Oblique magnetic fields and the role of frame dragging near rotating black hole

Magnetic null points can develop near the ergosphere boundary of a rotating black hole by the combined effects of strong gravitational field and the frame-dragging mechanism. The induced electric component does not vanish an efficient process of particle acceleration can occur. Furthermore, the effect of imposed (weak) magnetic field can trigger an onset of chaos. The model set-up appears to be relevant for low-accretion-rate nuclei of some galaxies which exhibit episodic accretion events (such as the Milky Way's supermassive black hole) embedded in a large-scale magnetic field of external origin. We review our recent results and we give additional context for future work with the focus on the role of gravito-magnetic effects caused by rotation of the black hole. While the test motion is strictly regular in the classical black hole space-time, with and without effects of rotation or electric charge, gravitational perturbations and imposed external electromagnetic fields may lead to chaos.


Introduction
Cosmic black holes can act as agents of acceleration to very high energies of electrically charged particles in their immediate vicinity.At the same time, the motion of particles can exhibit transition from regular to chaotic motion.In this overview we describe properties of a system consisting of a rotating black hole immersed in a large-scale magnetic field, i.e., an external (weak) field that is organized on length-scales exceeding the gravitational radius of the black hole, R g ≡GM • /c 2 ∼ 1.5 × 10 13 M 8 cm (M 8 ≡M • /10 8 M ⊙ , where M • denotes the mass of the black hole).Electrically charged particles in the immediate neighborhood of the horizon are influenced by strong gravity acting together with magnetic and induced electric components.
We relax several constraints which were often imposed in previous works: the magnetic field does not have to share a common symmetry axis with the spin of the black hole but they can be inclined with respect to each other, thus violating the axial symmetry.Also, the black hole does not have to remain at rest but it can instead perform fast translational motion together with rotation.We demonstrate that the generalization brings new effects.Starting from uniform electro-vacuum fields in the curved spacetime, we find separatrices and identify magnetic neutral points forming in certain circumstances.We suggest that these structures can represent signatures of magnetic reconnection triggered by frame-dragging effects in the ergosphere.We further investigate the motion of charged particles in these black hole magnetospheres.We concentrate on the transition from the regular motion to chaos, and in this context we explore the characteristics of chaos in relativity.We apply recurrence plots as a suitable technique to quan-tify the degree of chaoticness near a black hole.
Various parts of the scientific content of the present summary were published in refs.[15][16][17]23] and in the Thesis [20].Investigation of the particle motion in the astrophysical corona was preceded by the study of the topology of off-equatorial potential lobes performed by [25].Discussion of the charged particle motion in such lobes was also a subject of the contribution [24].Initial steps of the investigation of electromagnetic fields around drifting Kerr black hole were described by [19].Here we bring together different aspects of the motion in the context of weakly magnetised black holes and, in the last section, we give an outlook and future prospects of our research line together with open questions.

