Chaos in a gravitational field with dipoles

In this paper we investigate the dynamics of a test particle in the gravitational field with dipoles. At first we study the gravitational potential by numerical simulations, we find that, for appropriate parameters, there are two different cases in the potential curve: one is the one-well case with a stable critical point, and the other is the three-well case with three stable critical points and two unstable critical points. By performing Poincare sections for different values of the parameters and initial conditions, we find a regular motion and a chaotic motion. From these Poincar6 sections,we further confirm that the chaotic motion of the test particle originates mainly from the dipoles.     

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.     

Particle motion and electromagnetic fields of rotating compact gravitating objects with gravitomagnetic charge

The exact solution for the electromagnetic field occuring when the Kerr–Taub–NUT compact object is immersed (i) in an originally uniform magnetic field aligned along the axis of axial symmetry (ii) in dipolar magnetic field generated by current loop has been investigated. Effective potential of motion of charged test particle around Kerr–Taub–NUT gravitational source immersed in magnetic field with different values of external magnetic field and NUT parameter has been also investigated. In both cases presence of NUT parameter and magnetic field shifts stable circular orbits in the direction of the central gravitating object. Finally we find analytical solutions of Maxwell equations in the external background spacetime of a slowly rotating magnetized NUT star. The star is considered isolated and in vacuum, with monopolar configuration model for the stellar magnetic field.     

Chaos feature of test particle in the gravitational field with a quadrupole

In this paper we investigate the dynamics of a test particle in the gravitational field with a quadrupole. By constructing Poincaré sections for different values of the parameters and initial conditions, we find a chaotic evolution. From these Poincaré sections, we further confirm that the chaotic evolution of the test particle originates from the quadrupole.     

The Gravitational Waves from Test Particles around Black Holes Immersed in a Strong Magnetic Field

In this paper, we calculate the gravitational waveform from free test particles around Schwarzschild black holes immersed in a uniform strong magnetic field. By comparing with the cases of the Schwarzschild black holes, we find that for stable circle orbits, magnetic field can amplify amplitude and frequency of gravitational waves (here after GWs). For other general orbits, the uniform magnetic field also can amplify amplitude of GWs, enhance energy radiation of GWs and make it to shift to higher frequency. Another obvious influence of magnetic field B is that it can change the form of h _× component of GWs.     

Orbits in the field of a gravitating magnetic monopole

Orbits of test particles and light rays are an important tool to study the properties of space-time metrics. Here we systematically study the properties of the gravitational field of a globally regular magnetic monopole in terms of the geodesics of test particles and light. The gravitational field depends on two dimensionless parameters, defined as ratios of the characteristic mass scales present. For critical values of these parameters the resulting metric coefficients develop a singular behavior, which has profound influence on the properties of the resulting space-time and which is clearly reflected in the orbits of the test particles and light rays.     

Phase-plane analysis of test particle orbits in regular black holes

We investigate phase-plane analysis of general relativistic orbits in a gravitational field of the Reissner–Nordstr?m-type regular black hole spacetime. We employ phase-plane analysis to obtain different phase-plane diagrams of the test particle orbits by varying charge q and dimensionless parameter β, where β contains angular momentum of the test particle. We compute numerical values of radii for the innermost stable orbits and corresponding values of energy required to place the test particle in orbits. Later on, we employ similar analysis on an Ayón–Beato–García(ABG) regular black hole and a comparison regarding key results is also included.     

Dynamics of extended bodies with spin-induced quadrupole in Kerr spacetime: generic orbits

We discuss motions of extended bodies in Kerr spacetime by using Mathisson–Papapetrou–Dixon equations. We firstly solve the conditions for circular orbits, and calculate the orbital frequency shift due to the mass quadrupoles. The results show that we need not consider the spin-induced quadrupoles in extreme-mass-ratio inspirals for space-based gravitational wave detectors. We quantitatively investigate the temporal variation of rotational velocity of the extended body due to the coupling of quadrupole and background gravitational field. For generic orbits, we numerically integrate the Mathisson–Papapetrou–Dixon equations for evolving the motion of an extended body orbiting a Kerr black hole. By comparing with the monopole–dipole approximation, we reveal the influences of quadrupole moments of extended bodies on the orbital motion and chaotic dynamics of extreme-mass-ratio systems. We do not find any chaotic orbits for the extended bodies with physical spins and spin-induced quadrupoles. Possible implications for gravitational wave detection and pulsar timing observation are outlined.     

