Chaos of Charged Particles in Quadrupole Magnetic Fields Under Schwarzschild Backgrounds

Abstract

A four-vector potential of an external test electromagnetic field in a Schwarzschild background is described in terms of a combination of dipole and quadrupole magnetic fields. This combination is an interior solution of the source-free Maxwell equations. Such external test magnetic fields cause the dynamics of charged particles around the black hole to be nonintegrable, and are mainly responsible for chaotic dynamics of charged particles. In addition to the external magnetic fields, some circumstances should be required for the onset of chaos. The effect of the magnetic fields on chaos is shown clearly through an explicit symplectic integrator and a fast Lyapunov indicator. The inclusion of the quadrupole magnetic fields easily induces chaos, compared with that of the dipole magnetic fields. This result is because the Lorentz forces from the quadrupole magnetic fields are larger than those from the dipole magnetic fields. In addition, the Lorentz forces act as attractive forces, which are helpful for bringing the occurrence of chaos in the nonintegrable case.

1. Introduction

Many observational evidences have supported the presence of magnetic fields around astrophysical black holes. For example, a strong magnetic field exists in the vicinity of the supermassive black hole at the center of the Galaxy [1]. Recently, the existence of a highly regular, strong magnetic field in the vicinity of the central, supermassive black hole of 3C 84 was shown in Ref. [2]. These magnetic fields are due to the dynamics of ionized matter and plasma in accretion disks surrounding the black holes. They might be helpful for transferring the energies from the accretion discs to relativistic jets.

An exterior asymptotically uniform test magnetic field around a black hole with mass M does not change the gravitational background when its strength is much smaller than the value ${10}^{19}{M}_{\odot }/M$ Gauss [3], where $M_{\odot }$ is the mass of the Sun. This means that the metric tensor of the black hole geometry does not have any modification. In other words, such a relatively weak test magnetic field leads to a negligible effect on the motion of neutral particles around the black hole. However, it would exert a very large influence on the motion of a charged test particle near the black hole. In fact, the relative Lorenz force is still large if the ratio of the particle’s charge to the particle’s mass is large. The motion of charged test particles should be integrable in a gravitational field like one of the Schwarzschild black hole, the Reissner–Nordström one, the Kerr one, and the Kerr–Newman one, but is most likely nonintegrable in combined black hole gravitational fields and electromagnetic fields. There have been a large variety of papers claiming the onset of chaotic dynamics of charged particles around some black holes immersed in external magnetic fields (e.g., [4,5,6,7,8,9,10,11]). The chaotic dynamics close to the black hole horizons is useful for providing a mechanism for the transmutational energy interchange, which causes charged particle acceleration to create relativistic jets [8].

The external test electromagnetic fields around the black holes are solutions of the vacuum Maxwell equations. One path for the obtainment of the solutions is from linear combinations of the spacetime Killing vectors as vector potentials. This refers to the so-called Wald’s solutions [12], which require that the black holes should be stationary, axisymmetric, uncharged, and vacuum. However, linear combinations of all Killing vectors with constant coefficients, as uniform magnetic fields surrounding charged or nonvacuum black holes (of modified gravity), would fail to satisfy the source-less Maxwell field equations. In this case, the Wald’s solutions should be generalized by the coefficients taken as functions of the coordinates [13]. These generalized vector potentials are expressed in powers of the magnetic field and the rotation parameter.

A black hole itself can only have a monopole electromagnetic field in general. The Wald’s solutions are only special solutions of the vacuum Maxwell equations. In fact, the structures of magnetic fields from white dwarfs, neutron stars, black hole magnetospheres, or external current loops of charged matter like accretion disks around magnetic compact stars are described by general multipolar solutions of the vacuum Maxwell equations in curved spacetime backgrounds [14,15,16,17,18]. A class of general multipolar solutions are exterior solutions, which are expressed in terms of an infinite series on the reciprocal of the radial distance [14,15,16,17,18,19]. They vanish at infinity and are based on magnetic coupling to accretion discs in relation to dynamo action, jets, and hydromagnetic winds. The multipole moments of the currents circulating in the discs at larger distances from the horizons of the holes are described via these exterior solutions. Another class of general multipolar solutions are interior solutions, which are expressed as polynomials of the radial distance. They do not vanish at infinity and should be considered in finite domains of the radial distances in the inner regions of the magnetic fields [14,15,16,17,18,19]. They are in relation to magnetohydrodynamic models of jet formation and collimation in active galactic nuclei and microquasars. They describe the contribution of the currents circulating at the inner edge of the accretion discs at slightly larger distances from the horizons of the holes. Either the exterior solutions or the interior ones are combinations of different multipole magnetic fields.

