Homoclinic orbits in Kerr-Newman black holes

Recently successive observations of gravitational waves emitted by the merging of binary systems provide one of the long-awaited confirmations of General Relativity (GR) [1, 2, 3]. The capture of the spectacular images of the supermassive black holes M87* at the center of the M87 galaxy [4] and Sgr A* at the center of our galaxy [5] is also a great achievement that provides direct evidence of the existence of the black holes, the solutions of Einstein’s field equations [6, 7]. Spiraling of stellar mass compact objects into supermassive black holes due to the backreaction of gravitational wave (GW) emission is predicted to be a key source of low frequency gravitational waves, which will be targeted for the planned space-based Laser Interferometer Space Antenna (LISA) [8, 9]. Some of the science goals toward the direct detection of these extreme mass ratio inspirals (EMRIs) are to precisely determine the properties of the EMRIs black hole and its inspiraling components to extract the black hole’s astrophysical environments [10, 11].

The zeroth order approximation in the extreme mass ratio is considered as the small body travels along the geodesic of the background spacetime of the massive black hole. The beginning of the extensive study of the timelike geodesics near the black holes dates back to a remarkable discovery from Carter of the so-called Carter constant [12]. In the family of the Kerr black holes the geodesics of the particle due to the spacetime symmetry of the Kerr family possesses two conserved quantities, the energy $E_{m}$ and the azimuthal angular momentum $L_{m}$ of the particle. Nevertheless, the existence of the third conserved quantity Carter constant renders the geodesic equations as a set of first-order differential equations. Later, the introduction of the Mino time [13] further fully decouples the geodesics equations with the solutions expressed in terms of the elliptical functions. In this work, we would like to particularly focus on the geodesic dynamics in the case of Kerr-Newman black holes. The Kerr-Newman metric of the solution of the Einstein-Maxwell equations represents a generalization of the Kerr metric, and describes spacetime in the exterior of a rotating charged black hole where, in addition to gravitation fields, both electric and magnetic fields exist intrinsically from the black holes. Although one might not expect that astrophysical black holes have a large residue electric charge, some accretion scenarios were proposed to investigate the possibility of spinning charged back holes [14]. Moreover, theoretical considerations, together with recent observations of structures near Sgr A* by the GRAVITY experiment [15], indicate the possible presence of a small electric charge of central supermassive black hole [16, 17]. Thus, it is still of great interest to explore the geodesic dynamics in the Kerr-Newman black hole.

In our previous paper [18], we have studied the null and timelike geodesics of the light and the neutral particles respectively in the exterior of Kerr-Newman black holes. We classify the roots for both angular and radial potentials, and mainly focus on those of the radial potential with an emphasis on the effect from the charge of the black holes. We then obtain the solutions of the trajectories in terms of the elliptical integrals and the Jacobi elliptic functions for the geodesics, which are manifestly real functions of the Mino time that the initial conditions can be explicitly specified. In this paper, we will mainly focus on the work of [19], which has showed the exact Homoclinic solutions on the equatorial plane in the Kerr black holes, and extend the solutions to a general nonequatorial orbit in the Kerr-Newman black hole with an additional charge. By taking the zero charge limit, the obtained solutions can reduce to those of the nonequatorial homonclinic orbits in the Kerr black holes of more relevance to astrophysical black holes. A homoclinic orbit is the separatrix between bound and plunging geodesics, and is an orbit that asymptotes to an energetically bound, unstable spherical orbit. There are several approaches to model the EMRIs through a series of Kerr geodesics [23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. Due to the dissipation effects from the emission of gravitational radiation on EMRIs, the trajectories of the stellar mass compact object will transit from an inspiral to a plunge through a homoclinic orbit. The exact solutions for the homoclinic orbits could be very useful for analytic or numerical studies of the transition from inspiral to plunge [20, 21]. A thorough knowledge of the underlying dynamics of EMRIs becomes essential to shape the gravitational waveform emitted by them. Also, the added noise provided by the random kicks on the homoclinic orbit will expectedly give the chaotic behavior on the particle moving near the horizon [22]. On top of the fantastic observational effects, it is conjectured that there exists the bound of the Lyapunov exponent based upon the AdS/CFT correspondence [33]. The finding of the exact solution of the homoclinic orbits in the Kerr and Kerr-Newman black holes will be the first step to study the potential chaotic motion when the particle travels near the horizon by explicitly computing the Lyapunov exponents to justify or falsify the conjecture. In this paper, we will start from the analytical solutions of the bound motion derived in [18], and find the solutions of the homoclinic orbits where the particle starts off from the position of the largest root of the radial potential and spent tremendous time moving toward the position of the double root of the radial potential. The parameter space of the homoclinic solutions will also be explored.

