Inspiral and Plunging Orbits in Kerr-Newman Spacetimes

Recent detections of gravitational waves emitted from the merger of binary systems have confirmed Einstein’s century-old prediction as a consequence of general relativity [1, 2, 3]. The capture of the spectacular images of supermassive black holes M87* at the center of M87 galaxy [4] and Sgr A* at the center of our galaxy [5] leads to another scientific achievement of a direct evidence of the existence of black holes. The black hole is one of the mysterious stellar objects, and it is the solution derived from the Einstein’s field equations [6, 7]. In astrophysics, extreme mass-ratio inspirals (EMRIs), which consist of a stellar mass object orbiting around a massive black hole, have recently received considerable attention. The goal is to analyze gravitational wave signals to accurately test the predictions of general relativity in its strong regime. Gravitational wave signals generated through EMRIs, which are key sources of low frequency gravitational waves and to be observed in the planned space-based Laser Interferometer Space Antenna (LISA), provides an opportunity to measure various fascinating properties of supermassive black holes [8, 9, 10, 11, 12].

The present work is motivated by EMRIs, which can be approximated as a light body travels along the geodesic of the background spacetime of a massive black hole. In particular, the recent studies in [13, 14] have been devoted to inspirals of the particle on the equatorial plane asymptotically from the innermost stable circular orbits (ISCO) of Kerr black holes. They also derive a simple expression for the equatorial radial flow from the ISCO relevant to the dynamics of the accretion disk. These exact solutions may have applications to the generated gravitational waveforms arising from EMRIs [10] as well as constructing theories of black hole accretion [15, 16, 18, 17, 19]. Additionally, the work by [20, 21] broadens the investigation of equatorial motion to encompass generic non-equatorial orbits in Kerr black holes. In the family of the Kerr black holes due to the underlying spacetime symmetry the dynamics 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, discovered in the sixties and known nowadays as Carter constant, renders the geodesic equations as a set of first-order differential equations [22]. Later, the introduction of the Mino time [23] further fully decouples the equations of motions with the solutions given in terms of the elliptical functions [24, 25]. In our previous paper [26], we have studied the null and time-like geodesics of the light and the neutral particles respectively in the exterior of Kerr-Newman black holes. We then obtain the solutions of the trajectories written in terms of the elliptical integrals and the Jacobi elliptic functions [27], in which the orbits are manifestly real functions of the Mino time and also the initial conditions can be explicitly specified [28]. In this work, we will mainly focus on the infalling particles into the Kerr-Newman black holes in general nonequatorial motion. Theoretical considerations, together with recent observations of structures near Sgr A* by the GRAVITY experiment [29], indicate possible presence of a small electric charge of central supermassive black hole [30, 31]. Thus, it is of great interest to explore the geodesic dynamics in the Kerr-Newman black hole [32].

Layout of the paper is as follows. In Sec. II, a mini review of the time-like geodesic equations is provided with three conserved quantities of a particle, the energy, azimuthal angular momentum, and Carter constant. The equations of motion can be recast into integral forms involving two effective potentials. In particular, the roots of the radial potential determine the different types of the infalling trajectories of particles to black holes [33]. Sec. III focuses on the parameter regime of the conserved quantities to have triple roots giving the radius of the innermost stable spherical orbits (ISSO) [34]. The analytical solutions of the infalling orbits are derived for the case that the particle starts from the coordinate $r$ slightly less than that of the ISSO. The two additional infalling orbits of interest are determined by the roots of the radial potential, consisting of a pair of complex roots and two real roots. In Sec. IV, we consider bound motion with one of the real roots inside the horizon and the other outside the horizon. In this case, the particle motion is bound by the turning point from the real root outside the horizon. In Sec. V we will consider unbound motion, in which both real roots are inside the event horizon. The exact solutions for plunging trajectories are given and illustrative examples for such trajectories are plotted. In VI the conclusions are drawn. For the clarity of notation and the completeness of the paper, Appendixes A and B provide some of relevant formulas derived in the earlier publications [26, 36].

