Inspiral and plunging orbits in Kerr-Newman spacetimes

We present the analytical solutions for the trajectories of particles that spiral and plunge inward of the event horizon along the timelike geodesics following general nonequatorial paths within Kerr-Newman spacetimes. Our studies encompass both bound and unbound motions. The solutions can be written in terms of the elliptical integrals and the Jacobian elliptic functions of manifestly real functions of the Mino time. They can respectively reduce to the Kerr, Reissner-Nordström, and Schwarzschild black holes in certain limits of the spin and charge of the black holes, and can be compared with the known ones restricted in equatorial motion. These explicit solutions may have some implications for the gravitational wave emission from extreme mass-ratio inspirals.

Figure 1

The main graphics shows the parametric plot of ${\lambda }_m(r_{\mathrm{isso}})$ versus ${\eta }_m(r_{\mathrm{isso}})$. The triple roots $r_{\mathrm{isso}}$ are the solutions of the equations $R_m^{\prime \prime }(r)=R_m^{\prime }=R_m(r)=0$. The inset illustrates the behavior of the radial potential $R_m$ with the parameters located at A and B. The case of B has ${\eta }_m=0$, which is an example of equatorial motion.

Figure 2

An illustrative example of nonequatorial orbit with parameters A of Fig. 1. In this case, the particle starts from $r_i\le r_{\mathrm{isso}}$ and inspirals into the black hole after many azimuthal and longitudinal revolutions. From the top view one notices the very different time scales of the spiralling and plunging phases.

Figure 3

Illustration of the orbit on the equatorial plane with the parameters of B in Fig. 1. The particle starts from $r_i<r_{\mathrm{isso}}$ and inspirals into the black hole horizon.

Figure 4

The plots show the variation of azimuthal angle $\mathrm{\Delta }{\varphi }_m^I\equiv {\varphi }_m^I-{\varphi }_{mi}^I$ as a function of $r$ for the inspire bound motion for various black-hole models, including Kerr-Newman ($Q=0.7M,a=0.7M,{\lambda }_m=-3.84M,{\eta }_m=0M$), Kerr-extremal ($Q=0M,a=M,{\lambda }_m=-\frac{22\sqrt{3}}{9}M,{\eta }_m=0M$) and RN ($Q=0.7M,a=0M,{\lambda }_m=3.2M,{\eta }_m=0M$). In the figure, the blue curve and red curves illustrate the direct and retrograde orbits, respectively. See the text for more discussion.

Figure 5

The graphics shows the portion of parameter space bound by the double root solution, $r_{m2}=r_{m3}$. The equation $R_m(r)=0$ with parameters in the blue zone have complex roots, $r_{m2}=r_{m3}^{*}$, so that, a particle in this region, say C or D, and starts from $r_i<r_4$ will plunge directly into the black hole horizon. The inset shows the behavior of the radial potential $R_m(r)$ for the case of the parameters located in C and D.

Figure 6

Illustration of an orbit off the equatorial plane with the parameters of C in Fig. 5. In this case the particle starts from $r_i<r_4$ and plunges directly into the black hole horizon.

Figure 7

Illustration of an orbit on the equatorial plane with the parameters of D in Fig. 5. The particle initiates its journey at point $r_i$, moves outward, reaches the turning point at $r_{m4}$, and then reverses its course, plunging back into the black hole.

Figure 8

The plots show the $\mathrm{\Delta }{\varphi }_m^B$ as a function of $r$ for bound motion with the various black holes including Kerr-Newman ($Q=0.7M,a=0.7M,{\lambda }_m=M,{\eta }_m=0$), Kerr-extremal ($Q=0M,a=M,{\lambda }_m=-M,{\eta }_m=0$) and RN ($Q=0.7M,a=0M,{\lambda }_m=M,{\eta }_m=0$). In this plot, the red and blue curves illustrate the examples of direct and retrograde orbits, respectively.

Figure 9

The graphics shows the portion of parameter space limited by the double root solution, $r_{m3}=r_{m4}$ and $r_{m1}<r_{m2}<r_{-}<r_{+}$. For the region of the parameter space for E and F, the roots $r_{m3}$ and $r_{m4}$ are complex, $r_{m3}=r_{m4}^{*}$ and $r_{m1}<r_{m2}<r_{-}$. The inset shows the details of the roots of illustrative cases E and F in the main figure. See the text for more discussion.

Figure 10

Illustration of a general nonequatorial plunging orbit exemplifies with the parameters E in Fig. 9. In this case ${\eta }_m\ne 0$ and the particle starts from spatial infinity and infalls directly into the black horizon.

Figure 11

Illustration of an equatorial inspiral orbit with the parameters in the F region in Fig. 9 for ${\eta }_m=0$ where the particle starts from spatial infinity and plunge directly into the horizon.

Figure 12

The plots show the $\mathrm{\Delta }{\varphi }_m^U$ as a function of $r$ for unbound infalling motion, with the various black hole including Kerr-Newman ($Q=0.7M,a=0.7M,{\lambda }_m=M,{\eta }_m=0$), Kerr extremal ($Q=0M,a=M,{\lambda }_m=-M,{\eta }_m=0$) and RN ($Q=0.7M,a=0M,{\lambda }_m=M,{\eta }_m=0$). The blue and red curves show the examples of direct and retrograde orbits, respectively.