In Sec. II, we give a brief review of the timelike geodesic equations from which to define the radial and the angular potentials in terms of the Carter constant $C_{m}$ and the azimuthal angular momentum $L_{m}$ of the particle normalized by the energy $E_{m}$. The parameter space to have the general nonequatorial homoclinic solutions is explored. Later the results of the homoclinic orbits are adapted from Appendixes which show the solutions of the $\theta$ and $r$ dependence in general bound motion. We then particularly consider the zero azimuthal angular momentum homoclinic orbits, discussing the frame dragging effects in Sec. III. In Sec. IV the phase difference between the nonequatorial homoclinic orbit and the spherical motion of the unstable orbit is considered. In Sec. V the obtained solutions will reduce to the equatorial orbits to compare with [19] in the Kerr black holes as the charge of the black hole in our case is set to zero. All results will be summarized in the closing section.

In Sec. II we have shown that the Kerr-Newman spacetimes permit homoclinic orbits in a limited region of parameter space $\lambda_{m}$ and $\eta_{m}$. In particular, a particle with zero azimuthal angular momentum, $\lambda^{A}_{m}=0$, but $\eta_{m}^{\rm isso}<\eta^{A}_{m}<\eta_{m}^{\rm ibso}$, can realize homoclinic motion. Using the exact solutions discussed in Sec. II we show in Figs. 5 and 6 two illustrations for this type of motion. The particles share the same set of parameters $\lambda_{m}$ and $\eta_{m}$, but with different initial conditions. With the zero azimuthal angular momentum, the motions show the change of $\phi$ simply due to the frame dragging effects from the black hole spin. Additionally, the charge of the black hole will give another boost to the frame dragging to be seen later.

The analytical solutions with zero azimuthal angular momentum have additional simplifications for the homoclinic orbits. The roots of the angular potential in (57) reduce to

The relevant root is $u_{{m+}}=1$ giving the two turning points at the north and south poles of a sphere, $\theta_{m+}=\cos^{-1}(-\sqrt{u_{m+}})={\pi}$ and $\theta_{m-}=\cos^{-1}(\sqrt{u_{m+}})={0}$, respectively [18]. Substitute (40) into the solution of $\theta(\tau_{m})$ in (60) and notice that $u_{m+}/u_{m-}\ll 1$, since $\eta_{m}$ is relatively large value as seen in Fig.2, also $a$ and $1-\gamma_{m}<1$. We can thus have the approximate solutions given by

By inspection, the angular velocity in Mino time of the evolution around the $\theta$ can be read off as

In Fig. 5, we choose the initial conditions $\theta_{i}=\frac{\pi}{2}$ and $\nu_{\theta_{i}}=1$, departing from the equator. Thus, with ${\mathcal{G}_{\theta_{i}}}=0$ in (59), the solution just becomes $\theta(\tau_{m})=\cos^{-1}\left(-{\rm sin}\left(\sqrt{\eta_{m}}\tau_{m}\right)\right)$. And, in Fig 6 the particle departs from the north pole with $\theta_{i}=0$ and $\nu_{\theta_{i}}=1$, giving ${\mathcal{G}_{\theta_{i}}}=\frac{\pi}{2\sqrt{\eta_{m}}}$. The evolution of coordinate $\theta$ follows $\theta(\tau_{m})=\cos^{-1}\left({\rm cos}\left(\sqrt{\eta_{m}}\tau_{m}\right)\right)$.

As for the azimuthal motion, we notice that the equation of motion in (9) reduces to

simply due to the frame dragging. In particular, the angular speed goes to a constant as $r\rightarrow r_{\rm mss}$ when $\tau_{m}\rightarrow\infty$ ($t\rightarrow\infty$), given by

which can also be obtained directly from the analytical formulas (16) and (33) in the limits of $\lambda_{m}=0$ and $\tau_{m}\rightarrow\infty$ ($t\rightarrow\infty$). This allows one to plot the ratio of $\omega_{\phi}$ to $\omega_{\theta}$ shown in Fig. 7. It is expected that the larger value of the black hole spin $a$ gives relatively large dragging effects and moreover, the charge $Q$ of the black hole gives the boost to it. This is consistent with the conclusion in the study of the particle boomerang in [18]. In there, we consider the spherical orbits of the particle with the radius of $r_{\rm mss}$ of the double root of the radial potential. For sufficiently large value of black hole spin $a$ there can be a particle boomerang that leaves the north polar axis ($\theta=0$) and travels along a constant-$r$ orbit to return to the north polar axis in precisely the opposite direction to that at which it leaves. The dragging of inertial frames rotates the particle propagation direction by an angle $\Delta\phi=\pi$ during one orbit that returns to the same location on the north polar axis. However, in the presence of the charge of the black hole in the Kerr-Newman black holes, the finite charge effect will enhance the frame dragging so as to reduce the needed value of the spin a to sustain $\Delta\phi=\pi$.