Motivation for magnetic neutral points as a trigger of particle acceleration
Observations of microquasars, pulsars, gamma-ray bursts indicate that the astrophysical jets play an important role almost everywhere, i.e., in different kinds of compact objects, ranging from stellar-mass black holes to supermassive black holes in active galactic nuclei (AGNs) as well as the starving supermassive black hole in the Milky Way's center [29].There is plenty of observational evidence suggesting that the initial acceleration of jets takes place very near black holes (and other compact object) and it proceeds via electromagnetic forces.Jets and accretion disks in the vicinity of compact objects probably create a magnetically driven symbiotic system [e.g., ref . 10].
The current promising model of the dynamics (i.e.launching, accelerating, and collimating) of the astrophysical jets is based on the magnetohydrodynamics PREPRINT (MHD).The results of the simulations employing general relativistic MHD equations [28] correlate with observations of M87 [14] where the formation and the collimation of the jet were analyzed.
Moreover, the 3D relativistic MHD simulations carried out by [12] demonstrate the essential role which the accretion disk's coronae play in the collimation and acceleration of jets.Indeed the dominant force accelerating the matter outward in a given numerical model originates from the coronal pressure.Regions above and below the equatorial plane become dominated by the magnetic pressure and large-scale magnetic fields may also develop by the dynamo action.
Recent relativistic (numerical) study by [45] reveals the formation of ordered jet-like structure of an ultrastrong magnetic field in the merger of binary neutron stars.Such system thus might serve as an astrophysical engine for observed short gamma ray bursts.
Small-scale (turbulent) magnetic fields have been also employed in accretion physics, where the magnetorotational instability (MRI) is believed to operate in accretion flows, generating the effective viscosity necessary for the accretion process [2].Magnetic reconnection is likely to be responsible for rapid flares that are observed in X-rays.Finally, Faraday rotation measurements suggest that tangled magnetic fields are present in jets [4].
Observations of the Galactic center (GC) reveal the presence of another remarkable large-scale magnetic structure -nonthermal filaments (NTFs).NTFs cross the Galactic plane and their length reaches tens of parsecs while they are only tenths of parsec wide.The strength of the magnetic field within the NTF may approach ≈ 1 mG while the typical interstellar value is ≈ 10 µG [30].Initially, it was thought that NTFs trace the pervasive poloidal magnetic field present throughout the GC [39].Later, however, it became apparent that the structure of the magnetic field in the central region of the Galaxy is more complex [11].See, e.g., refs.[30,40] and the figure 1 for the snapshots of the GC at 90 cm (330 MHz) that exhibit NTFs.
Overall it is quite likely that electromagnetic mechanisms play a major role and operate both near supermassive black holes in quasars as well as stellar-mass black holes and neutron stars in accreting binary systems.Besides that a faint magnetic field is present throughout the interstellar medium, being locally intensified in NTFs.

Electro-vacuum fields
A theoretical survey of the properties of vacuum electromagnetic (EM) fields may be regarded as the fundamental starting point in studying the dynamics of diluted astrophysical environments.At the next stage we will consider the motion of non-interacting particles exposed to these fields.The structure of a particular astrophysically motivated EM field emerging in the vicinity of rotating black hole has been studied in detail [see refs.5, 6, 20, and further bibliography cited therein].Subsequently we also discuss the motion of charged particles exposed to the field representing a special case of a general solution explored before.Primarily we concern ourselves with the stable orbits occupying off-equatorial potential lobes.Particles on these orbits are relevant for the description of astrophysical corona comprising of diluted plasma residing outside the equatorial plane in the inner parts of accreting black hole systems.Gaseous corona is supposed to play a key role in the formation of observed X-ray spectra of both active galactic nuclei (AGNs) and microquasars [9].Power law component of the spectra is believed to result from the inverse Compton scattering of the thermal photons emitted in the inner parts of the disk.Relativistic electrons residing in the corona serve as a scatterers in this process.Their dynamic properties (e.g.resonances) thus shall have imprint on the observed spectra.
The role of magnetic fields near strongly gravitating objects has been subject of many investigations [e.g.44].They are relevant for accretion disks that may be embedded in large-scale magnetic fields, for example when the accretion flow penetrates close to a neutron star [31,36].Outside the main body of the accretion disk, i.e. above and below the equatorial plane, the accreted material forms a highly diluted environment, a corona, where the density of matter PREPRINT vol.
no. / Oblique Magnetic Fields and the Role of Frame Dragging is low and the mean free path of particles is large in comparison with the characteristic length-scale, i.e. the gravitational radius of the central body, r = R g .The origin of the coronal flows and the relevant processes governing their structure are still unclear.In this context we discuss motion of electrically charged particles outside the equatorial plane.