We start from a summary of the equations of motion for the particle in the Kerr-Newman black hole exterior. We work with the Boyer-Lindquist coordinates $(t,r,\theta,\phi)$. The spacetime of the exterior of the Kerr-Newman black hole with the gravitational mass $M$, angular momentum $J$, and angular momentum per unit mass $a=J/M$ is described by the metric

where $\Sigma=r^{2}+a^{2}\cos^{2}\theta$ and $\Delta=r^{2}-2Mr+a^{2}+Q^{2}$. The roots of $\Delta(r)=0$ determine outer/inner event horizons $r_{+}/r_{-}$ as

We assume that $0<a^{2}+Q^{2}\leq M^{2}$ throughout the paper.

For the asymptotically flat, stationary, and axial-symmetric black holes, the metric is independent of $t$ and $\phi$. Thus, the conserved quantities are energy $E_{m}$ and azimuthal angular momentum $L_{m}$ of the particle along a geodesic. These can be constructed through the four momentum $p^{\mu}=mu^{\mu}=m\,dx^{\mu}/d\sigma_{m}$, defined in terms of the proper time $\sigma_{m}$ and the mass of the particle $m$ as

Additionally, another conserved quantity is the Carter constant explicitly obtained by

Together with the time-like geodesics, $u^{\mu}u_{\mu}=m^{2}$, one obtains the equations of motion

where we have normalized $E_{m}$, $L_{m}$, and $C_{m}$ by the mass of the particle $m$

The symbols $\pm_{r}={\rm sign}\left(u^{r}\right)$ and $\pm_{\theta}={\rm sign}\left(u^{\theta}\right)$ are defined by the 4-velocity of the particle. Moreover, the radial potential $R_{m}({r})$ in (6) and and angular potential $\Theta_{m}(\theta)$ in (7) are obtained as

As well known [23], the set of equations of motion (6)-(9) can be fully decoupled by introducing the so-called Mino time $\tau_{m}$ defined as

For the source point $x_{i}^{\mu}$ and observer point $x^{\mu}$, the integral forms of the equations above can be rewritten as [28]

where the integrals $I_{mr}$, $I_{m\phi}$, and $I_{mt}$ involve the radial potential $R_{m}(r)$

The angular integrals are

The radial potential $R_{m}(r)$ is given by a quartic polynomial and the types of its roots play essential roles in the classification of different orbits. In the previous work [26, 36], we have shown the exact solutions to some of the cases of both null and time-like geodesics. For the present work, we will mainly focus on the infalling orbits of the bound and the unbound motion at the black hole exterior. In this context, we will consider three types of such orbits in the subsequent sections. The discussion of the angular potential $\Theta(\theta)$ and the integrals involved has been presented in Ref. [26]. For the sake of completeness, we will provide a short summary in Appendix A.

Before ending this section, let us introduce a few notations that will be used in the subsequent sections. Related to $R_{m}(r)$ we define the integrals

In terms of $I_{1}$, $I_{2}$, and $I_{\pm}$ we can rewrite (18) and (19) as follows

According to previous studies of radial potential [26], by examining different ranges of the parameters $\lambda_{m}$ and $\eta_{m}$ for bound motion ($\gamma_{m}<1$), it is clear that there are two distinct categories of infalling motion that traverse the horizon and enter the black hole. One is that the particle starts from $r_{i}\leq r_{\rm isso}$, where the radius of the ISSO orbit $r_{\rm isso}$ is within the parameters located at A and B in Fig. 1, spirals and then plunge into the horizon of the black holes [37, 38]. The other one is starting from $r_{i}<r_{m4}$, with the parameters of C and D in Fig. 5, and plunge through the horizon of the black holes.

We first consider the particle starts from $r_{i}\leq r_{\rm isso}$. The solutions along the $r$ direction can be obtained from the inversion of (14) with the integral $I_{mr}$ in (17), where the radial potential (11) is given in the case of the triple root located at the ISSO radius, namely $r_{m2}=r_{m3}=r_{m4}=r_{\rm isso}$, and the initial $r$ is set at $r_{i}\leq r_{\rm isso}$. So, from the integral (14) and (17), we can have the Mino time $\tau_{m}$ as the function of $r$

The particle moves toward the horizon with $\nu_{r_{i}}=-1$. Thus, the inverse of (27) leads to

where

The solution (28) of the coordinate $r$ involves the triple root $r_{\rm isso}$ of the radial potential, which can be determined as follows.