On the accuracy of the IWM–CFC approximation in differentially rotating relativistic stars

A circular restricted three-body problem describes the motion of a test particle around two massive bodies in circular orbits. In this system, orbital decay caused by a gravitational radiation reaction between the two primary bodies is considered but the direct effect of gravitational radiation on the test particle is neglected. We adopt distance- and time-scale transformations to Newtonian problems so that systems without orbital decay will not depend on separation between the primaries but systems with orbital decay will depend on this separation. If a regular or chaotic orbit is given in a Newtonian system, the starting separation of the primaries varies according to the corresponding decay system. Thus, insights into the chaotic behaviour of a third body in a decay case are provided. For a large initial separation between the primaries, the chaos that exists in a Newtonian problem may be retained for a long enough time scale of dissipative evolution before the primaries coalesce. The final state of a third body is escape attributed to orbital decay.     

Analytical study of chaos and applications

We summarize various cases where chaotic orbits can be described analytically. First we consider the case of a magnetic bottle where we have non-resonant and resonant ordered and chaotic orbits. In the sequence we consider the hyperbolic Hénon map, where chaos appears mainly around the origin, which is an unstable periodic orbit. In this case the chaotic orbits around the origin are represented by analytic series (Moser series). We find the domain of convergence of these Moser series and of similar series around other unstable periodic orbits. The asymptotic manifolds from the various unstable periodic orbits intersect at homoclinic and heteroclinic orbits that are given analytically. Then we consider some Hamiltonian systems and we find their homoclinic orbits by using a new method of analytic prolongation. An application of astronomical interest is the domain of convergence of the analytical series that determine the spiral structure of barred-spiral galaxies.     

Charged particle trajectories in a magnetic field on a curved space-time

In this paper we have studied the motion of charged particles in a dipole magnetic field on the Schwarzscbild background geometry. A detailed analysis has been made in the equatorial plane through the study of the effective potential curves. In the case of positive canonical angular momentum the effective potential has two maxima and two minima giving rise to a well-defined potential well rear the event horizon. This feature of the effective potential categorises the particle orbits into four classes, depending on their energies. (i) Particles, coming from infinity with energy less than the absolute maximum ofV _eff, would scatter away after being turned away by the magnetic field. (ii) Whereas those with energies higher than this would go into the central star seeing no barrier. (iii) Particles initially located within the potential well are naturally trapped, and they execute Larmor motion in bound gyrating orbits. (iv) and those with initial positions corresponding to the extrema ofV _eff follow circular orbits which are stable for non-relativistic particles and unstable for relativistic ones. We have also considered the case of negative canonical angular momentum and found that no trapping in bound orbits occur for this case. In the case when particles are not confined to the equatorial plane we have found that the particles execute oscillatory motion between two mirror points if the magnetic field is sufficiently high, but would continuously fall towards the event horizon otherwise. An erratum to this article is available at .     

Field effects and the critical end point in polymeric nematics

We calculate the effect of an external field on nematic polymer liquid crystals with polarizable dipoles along the chain backbone. In close analogy to conventional nematics we find a line of first-order transitions from nematic to paranematic as the field is increased, culminating in a critical end point as the critical field is attained. This critical point is related to those found by other authors. The results hold for electric and magnetic fields, the latter coupling to diamagnetic anisotropy. The electric case will be complicated if permanent dipoles are present. We demonstrate the equivalence of these results to the Landau approach in a field. Shortcomings in previous work on nematic polymers in fields are analysed.     

Charged string loops in Reissner–Nordström black hole background

We study the motion of current carrying charged string loops in the Reissner–Nordström black hole background combining the gravitational and electromagnetic field. Introducing new electromagnetic interaction between central charge and charged string loop makes the string loop equations of motion to be non-integrable even in the flat spacetime limit, but it can be governed by an effective potential even in the black hole background. We classify different types of the string loop trajectories using effective potential approach, and we compare the innermost stable string loop positions with loci of the charged particle innermost stable orbits. We examine string loop small oscillations around minima of the string loop effective potential, and we plot radial profiles of the string loop oscillation frequencies for both the radial and vertical modes. We construct charged string loop quasi-periodic oscillations model and we compare it with observed data from microquasars GRO 1655-40, XTE 1550-564, and GRS 1915+105. We also study the acceleration of current carrying string loops along the vertical axis and the string loop ejection from RN black hole neighbourhood, taking also into account the electromagnetic interaction.     