Regular and chaotic dynamics
Accretion onto black holes and compact stars brings material in a zone of strong gravitational and electromagnetic fields.We study dynamical properties of motion of electrically charged particles forming a highly diluted medium (a corona) in the regime of strong gravity and large-scale (ordered) magnetic field.
We start our discussion from a system that allows regular motion, then we focus on the onset of chaos.To this end, we investigate the case of a rotating black hole immersed in a weak, asymptotically uniform magnetic field.We also consider a magnetic star, approximated by the Schwarzschild metric and a test magnetic field of a rotating dipole.These are two model examples of systems permitting energetically bound, off-equatorial motion of matter confined to the halo lobes that encircle the central body.Our approach allows us to address the question of whether the spin parameter of the black hole plays any major role in determining the degree of the chaoticness.
The both dynamic systems may be regarded as different instances of the originally integrable systems which were perturbed by the electromagnetic test field.Complete integrability of geodesic motion of a free particle in Schwarzschild spacetime is easy to verify [37].To some surprise it was later found that also free particle motion in Kerr spacetime and even the charged particle motion in Kerr-Newman is completely integrable [7] since separation of the equations of motion is possible as there exists additional integral of motion -Carter's constant L. Trajectories found in such a system are purely regular.
In the non-integrable system, however, both regular and chaotic trajectories may coexist in the phase space.Standard method of a qualitative survey of the non-linear dynamics is based on the construction of Poincaré surfaces of section which allow us to visually discriminate between the chaotic and regular regimes of motion.
On the other hand, quantifying chaos by Lyapunov Characteristic Exponents (LCEs), as its standard and commonly used indicator, becomes problematic in General Relativity (GR) because LCEs are not invariant under coordinate transformations.Besides that the usual method of computing LCEs involves evaluation of distances between the neighbouring trajectories, which becomes intricate in GR.Although there are operational workabouts to partially overcome these difficulties [e.g.49] the need for a consistent treatment is apparent.The geometrical ap-proach suggested recently by [46] could eventually provide a covariant method of evaluation of the Lyapunov spectra in GR.
In this context we adopt a different tool to investigate the dynamic system -Recurrence Analysis [34].To characterize the motion, we construct the Recurrence Plots (RPs) and we compare them with Poincaré surfaces of section.We describe the Recurrence Plots in terms of the Recurrence Quantification Analysis (RQA; see fig. 10 later in the text), which allows us to identify the transition between different dynamical regimes.We demonstrate that this new technique is able to detect the chaos onset very efficiently, and to provide its quantitative measure.The chaos typically occurs when the conserved energy is raised to a sufficiently high level that allows the particles to traverse the equatorial plane.We find that the role of the black-hole spin in setting the chaos is more complicated than initially thought.

4.
Near-horizon structure of oblique electromagnetic test fields: magnetic null points and layers

Effects of Kerr metric onto weak magnetic fields
In ref. [20] we went through various issues concerning the structure of the electromagnetic field which arises from the interplay between the frame-dragging effect and the uniform magnetic field with general orientation with respect to the rotation axis of the Kerr source.We further generalized the model by allowing the black hole to move translationally in a general direction with respect to the magnetic background.Components of the electromagnetic tensor F µν describing resulting field were given explicitly (in a symbolic way regarding their length) in the terms of the former non-drifting solution.Special attention was paid to the comparison of various definitions of the electric/magnetic field.We also reviewed the construction of three distinct frames attached to the physical observers.
We have been interested in the solutions describing an originally uniform magnetic field under the influence of the Kerr black hole (see figures 2-3).Since the Kerr metric is asymptotically flat, this EM field reduces to the original homogeneous magnetic field in the asymptotic region.First such a test field solution was given by [48] for the special case of perfect alignment of the asymptotically uniform magnetic field with the symmetry axis.Using a different approach of Newman Penrose formalism a more general solution for an arbitrary orientation of the asymptotic field was inferred by [5].We departed from their solution to construct the EM field around the Kerr black hole which is drifting through the asymptotically uniform magnetic field.