The innermost stable circular orbits, quasi-harmonic oscillatory motions, and chaotic orbits of charged particles around (general relativity or modified gravity) black holes, immersed in the dipole magnetic field as an interior solution of the vacuum Maxwell equations, are widely taken into account in the literature, e.g., [4,5,6,7,8,9,10,11,20,21,22,23,24,25,26]. Nevertheless, the dynamics of charged particles near the Schwarzschild black hole with the quadrupole magnetic field as an interior solution of the vacuum Maxwell equations has been seldom noticed. In this paper, we mainly focus on the motion of charged particles in the combined gravitational field of the Schwarzschild black hole and the interior solution with a combination of dipole and quadrupole magnetic fields. In particular, the effect of the quadrupole magnetic field on the charged particle motion is compared with that of the dipole magnetic field on the charged particle motion.

The rest of this paper is organized as follows. In Section 2, two types of electromagnetic four-potentials in the Schwarzschild black hole background are introduced. In Section 3, we are interested in the motion of charged particles in the combined gravitational and electromagnetic background. Finally, the main results are summarized in Section 4.

2. Magnetic Fields in Schwarzschild Geometries

When the speed of light and the constant of gravity are taken as geometric units $c=G=1$, a Schwarzschild black hole with mass M is described in standard Schwarzschild–Droste coordinates $x^{\alpha }=(t,r,\theta ,\varphi )$ by the following line element:

$$\begin{array}{c}\hfill d{s}^2=g_{\alpha \beta }d{x}^{\alpha }d{x}^{\beta },\end{array}$$

(1)

where the four nonzero metric components are

$$\begin{array}{ccc}\hfill g_{tt}& =& {\displaystyle \frac{1}{g^{tt}}}=-\left(1-{\displaystyle \frac{2M}{r}}\right),\hfill \\ \hfill g_{rr}& =& {\displaystyle \frac{1}{g^{rr}}}={\left(1-{\displaystyle \frac{2M}{r}}\right)}^{-1},\hfill \\ \hfill g_{\theta \theta }& =& {\displaystyle \frac{1}{g^{\theta \theta }}}=r^2,\hfill \\ \hfill g_{\varphi \varphi }& =& {\displaystyle \frac{1}{g^{\varphi \varphi }}}=r^2{sin}^2\theta .\hfill \end{array}$$

This spacetime is static, axisymmetric, and asymptotically flat.

Assume that an electromagnetic field exists in the vicinity of the black hole that is too small to affect the spacetime geometry. This field should be a solution of the source-free Maxwell equations [16]:

$$\begin{array}{}\mathrm{(2)}& \begin{array}{cc}\hfill 0=& F_{\alpha \beta ;\gamma }+F_{\gamma \alpha ;\beta }+F_{\beta \gamma ;\alpha },\hfill \end{array}\mathrm{(3)}& \begin{array}{cc}\hfill 0=& J^{\mu }={\displaystyle \frac{1}{4\pi }}{F}_{;\nu }^{\nu \mu }\hfill \\ \hfill =& {\displaystyle \frac{1}{4\pi \sqrt{-g}}}{\displaystyle \frac{\partial }{\partial x^{\nu }}}\left(\sqrt{-g}{F}_{;\nu }^{\nu \mu }\right)\hfill \\ \hfill =& {\displaystyle \frac{1}{4\pi \sqrt{-g}}}{\displaystyle \frac{\partial }{\partial x^{\nu }}}\left(\sqrt{-g}{g}^{\mu \alpha }{g}^{\nu \beta }{F}_{\alpha \beta }\right).\hfill \end{array}\end{array}$$