According to [19], when the homoclinic orbits on the equatorial plane in Kerr black holes are considered, it is found a nontrivial fact that although as $r$ approaches to the unstable orbit, the accumulated phase $\phi$ and time spent go to infinity. Nevertheless, the phase difference between the homoclinic orbits and the circular motion of the unstable orbits with the same radius of the double root turns out to be finite at time $t\rightarrow\infty$, when both start at the same initial time. This will be also true for the homoclinic solution in the nonequatorial plane in the Kerr-Newman black holes where the circular motion is replaced by the spherical motion. As $\tau_{m}\rightarrow\infty$, giving $t\rightarrow\infty$ and $r\rightarrow r_{\rm mass}$, the radius of the unstable orbit, with the associated values of $\lambda_{\rm mass},\eta_{\rm mass}$ obtained from (21) and (22), the phase difference becomes

The general solutions of the homoclinic orbits and the spherical orbit contain the radial $r$ and angular $\theta$ dependence seen from Eqs.(16) and (19). The $\theta$ dependence cancels in computing $\Delta\phi_{\tau_{m}}^{\rm diff}$ and the only contributions come from the radial $r$ dependence. To obtain (IV), we have used the $r$ dependent solution of $\phi$ (33) to the homoclinic orbits. Also, the radius of the spherical orbit is set at $r=r_{\rm mass}$ with the $r$ dependent solution obtained from (9) in terms of the Mino time in (14). For $r_{\rm mss}$ lying between $r_{\rm isso}$ and $r_{\rm ibco}$ determined by (24) and (25), the corresponding $\lambda_{\rm isso},\eta_{\rm isso}$ and $\lambda_{\rm ibso},\eta_{\rm ibso}$ are determined by (21) and (22).

In Fig. 8, the phase difference is plotted as a function of $\beta_{\lambda_{\rm mass}}$ defined by $\beta_{\lambda_{\rm mass}}=\frac{{|\lambda_{\rm mass}}|-|\lambda_{\rm isco}|}{|\lambda_{\rm ibco}|-|\lambda_{\rm isco}|}$ for a fixed $\eta_{m}$, which is the same value of $\eta_{m}$ at B in Fig. 2 allowing $\lambda_{m}$ to vary. Since $|\lambda_{\rm isco}|\leq|\lambda_{\rm mass}|\leq|\lambda_{\rm ibco}|$, $0\leq\beta_{\lambda_{\rm mass}}\leq 1$. It is evident that the finite charge of the black hole will enhance the phase difference for both the direct motion ($\lambda_{\rm mass}>0$) and retrograde motion ($\lambda_{\rm mass}<0$). Also, for a fixed $Q$, $\Delta\phi^{\rm diff}$ increases with the spin parameter $a$ of the black hole. This means that the homoclinic orbit moving toward the unstable orbit can synchronize with the circular motion in terms of the Mino time but in the end with a phase difference. This may provide useful information to construct the full spectrum of the gravitation waves in the case of EMRIs and is one of the interesting applications from the analytical solutions we obtain in this paper.

This section is devoted to the equatorial homoclinic solution, exemplified by the case C in Fig. 2 of the parameter space $\lambda_{m}$ and $\eta_{m}$. The main purpose here is to reduce our exact formulas for nonequatorial orbits to the expressions obtained in the context of the equatorial case. We will show some important steps to find their consistency. Since the results of [19] are written as a function $r$, we begin with a useful relation between Mino time $\tau_{m}$ and $r$ given by the inverse of (31),

For the proper time ${\sigma_{m}(r)}$, following (7) and (14) with $\Sigma=r^{2}$ at the equatorial plane we then have

The term of $I^{H}_{2}$ given in (II.3) can be expressed by $r$ using (46). Rearranging them leads to

Plugging in $x^{H}(r)$ and $h$ from (38) and (39), and doing a little algebra we arrive at

This is the Kerr-Newman’s counterpart of equation (B66) in Ref. [19].

Taking the similar steps allows to obtain the equatorial homoclinic orbit in $\phi(r)$ coordinate. With the help of (9) evaluated at the equatorial plane and (14), we first rewrite

where ${I_{\phi}}$ is given by (33) and $\tau_{m}$ by (46). All the integrals for calculating ${\phi}(r)$ are available from (II.3). The result can be written as follows

where the coefficients $\mathcal{A}$ and $\mathcal{B_{\pm}}$ are $r$ independent quantities involving the black hole parameters and the constants of motion of the particle, reading as