General magnetized Weyl solutions: disks and motion of charged particles

We construct three families of general magnetostatic axisymmetric exact solutions of Einstein-Maxwell equations in spherical coordinates, prolate, and oblates. The solutions obtained are then presented in the system of generalized spheroidal coordinates which is a generalization of the previous systems. The method used to build such solutions is the well-known complex potential formalism proposed by Ernst, using as seed solutions vacuum solutions of the Einstein field equations. We show explicitly some particular solutions among them a magnetized Erez-Rosen solution and a magnetized Morgan-Morgan solution, which we interpret as the exterior gravitational field of a finite dislike source immersed in a magnetic field. From them we also construct using the well known “displace, cut and reflect” method exact solutions representing relativistic thin disks of infinite extension. We then analyze the motion of electrically charged test particles around these fields for equatorial circular orbits and we discuss their stability against radial perturbations. For magnetized Morgan-Morgan fields we find that inside of disk the presence of magnetic field provides the possibility of to find relativist charged particles moving in both prograde and retrograde direction.     

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.     

Hybrid stable-chaotic states in coupled quantum well stadia

We use tunnel current spectroscopy to investigate the quantum states of two GaAs quantum wells coupled by a low (100 meV) (AlGa)As tunnel barrier. A high tilted magnetic field is used to generate strongly chaotic electron motion in the two wells which act as coupled chaotic ‘stadia'. The effect of the tunnel barrier on the dynamics of the system depends on the magnitude of the applied bias voltage V. For V375 mV, the central potential barrier acts as a perturbation which modifies the trajectories of selected periodic orbits in the quantum well. Scattering off the central barrier also generates new periodic orbits involving multiple collisions on all three barriers. These orbits ‘scar' distinct sets of eigenstates which generate periodic resonant peaks in the current–voltage characteristics of the device. When the device is biased such that the injected electrons just surmount the central barrier, our calculations reveal novel hybrid scarred states with both stable and chaotic characteristics.     

脉冲式棘齿势场作用下囚禁离子的规则与混沌运动

考虑赝势近似下囚禁于Paul阱中的单离子与由脉冲式双激光驻波构成的棘齿势场的相互作用.应用积分方程方法得到系统的经典运动精确解，通过数值方法作出相空间轨道并计算由平均速度定义的流，结合分析与数值结果研究囚禁离子的规则与混沌运动特性.与单驻波型激光脉冲情形相比，发现两个重要的棘齿效应：一是脉冲式棘齿势场的作用导致参数空间混沌区域的改变，从而适当调节第二驻波参数，可使离子的规则运动变为混沌运动，或者混沌运动变为规则运动；二是通过分析平均流随激光参数的变化，发现棘齿势场的介入能使囚禁离子作平均意义下的单向输运，随着势场强度增加到混沌区域，流的强度明显减小并改变方向，系统进入混沌运动. 关键词： 脉冲式棘齿势 囚禁离子 混沌 输运     

Spherically symmetric configurations of General Relativity in presence of scalar fields: separation of circular orbits

We show that the test body stable circular orbits around the spherically symmetric black hole (BH) configuration can form disjoint structures in presence of a minimally coupled nonlinear scalar field. General conditions for the disjoint structures to exist are formulated. To present examples we construct a two-parametric family of exact solutions to Einstein equations with scalar fields for appropriate self-interaction potentials. For different values of the family parameters the solutions describe either BH or naked singularity (NS). We found numerically regions of the parameters when there exist two disjoint regions of stable circular orbits; such nonconnected structures indeed can exist in case of both BH and NS solutions.     

Chaos in the one-dimensional gravitational three-body problem

We have investigated the appearance of chaos in the one-dimensional Newtonian gravitational three-body system (three masses on a line with -1/r pairwise potential). In the center of mass coordinates this system has two degrees of freedom and can be conveniently studied using Poincare sections. We have concentrated in particular on how the behavior changes when the relative masses of the three bodies change. We consider only the physically more interesting case of negative total energy. For two mass choices we have calculated 18 000 full orbits (with initial states on a 100x180 lattice on the Poincare section) and obtained dwell time distributions. For 105 mass choices we have calculated Poincare maps for 10x18 starting points. Our results show that the Poincare section (and hence the phase space) divides into three well defined regions with orbits of different characteristics: (1) There is a region of fast scattering, with a minimum of pairwise collisions. This region consists of 'scallops' bordering the E=0 line, within a scallop the orbits vary smoothly. The number of the scallops increases as the mass of the central particle decreases. (2) In the chaotic scattering region the interaction times are longer, and both the interaction time and the final state depend sensitively on the starting point on the Poincare section. For both (1) and (2) the initial and final states consist of a binary + single particle. (3) The third region consists of quasiperiodic orbits where the three masses are bound together forever. At the center of the quasiperiodic region there is a periodic orbit discovered (numerically) by Schubart in 1956. The stability of the Schubart orbit turns out to correlate strongly with the global behavior.