Here, $J^{\mu }$ denotes a current, $g=det\left(g_{\alpha \beta }\right)=-r^4{sin}^2\theta$, and $F_{\mu \nu }=A_{\nu ,\mu }-A_{\mu ,\nu }$ is a tensor of the electromagnetic field. Note that $A_{\nu ,\mu }=\partial A_{\nu }/\partial x^{\mu }$, where $A_{\mu }$ is an electromagnetic four-vector potential. For the case of axial symmetry, the potential has only one nonzero component $A_{\varphi }$ as a function of $(r,\theta )$. Thus, the electromagnetic tensor exists with two nonzero components:

$$\begin{array}{c}\hfill F_{r\varphi }=A_{\varphi ,r},F_{\theta \varphi }=A_{\varphi ,\theta }.\end{array}$$

(4)

The equation $J^{\varphi }=0$ of Equation (3) is rewritten as

$$\begin{array}{ccc}& & {\left(\sqrt{-g}{g}^{rr}{g}^{\varphi \varphi }{F}_{r\varphi }\right)}_{,r}+{\left(\sqrt{-g}{g}^{\theta \theta }{g}^{\varphi \varphi }{F}_{\theta \varphi }\right)}_{,\theta }\hfill \\ & & =4\pi \sqrt{-g}{J}^{\varphi }=0,\hfill \end{array}$$

(5)

that is,

$$\begin{array}{ccc}& & r^2{\left[\left(1-{\displaystyle \frac{2M}{r}}\right)A_{\varphi ,r}\right]}_{,r}+sin\theta {\left({\displaystyle \frac{A_{\varphi ,\theta }}{sin\theta }}\right)}_{,\theta }\hfill \\ & & =-4\pi r^4{J}^{\varphi }{sin}^2\theta =0.\hfill \end{array}$$

(6)

This equation is separable and has a series solution [16]:

$$\begin{array}{c}\hfill A_{\varphi }(r,\theta )=\sum _{l=0}^{\infty }{\Re }_l\left(r\right){\mathrm{\Phi }}_l\left(\cos \theta \right),\end{array}$$

(7)

which satisfies two separable equations,

$$\begin{array}{ccc}\hfill \mathrm{\Lambda }{\Re }_l& =& r^2{\displaystyle \frac{d}{dr}}\left[\left(1-{\displaystyle \frac{2M}{r}}\right){\displaystyle \frac{d{\Re }_l}{dr}}\right],\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill \mathrm{\Lambda }{\mathrm{\Phi }}_l& =& -(1-u^2){\displaystyle \frac{d^2{\mathrm{\Phi }}_l}{d{u}^2}}.\hfill \end{array}$$

(9)

Note that $\mathrm{\Lambda }=l(l+1)$ and $u=cos\theta$.

Based on the Legendre polynomials $P_l\left(u\right)$, the angular functions ${\mathrm{\Phi }}_l$ are determined by ${\mathrm{\Phi }}_0\left(u\right)=0$ and ${\mathrm{\Phi }}_l\left(u\right)=l(P_{l-1}\left(u\right)-u{P}_l\left(u\right))$$(l=1,2,\dots )$. The radial functions ${\Re }_l\left(r\right)$ are an infinite series of $1/r$ [16]:

$$\begin{array}{c}\hfill f_l\left(r\right)=\sum _{n=l}^{\infty }{c}_n^{\left(l\right)}{\left({\displaystyle \frac{2M}{r}}\right)}^n,\end{array}$$

(10)

where $c_n^{\left(l\right)}$ ($n>l$) denotes a set of coefficients, expressed in terms of free parameters $c_l^{\left(l\right)}$ [18]. It is clear that $f_l\left(r\right)\to 0$ as $r\to \infty$. Equation (7) with Equation (10) is a class of exterior solutions of Equation (6) [19].