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Figure 2.These plots cover the immediate vicinity of a rotating black hole (black circle), i.e. the region within and just outside of ergosphere.Here, the drift-induced effects (i.e., the effects caused by the translation boost of the linear motion of the black hole) as well as the frame dragging effects (due to rotation) upon the structure of the aligned magnetic field (extreme spin, a = M ).Magnetic field lines are shown and their changing direction is indicated by arrows.Translational motion of the black hole is restricted to be parallel with the horizontal axis, vx = 0.5c, vy = vz = 0. We observe a narrow front that develops above the horizon, where the field lines have complex, multi-layered structure.The colour scale indicates the intensity of the field (in arbitrary units).See ref. [20] for further details and references.Kerr metric in Boyer-Lindquist coordinates x µ = (t, r, θ, ϕ) can be (introducing the geometrized units G = c = k = k C = 1) expressed as follows [37]: where PREPRINT vol.
no. / Oblique Magnetic Fields and the Role of Frame Dragging while a stands for the dimensionless spin of the black hole (|a| ≤ 1), and M ≡ M • for its mass.We compared several possible definitions of electric and magnetic field vectors.First we remained in the coordinate basis and defined coordinate, physical, renormalized and asymptotically motivated (AMO) components of the fields.These differ in their behaviour close to the horizon of the black hole and appear useful when exploring the field structures, but they only allow for a clear physical interpretation in the asymptotical region.A consistent definition of the electric and magnetic fields should, however, provide a natural physical interpretation of the observables measured by certain physical observer at any distance from the center.We let such an observer with four-velocity u µ equipped with the orthonormal tetrad e µ (α) measure the Lorentz force using his tetrad basis.Tetrad components of the vector fields defining the desired lines of force are given as the spatial part of the projection µ are 1-forms dual to the tetrad vectors e µ (α) .We also discus the choice of the tetrad in detail [20].Namely we employed standard ZAMO (zero angular momentum observer) tetrad and tetrad of the observer who is freely falling from the rest at infinity (FFOFI).In the equatorial plane we also used the astrophysicaly relevant tetrad attached to the circular Keplerian observer above the marginally stable orbit and freely inspiralling to the horizon under this orbit (keeping the Keplerian angular momentum and energy of the innermost orbit).However, in this text we only present the field structures measured by FFOFI.
Before exploring the rich structures arising from the drift and oblique background field we first revisited the issue of the expulsion of the aligned magnetic field out of the horizon of the extremal Kerr black hole (Meissner effect).Since the effect itself has already been discussed thoroughly in the literature [41], we concentrate on the observer aspect of the problem instead.By comparing alternative definitions of the magnetic vector field combined with the choice of the four-velocity profiles we came to the conclusion that (i) the Meissner effect is observer dependent and (ii) some definitions of the field lines do not fit well into the Boyer-Lindquist coordinate system since they artificially amplify the effect of the coordinate singularity at the horizon.
Namely we observed that in the ZAMO tetrad the field does not exhibit the Meissner effect while in the FFOFI frame the field is expelled.On the other hand in the renormalized field components the expulsion is observed for both ZAMO as well as for FFOFI test charges.In coordinate components the Meissner effect also appears but we decide not to use them because the coordinate basis is not normalized which causes artificial deformation of the field lines.On the other hand the properly normalized physical components appear problematic since they amplify the effect of the coordinate singularity at the horizon as mentioned above.We found them to be quite inconvenient for the use in Boyer-Lindquist coordinate system (at least in the region close to the horizon).Asymptotically-motivated (AMO) components which are observer independent as they reflect the F µν components directly were also employed and the resulting magnetic lines of force were identified with the section of the surfaces of the constant magnetic flux which represent yet another way to display the field.We note that in the FFOFI frame both magnetic and electric fields are expelled out of the horizon in the case of the extremal spin.
Upon introducing the perpendicular field component we observe that generally (i) the magnetic field is not expelled any more, and (ii) both the electric and magnetic fields acquire a tightly layered structure in the narrow zone just above to the horizon.Structure of the field is surprisingly complex in this region, self-similar patterns are observed regardless the choice of the observer proving that the layering is an intrinsic feature of the field rather than a merely observer's effect.

Separator reconnection in Kerr metric with boost
In the case of black hole's translational motion through the aligned field, we notice the complex layering of the field which we attribute to the transversal component arising from the Lorentz boost (figures 4-5).However, for a sufficiently rapid drift we observe a new effect emerging: as the layers gradually transform they give rise to the formation of the neutral points of both electric and magnetic fields (though not at the same location!).The field structure surrounding such point is characterised by four distinct domains (bundles of the field lines) divided by two separatrices intersecting at the neutral point.Such a topology is known to result from the separator reconnection, a mechanism that has been studied in the framework of resistive magnetohydrodynamics (MHD), see e.g.[43].In our electro-vacuum model, however, it arises entirely from the interaction of the strong gravitational field of the rotating BH with the background magnetic field, i.e., the gravitomagnetic effect.Charged matter injected into the magnetic separator site is prone to the acceleration by the electric field since its motion is not affected by the vanishing magnetic field and thus the acceleration is very effective.
From the astrophysical viewpoint we consider the topological changes that the drift causes upon the field structure, especially the formation of the neutral   points, as our main result demonstrating clearly that the strong gravitation of the rotating Kerr source may itself entangle the uniform magnetic field in a surprisingly complex way.We suggest that the gravity of the rotating black hole could act as a trigger for magnetic reconnection.

