Chaotic motion of scalar particle coupling to Chern-Simons invariant in Kerr black hole spacetime

We present firstly the equation of motion for the test scalar particle coupling to the Chern-Simons invariant in Kerr black hole spacetime through the short-wave approximation. We have analyzed the dynamical behaviors of the test coupled particles by applying techniques including Poinca\'e sections, power spectrum, fast Lyapunov exponent indicator and bifurcation diagram. It is shown that there exists chaotic phenomenon in the motion of scalar particle interacted with the Chern-Simons invariant in a Kerr black hole spacetime. With the increase of the coupling strength, the motion of the coupled particles for the chosen parameters first undergoes a series of transitions betweens chaotic motion and regular motion and then falls into horizon or escapes to spatial infinity. Thus, the coupling between scalar particle and Chern-Simons invariant yields the richer dynamical behavior of scalar particle in a Kerr black hole spacetime.


I. INTRODUCTION
Chaos is a kind of very complex motions with high sensitivity to initial conditions, which occurs in a definiteness system with nonlinear interactions. One of the most important feature of chaos is that the tiny errors in the chaotic motion grow at an exponential rate, which leads to that the motion differs totally from what it would be without these errors [1][2][3][4]. Thus, it is very difficult to make a long-term prediction to the motion of a chaotic dynamical system. This implies that chaotic systems own many novel properties not shared by the linear dynamical systems due to the nonlinear interactions, which triggers a lot of attention to study chaotic dynamics in various physical fields.
The Chern-Simons-modified gravity was firstly investigated in the non-dynamical formulation, where the Chern-Simons scalar field is not a dynamical field, but only an a priori prescribed function. In the nondynamical Chern-Simons theory, it is found that a valid solution of spacetime exists only if the Pontryagin density vanishes. Moreover, there also exist other theoretical problems [31][32][33]. Thus, it is quite artificial and is always treated only as a toy model used to obtain some insight in parity-violating theories of gravity. Based on this consideration, a lot of attention has been focused on the dynamical Chern-Simons modified gravity where the scalar field is assumed to own its kinetic term and dynamical evolution equation [31,34,35]. It is of interest to study the properties of black holes in the dynamical Chern-Simons modified gravity. In general, it is difficult to obtain an analytical black hole solution in this modified gravity because the Einstein's field equation and the dynamical equation of scalar field must be satisfied simultaneously. Thus, the analytical solution of a rotating in dynamical Chern-Simons gravity have been obtained only in the small-coupling and/or slow-rotation limit [31,[35][36][37][38][39]. The observable effects of small Chern-Simons coupling parameter in such black hole spacetimes have been studied including black hole shadow [40] and strong gravitational lensing [41]. Without any perturbational expansion, a numerical solution of a rotating black hole is obtained in [42], which is helpful to understand some important features emerging in the fast spinning and/or large coupling regimes in dynamical Chern-Simons modified gravity. Moreover, the stability of black holes under gravitational perturbations have also been studied in dynamical Chern-Simons modified gravity [43][44][45].
Most of above literature on dynamical Chern-Simons gravity focus mainly on the linear coupling case where the coupling term with Chern-Simons invariant is proportional to the scalar field. Inspired by the phenomenon of spontaneous scalarization recently discussed in the quadratic scalar-Gauss-Bonnet gravity [46][47][48][49], Gao et al [50] generalized it to the dynamical Chern-Simons modified gravity and considered the case where the Chern-Simons invariant is coupled to the quadratic function of the dynamical scalar field. And then they studied the scalar perturbation around a Kerr black hole and found that the black hole becomes unstable under linear perturbations in a certain region of the parameter space. In this paper, we want to study the effects of such quadratic coupling on the motion of a test scalar particle in a Kerr black hole background. In order to reach this purpose, we here assume the dynamical Chern-Simons scalar field as a perturbational field and then obtain the corrected Klein-Gordorn equation for the coupled scalar field. With the shortwave approximation, we can get the equation of motion of the coupled scalar particle from the above corrected Klein-Gordorn equation. The presence of the quadratic coupling between the scalar field and the Chern-Simons invariant results in that the equation of motion for the coupled scalar particle is not variable-separable, which means that the chaotic behavior could appear in the motion of the scalar particle. It is of interest to study how such coupling affects chaotic behaviors of scalar particles in black hole spacetimes.