The radial functions are also expressed in terms of polynomials with respect to r as

$$\begin{array}{c}\hfill g_l\left(r\right)=\sum _{n=2}^{l+1}{a}_n^{\left(l\right)}{\left({\displaystyle \frac{r}{2M}}\right)}^n,\end{array}$$

(11)

where the coefficients $a_n^{\left(l\right)}$ are written in [18] as

$$\begin{array}{c}\hfill a_{2+k}^{\left(l\right)}=\prod _{m=1}^k\left({\displaystyle \frac{m(m+1)-l(l+1)}{m(m+2)}}\right)a_2^{\left(l\right)}.\end{array}$$

(12)

Note that $k=1,\dots ,l-1$ are considered. Equation (7) with Equation (11) is a class of interior solutions of Equation (6) [19]. For $l=1$, Equation (7) with Equation (11) corresponds to the magnetic field potential

$$A_{\varphi 1}=g_1\left(r\right){\mathrm{\Phi }}_1\left(u\right)=a_2^{\left(1\right)}{\left({\displaystyle \frac{r}{2M}}\right)}^2{sin}^2\theta .$$

(13)

When $a_2^{\left(1\right)}=2B{M}^2$, the potential is

$$\begin{array}{c}\hfill A_{\varphi 1}={\displaystyle \frac{1}{2}}B{r}^2{sin}^2\theta ,\end{array}$$

(14)

where B represents the strength of the magnetic field. The result is consistent with the Wald potential [12], which arises from the space-like Killing vector ${\xi }_{\left(\varphi \right)}^{\alpha }=(0,0,0,1)$ through the relation $A^{\alpha }=(B/2){\xi }_{\left(\varphi \right)}^{\alpha }$. Then, the potential of Equation (14), as an interior solution of Equation (6), represents a uniform magnetic field. It is labelled as Magnetic Field 1 (MF1), and is related to the dipole magnetic field [16,18,19]. For $l=2$, Equation (7) with Equation (11) stands for the potential from the sum of the uniform magnetic field and the quadrupole one:

$$\begin{array}{ccc}\hfill A_{\varphi 2}& =& g_1\left(r\right){\mathrm{\Phi }}_1\left(u\right)+g_2\left(r\right){\mathrm{\Phi }}_2\left(u\right)\hfill \\ & =& a_2^{\left(1\right)}{\left({\displaystyle \frac{r}{2M}}\right)}^2{\sin }^2\theta +3a_2^{\left(2\right)}{\left({\displaystyle \frac{r}{2M}}\right)}^2\hfill \\ & & ·\left(1-{\displaystyle \frac{2r}{3M}}\right)\cos \theta {\sin }^2\theta .\hfill \end{array}$$

(15)

The term $g_2\left(r\right){\mathrm{\Phi }}_2\left(u\right)$ corresponds to a quadrupole magnetic field as an interior solution of Equation (6). Given $a_2^{\left(2\right)}=a_2^{\left(1\right)}=2B{M}^2$ (this choice is considered because the potential of Equation (13) in the case of $a_2^{\left(1\right)}=2B{M}^2$ is the Wald potential), the potential is of the expression

$$\begin{array}{c}\hfill A_{\varphi 2}={\displaystyle \frac{1}{2}}B{r}^2{\sin }^2\theta \left[1+\left(3-{\displaystyle \frac{2r}{M}}\right)cos\theta \right].\end{array}$$

(16)

Such a combination of the two magnetic fields is also an interior solution of Equation (6), marked as MF2. It is clear that the potentials $A_{\varphi 1}$ and $A_{\varphi 2}$ do not tend to zero as r gets sufficiently large. They should be limited to a finite interval of r.