From the double root solutions $R(r)=R^{\prime}(r)=0$ [26], two constants of motion in the case of spherical orbits are given by

The subscript ”ss” means the spherical orbits with $s=\pm$, which denotes the two types of motion with respect to the relative sign between the black hole’s spin and the azimuthal angular of the particle (see Section III C of [26]). Together with the two relations above, an additional equation from $R^{\prime\prime}(r)=0$ determines the triple roots. We have found the radius of $r_{\rm isso}$ satisfying the following equation

where

We proceed by evaluating the coordinates $\phi_{m}(\tau_{m})$ and $t_{m}(\tau_{m})$ using (15) and (16), which involve not only the angular integrals $G_{m\phi}$ and $G_{mt}$, but also the radial integrals (18) and (19). With the help of (25) and (26), we first rewrite (18) and (19) as

For the present case of the triple roots, the calculation of the integrals is straightforward and one can express $I_{n}^{I}$ and $I_{\pm}^{I}$ in terms of elementary functions,

It is worthwhile to mention that $\mathcal{I}_{\pm_{i}}^{I}$, $\mathcal{I}_{1_{i}}^{I}$, $\mathcal{I}_{2_{i}}^{I}$ are obtained by evaluating ${I}_{\pm}^{I}$, ${I}_{1}^{I}$, ${I}_{2}^{I}$ at $r=r_{i}$ of the initial condition, that is, ${{I}_{\pm}^{I}(0)={I}_{1}^{I}(0)={I}_{2}^{I}(0)=0}$. The solutions of $\phi^{I}(\tau_{m})$ and $t^{I}(\tau_{m})$ can be constructed from $I_{m\phi}$ (18), $G_{m\phi}$ (21) and $I_{mt}$ (19) and $G_{mt}$ (22) through (15) and (16). Together with the solutions along $r$ and $\theta$ directions in (28) and (90), they are infalling motions of the general nonequatorial orbits in the Kerr-Newman exterior. An illustrative example is shown in Fig. 2 with the parameters of the case A in Fig. 1. The particle starts by inspiraling around $r_{\rm isso}$ and then plunges into the black hole’s horizon.

For the particle initially at $r_{i}=r_{\rm isso}$, the solution (28) gives $r(\tau)=r_{\rm isso}$ obtained from $X^{I}\rightarrow-\infty$ in (29) when $r_{i}\rightarrow r_{\rm isso}$, meaning that the particle will stay in spherical motion with the $r_{\rm isso}$ radius. However, for $r_{i}<r_{\rm isso}$ of our interest, as $r$ reaches the outer horizon $r_{+}$, it takes finite Mino time $\tau_{m}$. Nevertheless, because of the $\tanh^{-1}$ function in (36), $I_{\pm}^{\rm isso}\rightarrow\infty$ as $r\rightarrow r_{+}$, giving the coordinate time $t\rightarrow\infty$ and the azimuthal angle $\phi\rightarrow\infty$ observed in the asymptotical flat regime. The above expressions can be further reduced to the Kerr black hole case by sending $Q\rightarrow 0$ in [20]. As for the contributions to the evolution of the angle $\phi$ in (15) from the integrals involving the radial potential $R_{m}(r)$, one can write the $\tanh^{-1}$ function in $I_{\pm}^{I}(\tau_{m})$ by $\log$ function through $\tanh^{-1}(x)=\frac{1}{2}\frac{\log(1+x)}{\log(1-x)}$. Then, the remaining terms in (34) directly proportional to $\tau_{m}$ are the same in [20]. Together with (93) of the integrals involving the $\theta$ potential $\Theta_{m}(\theta)$, the variables $z_{1}$ and $z_{2}$ defined in [20] can be related to the roots of $\Theta_{m}(\theta)$ by $z_{1}^{2}=u_{m+}$ and $z_{2}^{2}=u_{m-}a^{2}(1-\gamma_{m}^{2})$, leading to $k_{z}^{2}=\frac{u_{m+}}{u_{m-}}$, which gives the same expression in [20] from (93).