The paper is organized as follows. In Sect. II, we obtain the geodesic equation of a test scalar particle coupled to the Chern-Simons invariant in the Kerr black hole spacetime by the short-wave approximation. In Sect. III, we investigate the chaotic phenomenon in the motion of the scalar particle coupled to the Chern-Simons invariant by techniques including the fast Lyapunov indicator, power spectrum, Poincaré section and bifurcation diagram. We probe the effects of this coupling together with black hole rotation parameter on the chaotic behavior of a coupled scalar particle. Finally, we end the paper with a summary.

II. GEODESICS OF SCALAR PARTICLE COUPLING TO CHERN-SIMONS INVARIANT IN KERR BLACK HOLE SPACETIME
In this section, we will derive the equations of motion for a test scalar particle coupling to Chern-Simons invariant in the Kerr black hole spacetime. The simplest action containing the quadratic-scalar-Chern-Simons invariant coupling term can be expressed as [50] with the Chern-Simons invariant * RR = 1 2 where Φ is a scalar field with mass µ and the parameter α is a coupling constant with dimension of length squared. The tensors ǫ αβγδ and R µ νγδ are, respectively, the well-known Levi-Civita tensor and Riemann curvature tensor. In order to study the motion of a test scalar particle in a background spacetime, one can treat scalar field as a perturbation field and the corresponding scalar particle as a test particle. After this operation, one can find the coupling between scalar field and Chern-Simons invariant does not modify the background spacetime. One of solutions for the action (1) is Kerr metric, whose line element can be written in Boyer-Lindquist coordinates as where M , a denote the mass and the rotation parameter of black hole, respectively. The Chern-Simons invariant for Kerr black hole is * RR = 96aM 2 r cos θ 3r 2 − a 2 cos 2 θ r 2 − 3a 2 cos 2 θ Varying the action (1) with respect to Φ , one can find that the modified wave equation for scalar field becomes It is obvious that the curvature correction acts as an effective mass. With the short-wave approximation, one can get equation of the motion of a test scalar particle from the above corrected Klein-Gordon equation (6).
In this approximation, the wavelength of the scalar particle is much smaller than the typical curvature scale so that the wave aspect of scalar particle can be neglected. The scalar field Φ can be simplified as with a real and slowly varying amplitude f and a rapidly varying phase S. And then the derivative term f ;µ can be neglected because it is not dominated in this case. The wave vector, k µ = ∂ µ S, can be regarded as the usual momentum p µ of scalar particle. Thus, with the help of Eq. (7), the corrected Klein-Gordon equation (6) can be written as Here, we set the mass of the scalar particle µ = 1. The corresponding Hamiltonian for the test scalar particle becomes where x µ is spacetime coordinate and p µ is its corresponding canonical momentum. Comparing with the Hamiltonian of a free test particle, one find that there exits an extra potential due to the interaction with the Chern-Simons invariant. Thus, Hamilton canonical equations for the coupled particle can be expressed as Since g µν and * RR in Hamiltonian (9) are only functions of the coordinates r and θ, there exists two cyclic coordinates t and φ. It means that there exist two conserved quantities, i.e., the energy E and the z-component of the angular momentum L of the timelike particle, which is similar to that in the non-coupling case. With these two conserved quantities, the geodesic equation for the scalar particle coupling to Chern-Simons invariant in Kerr black hole spacetime can be written aṡ with a constraint condition As the Chern-Simons coupling vanishes, these equations reduce to those of the usual timelike particles in the Kerr black hole spacetime. In the non-zero coupling case, the complicated Chern-Simons invariant (5) results in that the equation (14) is not variable-separable, which differs from that in the case without the Chern-Simons coupling. Thus, chaotic behavior could appear in motion of a scalar particle due to the interaction with Chern-Simons invariant. In the next section, we will investigate the effect of the coupling parameter α on motion of a coupled scalar particle in a Kerr black hole background.

