1. Introduction
The Kerr metric that describes a rotating black hole is a solution to Einstein’s field equations of general relativity. The observed event-horizon-scale images of the supermassive black hole candidate in the center of the giant elliptical galaxy M87 are consistent with the dark shadow of a Kerr black hole predicted by general relativity [
1]. The motion of a particle in the vicinity of a Kerr black hole is integrable because of the existence of four conserved quantities, namely the energy, angular momentum, rest mass, and azimuthal motion, of the particle. The azimuthal motion corresponds to the Carter constant [
2], which is obtained from the separation of variables in the Hamilton–Jacobi equation.
Observational evidence demonstrates the existence of strong magnetic fields in the vicinity of the supermassive black hole at the center of the Galaxy [
3]. The external magnetic fields, which can be considered as tidal environments, are generally believed to play a crucial role in the transfer of the energy from the accretion disk to jets. Radiation reactions, depending on the external magnetic field strength, cause the accretion of charged particles from the accretion disk to shift towards the black hole. An inductive charge introduced by Wald [
4] generates an induced electric field due to a contribution to the Faraday induction from the parallel orientation of the spin of the black hole and the magnetic field. When the inductive charge takes the Wald charge, the potential difference between the horizon of the black hole and infinity vanishes, and the process of selective accretion is completed [
5,
6]. The effects of the magnetic fields involving the induced electric field are so weak in comparison to the gravitational mass effects that they do not change the spacetime metrics. However, they can essentially affect the motion of charged test particles in accreting matter if the ratio of the electric charge and mass of the particle is large. In most cases, the fourth invariable quantity related to the azimuthal motion of the particles is absent when the external electromagnetic fields are considered near the black hole. Thus, the dynamics of charged test particles in the black holes with external electromagnetic fields is nonintegrable.
Although the magnetic fields in the vicinity of the black holes destroy the integrability of these spacetimes in many problems, the radial motions of the charged particles on the equatorial plane are still integrable and solvable. It is mainly studied by means of an effective potential. The effective potential seems simple, but it describes many important properties of the spacetimes. In particular, unstable circular orbits, stable circular orbits, and innermost stable circular orbits (ISCOs) on the equatorial plane are clearly shown through the effective potential. It is interesting to study these equatorial orbits in the theory of accretion disks. An accreted material with sufficient angular momentum relative to an axisymmetric massive central body will be still attracted by the central body, but such force will be compensated due to the large angular momentum. This easily forms an accretion disk. However, accreted material without sufficient angular momentum will fall into the central body [
7,
8,
9]. Electromagnetic fields could influence dynamics of charged particles in accreting matter; therefore, the ISCOs in the field of a magnetized black hole shift towards the horizon for a suitable spin direction. In other words, the inner boundary of the accretion disk goes towards the central body. In view of the importance of the topic on the effective potential and stable circular orbits on the equatorial plane, the topic has been taken into account in a large number of literature studies [
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28]. These problems discussed in the existing works are based on the equatorial plane. In some extended theories of gravity, such as Brans–Dicke gravity, scale-dependent gravity, and asymptotically safe gravity in the context of black hole physics [
29,
30,
31,
32,
33,
34,
35], the effective potentials, unstable circular orbits, stable circular orbits, and ISCOs on a plane slightly different from the equatorial can be discussed similarly.
When the external magnetic fields destroy the spacetime’s symmetry (precisely speaking, the external magnetic fields lead to the absence of the fourth constant related to the particles’ azimuthal motion), the generic motion of charged particles on the non-equatorial plane can be chaotic in some circumstances. If the external magnetic fields do not destroy the symmetry, no chaotic dynamics is possible. For example, charged particle motions in the Kerr–Newman black hole spacetime are regular and nonchaotic because of the existence of four integrals leading to the integrability of the system [
36]. Chaos describes a dynamical-system-sensitive dependence on initial conditions. The theory of chaotic scattering in the combined effective potential of the black hole and the asymptotically uniform magnetic field is useful to explore the mechanism hidden behind the charged particle ejection [
5]. The energy of the charged particle in such combined fields is split into one energy mode along the magnetic field line’s direction and another energy mode at the perpendicular direction. The chaotic charged particle dynamics in the combined gravitational and magnetic fields leads to an energy interchange between the two energy modes of the charged particle dynamics. As a result, it can provide sufficient energy to ultra-relativistic motion of the charged particle along the magnetic field lines. Based on the importance of studies of the chaotic motion in the gravitational field of a black hole combined with an external electromagnetic field, many authors [
5,
6,
12,
20,
23,
37,
38,
39,
40,
41,
42,
43,
44,
45,
46] are interested in this field.