Why are the two kinds of potentials $A_{\varphi 1}$ and $A_{\varphi 2}$ considered in this paper? There are several reasons. Although the functions $f_l\left(r\right)$ and $g_l\left(r\right)$ have been given in [18], one cannot clearly know what physical meanings these coefficients $c_n^{\left(l\right)}$ and $a_n^{\left(l\right)}$ have. How to choose these coefficients is almost unclear as well. Only the choice $a_2^{\left(1\right)}=2B{M}^2$ of Equation (13) has appeared in many references, such as [5,6,7,8,9,10,20,21,22,23,24,25,26]. It was also reported in [18] that circular equatorial orbits always exist in the odd-multipole magnetic fields given by Equations (7) and (10). However, the coefficients $c_n^{\left(l\right)}$ associated with the magnetic field strength B should be extremely large so that the functions $f_l\left(r\right)$ have no negligible contributions for the radial distance r larger than the black hole horizon $R_h$. Such extremely strong magnetic fields would be very likely to lead to changes in the spacetime geometries. On the other hand, the coefficients $a_n^{\left(l\right)}$ associated with the magnetic field strength B should be small enough because $r^{l+1}$ becomes extremely large for $l\ge 3$ and $r\gg R_h$. The role of such extremely small coefficients $a_n^{\left(l\right)}$ makes the functions $g_l\left(r\right)$ have somewhat important contributions.

The nonzero components of the magnetic fields described by the two kinds of potentials are written in [18] as

$$\begin{array}{c}\hfill B_r={\displaystyle \frac{F_{\theta \varphi }}{r^2\sin \theta }},B_{\theta }=-{\displaystyle \frac{F_{r\varphi }}{\sin \theta }}\left(1-{\displaystyle \frac{2M}{r}}\right).\end{array}$$

(17)

For $A_{\varphi 1}$, the two nonzero components of the magnetic field are

$$\begin{array}{c}\hfill B_{r1}=B_z\cos \theta ,B_{\theta 1}=-B_z(r-2M)\sin \theta ,\end{array}$$

(18)

where $B_z=B$ denotes the constant magnetic field strength along the positive z-axis. The potential $A_{\varphi 2}$ corresponds to the two nonzero components of the magnetic field:

$$\begin{array}{ccc}\hfill B_{r2}& =& B_z\left[\cos \theta +\left({\displaystyle \frac{3}{2}}-{\displaystyle \frac{r}{M}}\right)\left(3{\cos }^2\theta -1\right)\right],\hfill \end{array}$$

(19)

$$\begin{array}{ccc}\hfill B_{\theta 2}& =& -B_z(r-2M)\sin \theta \left[1+3\left(1-{\displaystyle \frac{r}{M}}\right)\cos \theta \right].\hfill \end{array}$$

(20)

Although the weak external magnetic fields exert negligible influences on the spacetime geometries, they play an important role in the motion of charged particles near the Schwarzschild black holes.

3. Dynamics of Charged Particles in Combination of Dipole and Quadrupole Magnetic Fields

The motion of charged particles around the Schwarzschild black holes in the combination of dipole and quadrupole magnetic fields is mainly focused on here. For comparison, the dynamics of charged particles in the dipole magnetic field is also considered.

3.1. Dynamical Equations and Numerical Schemes

The motion equations of a charged test particle with charge q and mass $m_p$ are governed in terms of the Hamiltonian system

$$\begin{array}{ccc}\hfill H& =& {\displaystyle \frac{1}{2m_p}}{g}^{\mu \nu }(p_{\mu }-q{A}_{\mu })(p_{\nu }-q{A}_{\nu })\hfill \\ & =& {\displaystyle \frac{1}{2m_p}}[g^{tt}{p}_t^2+g^{rr}{p}_r^2+g^{\theta \theta }{p}_{\theta \theta }^2\hfill \\ & & +g^{\varphi \varphi }{(p_{\varphi }-q{A}_{\varphi })}^2].\hfill \end{array}$$

(21)

For MF1, $A_{\varphi }=A_{\varphi 1}$. For MF2, $A_{\varphi }=A_{\varphi 2}$. Because this Hamiltonian does not explicitly depend on the coordinates t and $\varphi$, two momentum components $p_t$ and $p_{\varphi }$ are constant. They are expressed as

$$\begin{array}{ccc}\hfill p_t& =& m_p{g}_{tt}\dot{t}=-E,\hfill \end{array}$$

(22)

$$\begin{array}{ccc}\hfill p_{\varphi }& =& m_p{g}_{\varphi \varphi }\dot{\varphi }+q{A}_{\varphi }=L,\hfill \end{array}$$