Motion of electrically charged matter
We first construct the super-Hamiltonian H [37], where m and q are the rest mass and charge of the test particle, π µ is the generalized (canonical) momentum, g µν is the metric tensor, and A µ denotes the vector potential of the electromagnetic field.The latter is related to the electromagnetic tensor F µν by The Hamiltonian equations are given as where λ = τ /m is the affine parameter, τ denotes the proper time, and p µ is the standard kinematical four-momentum for which the first equation reads  In the left panel we set Ẽ = 1.58 and we observe ordered off-equatorial motion.For the energy of Ẽ = 1.65 cross-equatorial regular motion is observed (middle panel).Finally in the right panel with Ẽ = 1.75 we observe chaotic motion whose trajectory would ergodically fill whole allowed region after the sufficiently long integration time.Spin of the black hole is a = 0.9 M and its event horizon is depicted by the bold line.
p µ = π µ − qA µ .In the case of stationary and axiallysymmetric systems, we identify two constants of motion, namely, the energy E ≡ −π t and angular momentum L ≡ −π ϕ .The above-given equations are employed to obtain particle trajectories.Several numerical integrators were tested for their accuracy confirming the supremacy of the symplectic solver (see fig. 11).
In order to locate off-equatorial orbits we derived effective potential in the following form: where +2g and where we introduce specific quantities L ≡ L/m, Ẽ ≡ E/m, and the specific charge q ≡ q/m.We studied the regular and chaotic motion of electrically charged particles near a magnetized rotating black hole or a compact star (see figure 9).We employed the method of recurrence analysis in the phase space, which allowed us to characterize the chaoticness of the system in a quantitative manner.Unlike the method of Poincaré surfaces, the Recurrence Plots have not yet been widely used to study the chaotic systems in the regime of strong gravity.
The main motivation for these investigations is the question of whether the matter around magnetized compact objects can exhibit chaotic motion, or if instead the system is typically regular.One of the main applications of our considerations concerns the putative envelopes of charged particles enshrouding the central body in a form of a fall-back corona, or plasma coronae extending above the accretion disk [25][26][27].
We concentrate ourselves on the specifications of the RP method in circumstances of a relativistic system, and so the assumed model cannot be considered as any kind of a realistic scheme for a realistic corona.We simply imposed a large-scale ordered magnetic field acting on particles in a combination with strong gravity.
Various aspects of charged particle motion were addressed throughout the Thesis [20], where further references can be found.First of all, we investigated the motion in off-equatorial lobes above the horizon of a rotating black hole (modeled by Kerr metric endowed by Wald's test magnetic field), as well as above the surface of a magnetic star (modeled by the Schwarzschild metric with a rotating dipolar magnetic field).In both cases we conclude that the motion of test particles is regular, which was confirmed for a representative number of orbits across the wide range of parameters over all topological types of off-equatorial potential wells.This result is somewhat unexpected because the off-equatorial orbits require a perturbation to be strong enough (in terms of strength of the electromagnetic field), so that it can balance the vertical component of the gravitational force.
Further, we investigated the response of the particle dynamics when the energy level Ẽ was raised gradually from the potential minimum to values allowing cross-equatorial motion.We examined various topological classes of the effective potential and came to the conclusion that the cross-equatorial orbits are typically chaotic, although very stable regular orbits may also persist for a certain intermediate energy range.The classical work of [13] should be recalled in this context since it also identifies the energy as a trigger for the chaotic motion in the analysed simple system.More recently the Hénon-Heiles system was revisited in the relativistic context by [47].
We also addressed the question of spin dependence of the stability of motion for Kerr black hole in the Wald field.We noticed that this is a rather subtle problem.The effective potential is by itself sensitive to the spin value a -hence, we had to link the potential value roughly linearly with the energy Ẽ to maintain the potential lobe at a given position.In other words, we did not find any clear and unique indication of the spin dependence of the motion chaoticness.Most trajectories exhibited regular behaviour, which is also in agreement with the previous results indicating that motion in off-equatorial lobes is generally regular.On the other hand, in the case of the crossequatorial motion we observed that, for higher spins, more chaotic features come into play when compared with the case of slow rotation.This trend might be also attributed to simultaneous adjustments of Ẽ.In other words, it appears impossible to give an unambiguous conclusion about the spin dependence of the particles dynamics.Instead, one has to deal with a complex, interrelated dependence.PREPRINT vol.
no. / Oblique Magnetic Fields and the Role of Frame Dragging  In the case of a Kerr black hole immersed in a large-scale magnetic field, we observed the effect of confinement of particles regularly oscillating around the equatorial plane.Escape of particles from the plane is allowed for a given range of initial conditions since the equipotentials do not close; they form an endless axial "valley" instead.The escaping trajectories create a narrow, collimated structure parallel to the axis.