III. CHAOTIC MOTION OF SCALAR PARTICLES COUPLING TO CHERN-SIMONS INVARIANT IN KERR BLACK HOLE SPACETIME
In order to probe motion of scalar particle coupling to Chern-Simons invariant in Kerr black hole spacetime, we must solve numerically differential equations (11)- (13). Here, we adopt to the corrected fifth-order Runge-Kutta method [51][52][53][54], where the high precision can be effectively ensured by correcting the velocities (ṙ,θ) at every integration step so that the numerical deviation is pulled back in a least-squares shortest path.
In Fig.1 components in the solution increase with α. Thus, the presence of the coupling makes the motion of particle more complicated. It is also confirmed by the phase curve in the (r,ṙ) plane of the phase space, which is shown in Fig.2. The phase path in the case with α = 0 is only a closed curve and then the corresponding solution is periodic, which means that the particle moves along the regular periodic orbit around the black hole for the choose initial condition. With the increase of α, we find that the degree of disorder and non-integrability of the motion of particle increases because the phase path becomes more complex and the region fulled by the path is enlarged. Especially, as α = 44, 46, 47, the complicated phase curves indicate that the motion are chaotic for the coupled scalar particle under the interaction with the Chern-Simons invariant. In Fig.3, we also present the power spectrum for the signals with different α plotted in figure (1). As α = 0, we find that the spectrum is discrete because that there are three peaks appeared at the positions with the frequency f = f 0 , 2f 0 and 3f 0 , respectively, where f 0 is the lowest frequency and its numerical value is 0.0048. With the increase of α, the frequency components in the spectrum increase gradually. As the coupling parameter α increases to 44, the distinct continuous spectrum appears and the corresponding motion of the coupling particle is chaotic. As α = 47, we find that the range of continuous spectrum widens and the strength of chaotic motion enhances.
The Fast Lyapunov indicators (FLI) is a kind of fast and effective tools to discern chaotic orbits of particles [55][56][57][58]. In the curved spacetime, the FLI with two particles method can be described by [59][60][61] where d(τ ) = |g µν ∆x µ ∆x ν |, ∆x µ denotes the deviation vector between two nearby trajectories at proper time τ . The sequential number of renormalization k is used to avoid the numerical saturation arising from the fast separation of the two adjacent orbits. It is well known that FLI(τ ) grows with exponential rate for chaotic motion, even for weak chaotic motion, but it grows algebraically with time for the regular resonant orbit and for the periodic one. In Fig.4, we present FLI(τ ) for the signals plotted in Fig.(1), which tells us that in the cases with α = 0, 15, 30, the FLI(τ ) increases linearly with τ and then the motions of the scalar particle are regular. However, in the cases with α = 44, 46, 47, the FLI(τ ) grows with exponential rate and then the corresponding motions are chaotic. It is consistent with the previous analysis.
Poincaré section is another effective method to discern chaotic motion. It is an intersection of particle's trajectory and a given hypersurface which is transversal to the trajectory in the phase space. According to the intersection point distribution in Poincaré section, the motions of particles are classified as three kinds for a dynamical system. The periodic motions and the quasi-periodic motions correspond to a finite number of points and a series of close curves in the Poincaré section, respectively. The chaotic motion solutions correspond to strange patterns of dispersed points with complex boundaries [52]. In Fig.(5), we show the change of the Poincaré sections (θ = π 2 ) with different Chern-Simons coupling parameter α for the solutions which is plotted in Fig. (1). We find that for α < 43 the phase path of the coupled scalar particle is a quasi- destroyed again and many discrete points appear. Obviously, the trajectory of the coupled particle is chaotic and the dynamical system in this case becomes non-integrable. As α = 45 ∼ 47, one can find that the tori is completely vanished and the pattern is composed of discrete points, which means that the chaotic behavior of the scalar particle becomes stronger under the coupling with the Chern-Simons invariant.