(23)

where E and L are the energy and angular momentum of the particle, respectively. In this case, the Hamiltonian is a system of two degrees of freedom:

$$\begin{array}{ccc}\hfill H& =& {\displaystyle \frac{1}{2m_p{r}^2{\sin }^2\theta }}{(L-q{A}_{\varphi })}^2-{\displaystyle \frac{1}{2m_p}}{(1-{\displaystyle \frac{2}{r}})}^{-1}{E}^2\hfill \\ & & +{\displaystyle \frac{1}{2m_p}}(1-{\displaystyle \frac{2}{r}})p_r^2+{\displaystyle \frac{p_{\theta }^2}{2m_p{r}^2}}.\hfill \end{array}$$

(24)

The rest mass in the timelike spacetime corresponds to a conserved Hamiltonian quantity of the form

$$H=-{\displaystyle \frac{m_p}{2}}.$$

(25)

To simplify the related expressions, we apply dimensionless operations to the Hamiltonian (24) via a series of scale transformations: $r\to rM$, $\tau \to M\tau$, $t\to Mt$, $E\to m_{p}E$, $p_r\to m_p{p}_r$, $L\to m_{p}ML$, $p_{\theta }\to m_{p}M{p}_{\theta }$, $q\to m_{p}q$, $B\to B/M$, and $H\to m_{p}H$. Note that $\tau$ is the proper time. Hereafter, $\beta =Bq$ is used. In this way, the mass factors M and $m_p$ are absent in all the above-mentioned expressions, and the Hamiltonian (24) becomes dimensionless. Of course, the correspondences between the dimensionless qualities and the practical physical qualities can also be seen from these scale transformations. For instance, the dimensionless angular momentum L corresponds to the practical angular momentum $m_{p}ML$. When the dimensionless strength of the magnetic field is B, the practical one should be $B/M$.

Because of the inclusion of the external magnetic fields in the dimensionless Hamiltonian (24), a fourth constant of motion no longer exists. This Hamiltonian is not integrable. Numerical integration schemes are a good technique for studying these Hamiltonian dynamics.

The dimensionless Hamiltonian can be split into four explicitly integrable parts for the presence of the external magnetic fields, as it can for the absence of the external magnetic fields in [27]. Hence, a second-order explicit symplectic integrator $S_2$ and a fourth-order one $S_4$ [28], which preserve the symplectic structure of the Hamiltonian, are easily available. See the paper given in [27] for more details on the construction of these explicit symplectic methods in the Schwarzschild spacetime.

Figure 1 tests the numerical performance of the two algorithms by means of three orbits in the two magnetic fields. Here, both symplectic algorithms $S_2$ and $S_4$ use a fixed time step $h=\Delta \tau$, where $h=1$. The parameters are taken as $E=0.995$ and $L=4.5$. Orbits 1 and 2 have their initial conditions $p_r=0$ and $\theta =\pi /2$. In MF1, $\beta =1.8\times {10}^{-3}$, and the initial separations are $r=50$ for Orbit 1 and $r=11$ for Orbit 2. The starting values of $p_{\theta }>0$ in the two orbits are solved from the theoretical result $\Delta H=H+1/2=0$. However, ΔH ≠ 0 from the numerical viewpoint, as shown in Figure 1a,b. The method S₂ shows no secular growth in the errors ΔH for both orbits. This is attributed to a key characteristic of a symplectic integrator. The secular growth of the errors is still absent for the algorithm S₄ acting on Orbit 2, but does not appear until Orbit 1 is integrated for a long enough time by S₄. There are no typical differences in the accuracies of ΔH between the two orbits for each of the algorithms. S₄ performs about three orders of magnitude better in the accuracy than S₂. Figure 1c,d shows that the two algorithms have similar performance for Orbits 2 and 3 in MF2, where the magnetic field parameter is β = 3.1 × 10⁻⁶ and the initial separation is r = 90 for Orbit 3.

