole. This analysis investigates stable as well as unstable circular orbits in all possible configurations of nonextremal and extremal cases. It is found that a negative tidal charge will increase the energy and angular momentum of a neutral test particle moving around a black hole. We obtain a continuous region of stability for both extremal and nonextremal cases. We conclude that the region of stability as well as radius of last stable circular orbit shows increasing behavior for Q < 0.     

Relativistic orbits of classical charged bodies in a spherically symmetric electrostatic field

The orbits of a relativistic charged body in a static, spherically symmetric electric field are calculated and classified in the classical theory. Contrary to the nonrelativistic problem, we find that there is a limiting minimal value for the angular momentumL _c . Should the actual angular momentum of a charged test body be lower than this limit, the test particle will spiral into the central point charge instead of having (precessing) Keplerian orbits.     

Charged particle motion in an electromagnetic field on Kerr background geometry

In this paper we study the trajectories of charged particles in an electromagnetic field superimposed on the Kerr background. The electromagnetic fields considered are of two types: (i) a dipole magnetic field with an associated quadrupole electric field, (ii) a uniform magnetic field. The contribution of the background geometry to the electromagnetic field is taken through the solutions of Petterson and Wald respectively. The effective potential is studied in detail for ther-motion of the particles in the equatorial plane and the orbits are obtained. The most interesting aspect of the study is the illustration of the effect of inertial frame dragging due to the rotation of the central star. This appears through the existence of nongyrating bound orbits at and inside the ergo surface. The presence of the magnetic field seems to increase the range of stable orbits, as was found in a previous study involving the Schwarzschild background.     

Bethe-ansatz solution of a model for a mixed valent impurity with two magnetic configurations: II. Numerical solution of the thermodynamic equations

A mixed valence impurity with two magnetic configurations of total angular momentumJ ₂ andJ ₁=J ₂+1/2, respectively, coupled by conduction electrons with total angular momentum 1/2 via a hybridization matrix element is considered. The thermodynamic Bethe-ansatz equations derived previously are solved numerically for various values ofJ ₂. Thef-level occupation, the entropy, the magnetic susceptibility and the specific heat are obtained as a function of temperature for variousf-level positions. The magnetic field dependence is also discussed in the limit of integer valence (exchange model).Supported by the CONICET, Argentina     

Fragment angular distributions from fission of238U,209Bi and197Au by 140-MeV4He Ions

The fission fragment angular distributions from reactions of 140-MeV⁴He ions with²³⁸U,²⁰⁹Bi and¹⁹⁷Au have been studied. From the anisotropies in the angular distributions, values for? _eff, the effective moment of inertia at the fission saddle point, have been estimated and compared with results obtained at lower bombarding energies. The derivation of? _eff values has included corrections for the effects of incomplete fusion mechanisms on the orbital angular momentum distribution of the fissioning nuclei and for neutron evaporation prior to fission. The results are also compared with heavy-ion-induced fission data for systems of similar fissility. Also, examination of the forward-backward symmetry of the²³⁸U angular distribution substantiates other results which show that the fraction of fission reactions which follow complete fusion of the target and projectile is less than 0.5 for 140-MeV⁴He-ion bombardment of²³⁸U.     

Testing spatial noncommutativity via Rydberg atoms

The possibility of testing spatial noncommutativity via Rydberg atoms is explored. An atomic di-pole of a cold Rydberg atom is arranged in appropriate electric and magnetic fields, so that the motion of the dipole is constrained to be planar and rotationally symmetric. Spatial noncommutativity leads the canonical angular momentum to possess fractional values. In the limit of vanishing kinetic energy, the dominate value of the lowest canonical angular momentum takes variant Planck's over h/2. Furthermore, in the limit of eliminating the magnetic field, the dominate value of the lowest canonical angular momentum changes from variant Planck's over h/2 to variant Planck's over h/4. This result is a clear signal of spatial noncommutativity. An experimental verification of this prediction is suggested.     

Limitations on the Penrose process

It is well known that the Penrose process (PP) is a way for extracting energy from a black hole (BH). An analytical and partially numerical study of the PP is presented for a particular case: an incoming particle, at rest at infinity, decays into two photons inside the ergoregion of a Kerr BH, assuming that all particles follow equatorial orbits. It is shown that this process cannot exist if the angular momentum of the BH is lower than a critical value. Considering the features of the non equatorial Kerr geodesics, it is conjectured that even when the incoming particle is not at rest at infinity and the trajectories are not equatorial, the critical value still holds. This lower limit on the angular momentum implies that not all the rotational energy of the BH could be extracted with the Penrose processes that we have considered.