Open questions and outlook: characterising possible observational effects of chaotic motion
In the following we present our to-do list comprising of the issues that arose during the previous study of the electromagnetic fields and charged particle dynamics.Most importantly we want to combine two major topics which were presented separately in ref. [20].Namely, we plan to investigate an electrically charge (ionized) particle motion governed by the generalized oblique and drifting EM fields.Besides that we shall go through several rather technical issues related closely to this problem, in particular, the re-V.Karas, O. Kopáček, D. Kunneriath

Acta Polytechnica
PREPRINT currence analysis of chaotic trajectories and the offequatorial orbits around magnetised black holes.

Models of gaseous corona
We shall extend our former axisymmetric model by considering oblique (misaligned with the rotation axis) magnetic fields in which the central body may be uniformly drifting in a general direction [21,22,38].Structure of the electromagnetic field is profoundly enriched and we suppose that similarly the dynamics of the particles will become considerably more complex.We plan to discuss the impact of new parameters upon the off-equatorial stable orbits and investigate how do they affect the dynamic regime of motion.We will try to identify a possible trigger of chaotic dynamics among new parameters.Besides standard methods the recurrence analysis will be employed since it proved useful in our previous work.We also plan to further elaborate ideas concerning the frequency analysis of the off-equatorial orbits.We have seen that fragmented curves we observed in the Poincaré surfaces of section corresponding with the trajectories bound in the closed equatorial lobes may be identified with the Birkhoff chains of islands of stability.Such resonant chain is characterized by a single value of a rotation number which is in principle detectable in terms of spectral analysis of the observed signal.Presence of the Birkhoff chains allows us to discriminate between perturbed and regular system.Moreover the position and the width of the chains reflects other properties of the system.A detailed discussion of this approach applied to the different type of system may be found in [33].However, in our analysis of the off-equatorial trajectories we observed more complicated structures in the surfaces of section which do not allow for the straightforward evaluation of the rotation number, nor the ratio of fundamental frequencies.Therefore we intend to adjust the method for the application to our scenario and infer the possible observational consequences for the system of gaseous corona.
In the integrable system the trajectories in the phase space reside on the surface of tori characterized by the values of integrals of motion which determine the characteristic frequencies of the orbit.Fundamental frequencies in the axisymmetric system (here we do not consider oblique nor drifting fields) are those of radial and latitudinal motion.Once the system is slightly perturbed (by the magnetic field in our model) the tori characterized by the irrational ratio of frequencies survive which is assured by the KAM theorem.
On the other hand, the Poincaré-Birkhoff theorem tells us that resonant tori with the rational frequency ratio must disintegrate into a chain of islands when the system is perturbed.Such resonant chains are in principle detectable in terms of spectral analysis of the observed signal.Presence of the Birkhoff chains allows us to discriminate between perturbed and reg-ular systems.Moreover the position and the width of the chains reflects other properties of the system.

Magnetic shift of the ISCO
The position of the inner edge of the accretion disk is usually identified with the marginally stable geodesic orbit r ms (also referred to as Innermost Stable Circular Orbit -ISCO) whose position is uniquely determined by the value of the black hole spin a [3].Common black hole spin measurement methods are based on this relation as they actually determine r ms to evaluate a [35].In this context we raise the question whether the presence of the magnetic field may change the position of ISCO noticeably.Recently a similar problem was addressed by [1] for the case of Schwarzschild source endowed with the dipole magnetic field.An introductory account of the influence of the uniform magnetic field aligned with the symmetry axis of Kerr black hole was brought by [42].We shall discuss the effect of the oblique uniform magnetic field around Kerr source upon the marginally stable orbit in detail.