One of the interesting cases is considering the equatorial motion by taking $\theta=\frac{\pi}{2}$ and $\eta_{m}\rightarrow 0$ limits in the results above. The particle starts from the coordinate $r$ slightly less than the radius of innermost circular motion $r_{\rm isco}$. In particular, $G_{m\phi}=\tau_{m}$ and the solution of $\phi^{I}_{m}$ in (15) simplifies to [26, 36]

where $I^{I}_{m\phi}$ is given by (34). In addition, one can replace Mino time $\tau_{m}$ by the coordinate $r$ through Eq. (27). Then the infalling solution of the angle $\phi_{m}$ on the equatorial plane can be expressed as a function of $r$,

Likewise, for the equatorial orbits, Eq. (16) with $G_{mt}=0$ gives

where $I^{I}_{mt}$ has been calculated in (35). Substituting $\tau^{I}_{m}$ to replace $r$, we find

As for the initial conditions one can determine $\phi^{I}_{mi}$ and $t^{I}_{mi}$ by $I^{I}_{m\phi}\left(\tau^{I}_{m}\left(r\right)\right)+\lambda_{m}\tau^{i}_{m}\left(r\right)$ and $I^{I}_{mt}\left(\tau_{m}\right)$ vanishing at the initial $r_{i}$. The corresponding trajectories are shown in Fig. 3, with the additional parameter $Q$ apart from $a$ of the black holes. This generalizes the solution in [13] for the Kerr black holes, where the particle starts from $r\lesssim r_{\rm isco}$ at $t_{m}(r)=-\infty$ and inspirals to the event horizon. In the limit of $Q\rightarrow 0$ where $r_{m1}\rightarrow 0$, $1-\gamma_{m}^{2}\rightarrow{2M}/{3r_{\rm isco}}$, and $(r_{\rm isco}-r_{+})(r_{\rm isco}-r_{-})\rightarrow r^{2}_{\rm isco}-2Mr_{\rm isco}+a^{2}$, the first term of Eq.(40) reduces to the corresponding term $\sqrt{\frac{r}{r_{\rm isco}-r}}$ of $\phi^{I}_{m}(r)$ in [13]. Together with $t_{\pm}=\sqrt{\frac{r_{\pm}}{r_{\rm isco}-r_{\pm}}}$ defined in [13], one can make the replacement

to reproduce the $\tanh^{-1}$ terms.

One of the limiting cases that can significantly simplify the above expressions is to consider the extremal limit of the Kerr black hole, $a\rightarrow M$. For $Q\rightarrow 0$ giving $r_{m1}=0$, and for the extremal black holes, the ISCO radius for direct orbits is on the event horizon. Here we focus on the extremal retrograde motion with $r_{\rm isco}=9M$, $\lambda_{m}=-22\sqrt{3}M/9$ and $\gamma_{m}=5\sqrt{3}/9$. Notice that in the extremal black holes, $r_{+}$ and $r_{-}$ collapse into the same value. Then the solutions can be reduced into the known ones [13, 14]

Another limiting case is considering the Reissner Nordstr$\ddot{o}$m (RN) black hole. Since $a\rightarrow 0$ leading to the spherically symmetric metric, the general motion can be studied by considering equatorial motions. The coefficients of the $\tanh^{-1}$ terms of the above expressions (40) all vanish. The expressions of $\phi^{I}_{m}\left(r\right)$ and $t^{I}_{m}\left(r\right)$ can be simplified as

Further simplification occurs in the extremal limit. For $M=\pm Q$ in the RN black holes, $r_{\pm}=M$, and with $r_{\rm isco}=4M$, $r_{m1}=4M/5$, $\lambda_{m}=2\sqrt{2}M$ and $\gamma_{m}=3\sqrt{6}/8$, (46) and (47) can have such a simple form