The detection of the chaotic behavior requires the adopted computational scheme with reliable results. Without a doubt, higher-order numerical integrators such as an eighth- and ninth-order Runge–Kutta–Fehlberg integrator with adaptive step sizes can yield high-precision numerical solutions. However, they are more computationally demanding than lower-order solvers. For Hamiltonian systems, the most appropriate solvers are symplectic integrators, which respect the symplectic nature of Hamiltonian dynamics and show no secular drift in energy errors [
47,
48,
49,
50,
51,
52,
53]. A symplectic integrator method for numerical calculation of charged particle trajectory is well known due to its small error in energy even for long integration times, which make it perfectly suited for the description of regular and chaotic dynamics through Poincaré sections calculations [
54]. Because the variables are inseparable in Hamiltonian systems associated with curved spacetimes, the standard explicit symplectic integrators do not work when these Hamiltonian systems are separated into two parts. In this case, completely implicit symplectic methods including the implicit midpoint method [
55,
56] and Gauss–Runge–Kutta methods [
41,
54,
57,
58] are often considered. Explicit and implicit combined symplectic methods [
59,
60,
61,
62,
63] take less cost than these completely implicit methods and then are also used. Recently, explicit symplectic integrators were proposed for nonrotating black holes when the Hamiltonians of these black holes have several splitting parts with analytical solutions as explicit functions of proper time [
64,
65,
66]. With the aid of time transformations, explicit symplectic integrators are easily available for the Kerr-type spacetimes [
67].
The authors of [
54] employed the Gauss–Legendre symplectic solver (i.e., s-stage implicit symplectic Runge–Kutta method) to study the regular and chaotic dynamics of charged particles around the Kerr background endowed with an axisymmetric electromagnetic test field with the aid of Poincaré sections. The authors of [
68] applied the time-transformed explicit symplectic integrators introduced in [
67] to mainly explore the effect of the black hole spin on the chaotic motion of a charged particle around the Kerr black hole immersed in an external electromagnetic field. Unlike Reference [
68], the present work particularly focuses on how a small change in the black hole’s snductive charge [
6] exerts influences on the effective potential, stable circular orbits and ISCOs on the equatorial plane, and a transition from order to chaos of orbits on the non-equatorial plane. The effects of other dynamical parameters such as the magnetic field parameter are also considered. For this purpose, we introduce a dynamical model for the description of charged particles moving around the Kerr black hole immersed in an external magnetic field in
Section 2. The effective potential, stable circular orbits, and ISCOs on the equatorial plane are discussed in
Section 3. The explicit symplectic integrators are designed for this problem, and the dependence of the orbital dynamical behavior on the parameters is shown in
Section 4. Finally, the main results are summarized in
Section 5.
2. Kerr Black Hole Immersed in External Magnetic Field
The Kerr black hole is the description of a rotating black hole with mass
M and angular momentum
a. In the standard Boyer–Lindquist coordinates
, its time-like metric is written as
, that is,
These nonzero components in this metric are found in the paper of [
69] as follows:
where
,
and
.
and
t are proper and coordinate times, respectively.
c is the speed of light, and
G denotes the gravitational constant.
Suppose the Kerr black hole is immersed in an external asymptotically uniform magnetic field, which has strength
B and yields an induced charge
Q. Set
and
as time-like and space-like axial Killing vectors. An electromagnetic four-vector potential can be found in References [
6,
70] and is written as
This potential has two nonzero covariant components
When
, the inductive charge is the Wald charge
, and
is a magnetic field corresponding to the Wald charge [
4]. The induced charge like the Wald charge
is so small that it has no contribution to the background geometry of the black hole [
71]. However, the induced charge can exert an important influence on the motion of a charged particle under some circumstances, as will be shown in later discussions.
The motion of the particles around the rotating black hole embedded in the external magnetic field is described by the Hamiltonian
where
and
are generalized momenta, and
is a function of
r and
[
68]:
Here,
,
, and
are functions of
r and
as follows:
is a constant energy of the particle, and
is a constant angular momentum of the particle.
and
are generalized momenta, which satisfy the relations
Because the 4-velocity
is always identical to the constant
, the Hamiltonian (5) remains invariant and obeys the constraint
In fact, this third invariable quantity corresponds to the rest mass of the particle.
For simplicity, c and G take geometrized units: . Dimensionless operations to the Hamiltonian (5) are carried out thorough a series of scale transformations: , , , , , , , , , and . Note that no scale transformation is given to the inductive charge Q. When these treatments are employed, M and m in all the above-mentioned expressions are eliminated or taken as geometrized units: . The horizon event of the black hole exists for . For convenience, we take and .