Application for Lyapunov spectra
Lyapunov characteristic exponents (LCEs) are the basic indicators of chaos which capture the divergent features of the chaotic orbits straightforwardly.The classical non-covariant definition of LCEs, however, meets difficulties when applied to curved spacetimes [18,32].Recently, [46] proposed a novel geometrical approach to the computation of the Lyapunov spectra which completely avoids the conventional method of solving the variational equations to obtain the Lyapunov vectors which are periodically Gram-Schmidt orthonormalized along the flow.New algorithm is covariantly formulated and thus seems to be highly convenient for the application in general relativistic systems.We plan to implement this method when inspecting the dynamics of charged particles, which should be beneficial to prove the new method fruitful.

Figure 1 .
Figure 1.Tentative motivation to study the effects of black holes embedded within magnetic fields of external origin: Milky Way's central region is penetrated by narrow Non-Thermal Filaments (NTFs) that may suggest the presence of organised magnetic fields.In these regions the strength of the ordered magnetic field can approach ≈ 1 mG.Length of NTFs reaches tens of parsecs.The inner region of Galactic center 0.8 • × 1.0 • is shown at wavelength 90 cm (330 MHz; the image taken by the Very Large Array (VLA); figure credits: [30, 40]).

Figure 3 .
Figure 3. Drift induced effects upon the structure of the aligned magnetic field increase profoundly in the case of extremely rapid motion vx = 0.99c.A neutral (null) point of the magnetic field occurs (left panel).We observe self-similar tightly folded layered structures which are considerably enhanced when compared to the case of slower motion presented in fig. 2 (right panel).

Figure 4 .
Figure 4. Extreme Kerr black hole which rotates along z-axis is immersed in the perpendicular magnetic field Bx > 0. Left panel shows the situation with zero drift speed while in the right panel we observe the impact which the drift motion of the black hole along the z-axis (vz = 0.7c) has upon the field structure.

Figure 5 .
Figure 5. Electric field lines (analogically to the magnetic lines) are expelled from the horizon of the non-drifting (stationary) extreme Kerr black hole in the case of aligned magnetic field (Bx = 0) on the background (left panel).However, for the inclined field (Bx/Bz = 0.3) the structure of the electric field becomes intricate and some field lines penetrate the horizon (right panel).

Figure 6 .
Figure 6.Potential lobes appearing above the horizon of the Kerr black hole (grey surface) immersed in the aligned magnetic field allowing for the off-equatorial motion of charged matter.Two exemplary trajectories of test particles are shown in the left panel -in the left lobe a chaotic orbit, while in the right lobe the regular, purely off-equatorial trajectory which differs from the former one only in the corresponding energy level (other parameters of the motion have been set to the same values).The latter trajectory is shown in the right panel in the stereometric projection.

Figure 7 .
Figure 7.A test particle ( L = 6M , qB0 = M −1 and q Q = 1) is launched from the locus of the off-equatorial potential minima r(0) = 3.68 M , θ(0) = 1.18 with u r (0) = 0 and various values of the energy Ẽ.In the left panel we set Ẽ = 1.58 and we observe ordered off-equatorial motion.For the energy of Ẽ = 1.65 cross-equatorial regular motion is observed (middle panel).Finally in the right panel with Ẽ = 1.75 we observe chaotic motion whose trajectory would ergodically fill whole allowed region after the sufficiently long integration time.Spin of the black hole is a = 0.9 M and its event horizon is depicted by the bold line.

Figure 8 .
Figure 8. Recurrence plots corresponding to the trajectories from fig.7.The regular orbit leads to the simple diagonal pattern (left panel) while the deterministic chaos manifests itself by the complex structure in the recurrence plot (right panel).See ref.[23] for further description of the recurrence analysis of the motion in the context of strong gravitational fields.

Figure 10 .
Figure 10.Recurrence quantification analysis (RQA) provides statistical measures of recurrence plots.Average length of the diagonal line L and the inverse of the longest diagonal line DIV are presented here as a function of the specific energy Ẽ (≡ E/m); 400 distinct trajectories are analysed, which only differ in the energy level, as specified in fig.7.A dramatic change of the behaviour is apparent at Ẽ ≈ 1.685 for both quantities.This allows us to identify very precisely the moment where the chaos sets on.