A notable point is that the error curves have more dramatic changes in Figure 1a,c than those in Figure 1b,d. This phenomenon implies whether Orbit 1 in MF1 and Orbit 2 in MF2 are possibly different from Orbit 2 in MF1 and Orbit 3 in MF2 in the orbital dynamical behavior. In what follows, we shall answer this question using $S_4$ rather than $S_2$ due to the preference of $S_4$ over $S_2$ in the accuracy.

3.2. Chaotic Dynamics of Charged Particles in the Two Electromagnetic Potentials

Based on the analysis of Poincaré sections, the dynamical nature of the orbits in Figure 1 can be determined clearly. Orbit 2 in MF1 of Figure 2a and Orbit 3 in MF2 of Figure 2d are regular because the intersection points (r,p_r) of the orbits intersected with the plane $\theta =\pi /2$ and $p_{\theta }>0$ form continuous smooth closed curves. However, Orbit 1 in MF1 of Figure 2a and Orbit 2 in MF2 of Figure 2d exhibit chaotic dynamics because the intersection points randomly fill two-dimensional regions in the phase space $r-p_r$.

The dynamical nature is also shown through the largest Lyapunov exponents (LEs), which measure the diverge or converge rate between nearby trajectories in phase space. They are defined in [29] by

$$\lambda =\underset{\tau \to \infty }{lim}{\displaystyle \frac{1}{\tau }}ln{\displaystyle \frac{d\left(\tau \right)}{d_0}},$$

(26)

where $d\left(\tau \right)$ and $d_0$ are the proper distances between two nearby orbits at the time $\tau$ and the starting time, respectively. Positive values of $\lambda$ correspond to the exponential divergence between nearby trajectories and sensitive dependence on initial conditions. They indicate the chaotic dynamics of bounded Orbit 1 in MF1 of Figure 2b and Orbit 2 in MF2 of Figure 2e. Nevertheless, zero values of the LEs indicate a power-law divergence between nearby trajectories and show the regular nature of Orbit 2 in MF1 of Figure 2b and Orbit 3 in MF2 of Figure 2e.

It is worth emphasizing that the true values of the LEs are from the limit values of $\lambda$ as $\tau \to \infty$. Although the integration times have no way to tend to infinity in practical computations, they are still required to be long enough, e.g., $\tau ={10}^8$. In this case, it takes more CPU time to carry out this task. Fast Lyapunov indicators (FLIs) [30] are quicker at distinguishing between regular and chaotic dynamics than LEs. They can also be calculated in terms of the distances between two nearby orbits [29] by

$$FLI={log}_{10}{\displaystyle \frac{d\left(\tau \right)}{d_0}}.$$

(27)

When the integration time is equal to $\tau ={10}^6$, the FLIs can clearly describe the regularity of Orbit 2 in MF1 of Figure 2c and Orbit 3 in MF2 of Figure 2f through the FLIs that increase algebraically slowly with the time ${log}_{10}\tau$. The chaoticity of bounded Orbit 1 in MF1 of Figure 2c and Orbit 2 in MF2 of Figure 2f is also shown by the FLIs that increase exponentially with time.

Since the technique of FLIs is faster for finding chaos than that of LEs, it is mainly employed to survey the effect of a small change of one or two parameters on a transition from regular dynamics to chaotic dynamics. The magnetic field parameter $\beta =0.00185$ and the initial separation $r=50$ are considered. Taking the angular momentum $L=4.5$, we have the correspondence of the energy E and the FLI in Figure 3a, where E is varied from 0.990 to 1.00 in an interval of 0.0001. Each of the FLIs is not obtained until $\tau ={10}^6$. All values of the energy E with $FLIs<5$ admit the regular dynamics, whereas those with $FLIs\ge 5$ indicate the chaotic dynamics. As $E>0.993$ increases, chaos is most likely to happen and is strengthened in MF1. However, the dependence of the FLI on the angular momentum L in Figure 3b shows that the possibility for the occurrence of chaos is large for a small value of L in MF1. These results are also supported by the dependence of the FLI on the two parameter spaces (E,L) in Figure 3c. On the other hand, strong chaos is always existent for the considered parameters in MF2 of Figure 3d–f. This extent of chaos seems to be independent of the values of the angular momentum L in MF2. However, the occurrence of chaos becomes difficult when the angular momentum L increases in MF1. This fact sufficiently shows that the inclusion of quadrupole magnetic fields dramatically enhances the extent of chaos of charged particles.