In Figs. (6) and (7), we present Poincaré section (r,ṙ) for the motion of the coupled particle with different initial conditions in the background of Kerr black hole spacetime. We find that for the fixed rotation parameter a = 0.8, the chaotic region first increases and then decreases with the Chern-Simons coupling parameter α.
We also note that in the case with α = 0, there are a series of close curves in the Poincaré section and then there exists only regular orbits for the scalar particle in the Kerr black hole spacetime, which agrees with the previous discussion. For the fixed coupling parameter α = 46, we find that the chaotic region first increases and then decreases with the increase of the rotation parameter a. Especially, as a = 0, all of the orbits of particles is regular, which can be explained by a fact that the Chern-Simons invariant * RR disappears and the motion of particle reduces to that of in the usual Schwarzschild black hole spacetime. Moreover, it is shown clearly in the Poincaré section that the numbers and positions of fixed points of the system depend on the rotation parameter a and the Chern-Simons coupling parameter α.
The bifurcation diagram can show how the motion of the scalar particle depend on the rotation parameter a of black hole and the Chern-Simons coupling parameter α. In figures (8) and (9) there is only a periodic solution and no bifurcation for the dynamical system, which means that the motions of the scalar particles are regular in these two cases. In the rotating black hole case, we find that the presence of the periodic, chaotic and escape motions of particles is dominated by the Chern-Simons coupling parameter α and the rotation parameter a. With the increase of the parameters α and a, the motion of the coupled scalar particle transforms among the single-periodic, multi-periodic and chaotic motions. From Fig.(8), one can find that with the increase of the rotation parameter a, the range of α chaotic motion appeared first increases and then decrease, and finally increases. Fig.(9) shows that the lower bound a of emerging chaotic orbits decreases with the Chern-Simons coupling parameter α. These results indicate that the effects of the Chern-Simons coupling parameter and the rotation parameter on the motion of the coupled scalar particle are very complex, which are typical features of bifurcation diagram for the usual chaotic dynamical system.
Thus, the dynamical behaviors of scalar particle, under the interaction with Chern-Simons invariant, becomes much richer in a Kerr black hole spacetime.

IV. SUMMARY
In this paper we present firstly the equation of motion for a test scalar particle coupled to the Chern-Simons invariant in the Kerr black hole spacetime through the short-wave approximation. We have analyzed the dynamical behaviors of the test coupled particles by applying techniques including Poincaé sections, power spectrum, fast Lyapunov exponent indicator and bifurcation diagram. Our results confirm that there exists chaotic phenomenon in the motion of scalar particle interacted with the Chern-Simons invariant in the background of a rotating black hole. The main reason is that the presence of the coupling leads to that the equation of motion is not variable-separable and the dynamical system is non-integrable. Moreover, we probe effects of the coupling parameter and black hole rotation parameter on chaotic behavior of a test coupled scalar particle. Our results show that the dependence of particle's motion on the coupling parameter α is determined by the initial conditions and the parameters of system. For the fixed rotation parameter a = 0.8, we find that the chaotic region in the Poincaré section first increases and then decreases with the Chern-Simons coupling parameter α. For the fixed coupling parameter α = 46, we find that the chaotic region first increases and then decreases with the black hole rotation parameter a. Moreover, as α = 0 or a = 0, we find that all of orbits of particles become regular, which is because the interaction with Chern-Simons invariant * RR disappears and then the motion of particle reduces to that of in the usual Kerr black hole spacetime. Thus, the coupling