3. Effective Potential and Stable Circular Orbits
Apart from the three integrals (10)–(12) in the dimensionless Hamiltonian (5), the fourth constant related to the particles’ azimuthal motion is absent in general when the external magnetic field forces are included. The absence of the fourth constant is mainly caused by the
term in Equation (
4) rather than the
term in Equation (
3). Because
is only a function of
r, it does not destroy the presence of the fourth constant. However,
is a function of
r and
, and therefore the Hamilton–Jacobi equation of Equation (
5) has no separable form of variables
r and
. This leads to the absence of the fourth constant. Of course, the
terms, being functions of
r and
in Equations (3) and (4), also have some contributions to the absence of the fourth constant. In other words, the main contribution to the absence of the fourth constant in the (5) comes from the external magnetic fields associated with
. The inductive charges associated with
also exert some influences on the absence of the fourth constant. Thus, the dimensionless Hamiltonian (5) is non-integrable. However, it can be integrable for some particular cases. For instance, radial motions of charged particles on the equatorial plane
are integrable. The radial motions are described in terms of effective potential
V, i.e., the expression of
E obtained from Equations (5) and (12) with
:
where
A,
B and
C are expressed as
The local minimal values of the effective potential correspond to stable circular orbits, which satisfy the relation
and the following conditions
When the equality sign (=) is taken in Equation (
15), the innermost stable circular orbit (ISCO) is present.
Taking parameters
,
, and
(if
and
, then
is the Wald charge), we plot the effective potentials for several different magnetic parameters
in
Figure 1. When the magnetic parameter
increases, the left shape of the effective potential goes away from the black hole, and the shape of the effective potential is not altered. The energies of the unstable or stable circular orbits become smaller. That is to say, the effective potential for a larger value of
is below that for a smaller value of
. However, the radii of the stable circular orbits in
Table 1 get larger as
increases.
An increase in the inductive charge parameter
does not alter the shape of the effective potential but makes the left shape of the effective potential go away from the black hole in
Figure 2. Meanwhile, the energies of the unstable or stable circular orbits decrease, but the radii of the stable circular orbits increase in
Table 2.
Figure 3 clearly describes the dependence of the effective potential on the black hole’s spin
a. The energies of the stable circular orbits increase when
a gets larger. The radii of the stable circular orbits always increase (see also
Table 3).
How does the effective potential vary as the particle’s angular momentum
L increases? The effective potential for a larger value of
L is always over that for a smaller value of
L, as shown in the Kerr spacetime of
Figure 4b,c. Note that there are critical values of
L corresponding to the ISCOs colored red in
Table 3, such as
for the Schwarzschild spacetime with
. When the angular momenta
L are larger than the critical values, the stable circular orbits are present in
Table 3. However, no stable circular orbits exist for
L less than the critical values. As
a or
L increases, the radii of the stable circular orbits also increase.
Although the radii of the stable circular orbits increase with an increase in
a, the radii of the ISCOs become smaller, as shown in
Table 3. In addition, the radii of the ISCOs depend on the sign of the particle’s angular momentum as well as the magnitude of the particle’s angular momentum. When
(for this case, the spin direction of the black hole is consistent with the particle’s angular momentum), the considered orbits are called direct orbits. When
(for this case, the spin direction of the black hole is opposite to the particle’s angular momentum), the considered orbits are called retrograde orbits [
38]. Given parameters
a,
and
, the radii of the ISCOs for the retrograde orbits are larger than those for the direct orbits. If any one of the parameters
a,
, and
increases, the radii of the ISCOs for the retrograde orbits or the direct orbits decrease. More details on the ISCOs are listed in
Table 4,
Table 5 and
Table 6.
5. Conclusions
In this paper, we mainly focused on studying the dynamics of charged particles around the Kerr black hole immersed in an external electromagnetic field, which can be considered as a tidal environment.
First, we discussed the radial motions of charged particles on the equatorial plane through the effective potential. We traced how the dynamical parameters exert influences on the effective potential. It was found that the particle energies at the local maxima values of the effective potentials increase with an increase in the black hole spin and the particle angular momentum, whereas they decrease as one of the inductive charge parameter and magnetic field parameter increases. In addition, the radii of stable circular orbits on the equatorial plane always increase. However, the radii of ISCOs decrease as any one of the black hole spin , inductive charge parameter , and uniform magnetic field parameter is increasing.
Then, we investigated the motions of charged particles on the non-equatorial plane using a time-transformed explicit symplectic integrator. The effects of small variations of the parameters on the orbital regular and chaotic dynamics are studied through the techniques of Poincaré sections and fast Lyapunov indicators. As a result, the dynamics have sensitive dependence on a small variation in any one of the inductive charge parameter, magnetic field parameter, energy, and angular momentum. Chaos is easily induced as the inductive charge parameter, magnetic field parameterm and energy increase but weakened as the angular momentum increases. When the dragging effects of the spacetime increase, the chaotic properties are not always enhanced or weakened under some circumstances.
The theoretical work may have potential astrophysical applications. The unstable or stable circular orbits and ISCOs would be helpful to study some accretion disks. The theory of chaotic scattering in the combined effective potential and the asymptotically uniform magnetic field would be applicable to explaining the mechanism hidden behind the charged particle ejection. The existence of the magnetic fields involving the induced electric field may be demonstrated through observational evidence.