The possibility for chaos would increase in MF1 when both the magnetic parameter $\beta$ and the energy E increase, as shown via the FLIs corresponding to the two parameters in Figure 4a. However, there is a great strong chaotic region in the two-parameter space $(E,\beta )$ for the case of MF2 in Figure 4b. The onset of strong chaos is not typically affected, regardless of whether the energy E and the magnetic parameter $\beta$ (larger than a certain value) are large or small. On the other hand, chaos gets easier in MF1 of Figure 5a with the increase of $\beta$ or the decrease of L. Chaos is almost allowed for the parameters L and $\beta$ obtained in MF2 of Figure 5b. The chaotic dynamics do not depend on the values of the magnetic parameter $\beta$ (larger than a certain value) and the angular momentum L in MF2. It is worth pointing out that chaos is impossible in Figure 4 and Figure 5 if the magnetic parameter $\beta$ is too small. This point is clearly shown from the integrable dynamics in the case of $\beta =0$.

In a word, the possibility and strength for chaos are larger for MF2 with the combination of dipole and quadrupole magnetic fields than those for MF1 with the dipole magnetic fields under some appropriate circumstances. This result is because the contribution of the quadrupole magnetic fields to the charged particle motions is larger than that of the dipole magnetic fields. This point can be seen clearly from Equation (16), where $2r-3\gg 1$ for $r\gg 2$ and $\theta =\pi /2$. Namely, the quadrupole term plays a dominant role in Equation (15) or (16), compared with the dipole one. In other words, the results of Figure 3d–f, Figure 4b, and Figure 5b have no typical differences between the exclusion of the dipole term and the inclusion of the dipole term in Equation (15) or (16). In addition, both the Lorentz forces from the dipole magnetic fields and those from the quadrupole magnetic fields act as attractive forces. Similarly, the energy E acting on the Newtonian gravitational potential of the black hole also brings an attractive force contribution to the particles. These attractive forces are helpful for inducing chaos under appropriate conditions from a statistical viewpoint. However, the angular momentum L gives rise to a repulsive force contribution. Consequently, its increase would weaken the degree of chaos.

4. Summary

The four-vector potentials of electromagnetic fields in the Schwarzschild background, as solutions of the source-free Maxwell equations, have two kinds of expressional forms. One kind of expression is exterior solutions expressed as an infinite series on the reciprocal of the radial distance, and another kind of expression is interior solutions expressed as polynomials of the radial distance. For convenience of practical computations, the latter kind of four-vector potential with a combination of dipole and quadrupole magnetic fields in a finite interval of the radial distance is considered an interior solution of the source-free Maxwell equations in this paper.

The motion of charged particles in the combination of dipole and quadrupole magnetic fields is nonintegrable. Explicit symplectic integrators that conserve the phase flow of the Hamiltonian system are very suitable for studying the long-term evolution of charged particle motions. One of the integrators is applied to investigate the effects of the magnetic fields on chaotic dynamics of charged particles with the fast Lyapunov indicator. Compared with the inclusion of the dipole magnetic fields, the quadrupole magnetic fields easily induce chaos under some circumstances. This is because the external test magnetic fields cause the dynamics of charged particles to be nonintegrable and are mainly responsible for chaos of charged particles. In addition, the quadrupole magnetic fields result in a greater contribution to the charged particle motions than the dipole magnetic fields. In fact, the Lorentz forces from the quadrupole magnetic fields as well as those from the dipole magnetic fields act as attractive forces, which are helpful for encouraging the occurrence of chaos in the nonintegrable case.