Motion of spinning particles in the Kerr-Newman black hole exterior. I. Periodic orbits

TL;DR

This work analyzes the exterior motion of a spinning test body around a Kerr-Newman black hole within the pole-dipole, linear-in-spin regime of the Mathisson-Papapetrou equations. By formulating the dynamics in Mino time and deriving a spin-corrected radial potential $R^s_m(r)$, the authors obtain analytic equatorial solutions in terms of elliptic functions for both aligned and misaligned spins, and compute the associated orbital frequencies. They then construct numerical kludge gravitational-wave signals from these trajectories, showing that waveform amplitudes and mode content are sensitive to turning points and spin configuration, with implications for EMRIs detectable by LISA. Overall, the results extend prior Kerr and Reissner-Nordström analyses to Kerr-Newman spacetimes, providing analytic tools and waveform templates for interpreting charged-black-hole environments in EMRI GW astronomy.

Abstract

The motion of a spinning particle in the exterior of a Kerr-Newman black hole is studied. The dynamics is governed by the Mathisson-Papapetrou equations in the pole-dipole approximation, including the spin-curvature coupling to leading order in the spin. In terms of conserved quantities, the dynamical equations in Mino time can be transformed into the integral form for both aligned and misaligned spins with respect to the orbital motion. These non-geodesic equations can be solved analytically with the solutions involving Jacobi elliptic functions. The radial potential is derived to study the particle's parameter space for various types of orbits, based on its roots corrected by the particle's spin. We consider equatorial motion oscillating between two turning points, which are the two outermost roots of the radial potential, in the misaligned case. In this case, there is an induced oscillatory motion out of the equatorial plane. In particular, the periods of the motion are obtained explicitly. When the orbits become a source of gravitational-wave emission, these periods of motion will play a key role in determining gravitational-wave signals in the frequency domain. Numerical kludge waveforms are constructed. The gravitational-wave amplitudes are found to be sensitive to the turning points of the orbits. The implications for gravitational-wave emission due to extreme mass-ratio inspirals (EMRIs) are discussed.

Figures (5)

• Figure 1: The plot shows the radial potential $R_m^s(r)$ and the parameter space $(\lambda_m, \gamma_m)$ for black hole spin $a/M = 0.5$, charge $Q/M = 0.6$, and particle spin $s_\parallel/m = 0.1$ in the case of the bound motion for $\gamma_m<1$. In the right inset, the green (orange) line represents the parameters for the unstable (stable) double root of $r_{m2} =r_{m3}$ ($r_{m3} =r_{m4}$) with the representative point A(C). Two double roots merge at point D of a triple root. The parameters at the point B cause the oscillation motion between two turning points, $r_{m3}$ and $r_{m4}$. The lower inset plot shows the details of the radial potential in the main figure with the parameters at points A-D.
• Figure 2: The plots show the dynamical variables of the orbits: (a) $t^B(\tau_m)$ from (\ref{['I_t']}), (b) $r(\tau_m)$ from the inversion of (\ref{['I_r^B']}), (c) $\delta \theta^B(\tau_m)$ from (\ref{['deltatheta2']}) and (\ref{['I_psi']}), and (d) $\phi^B(\tau_m)$ from (\ref{['I_phi']}) as a function of $\tau_m$ of the spinning particle traveling between two turning points for a misaligned spin, $s_\perp\neq 0$ with the initial conditions at $r_i=r_{m3}$ and $w=\pi/2$. The black hole parameters are chosen to be $a/M=0.5$, $Q/M=0.4$ ($a/M=0.5$, $Q/M=0.8$), and the particle parameters are chosen to be $\gamma_m=0.97$, $|\lambda_m|=0.42$, $s_\parallel/m= 0.1$, and $s/m=0.3$ for $\lambda_m>0$ (red) and $\lambda_m<0$ (blue) (for $\lambda_m>0$ (orange) and $\lambda_m<0$ (purple)). Note that the choice of spin is to emphasize the precession along the polar angle, but in fact, this condition $s_\parallel \gg s_\perp$ should be held.
• Figure 3: The 3D trajectory of the particle traveling between two turning points, $r_{m4}$ (the red dashed circle) and $r_{m3}$ (the dark red dashed circle), with $s_\parallel/m = 0$, $s/m = 0.3$ around the Kerr-Newman black hole with $a/M=0.5$ and $Q/M=0.8$. The induced oscillation motion about $\delta \theta^B$ is visible due to the misaligned spin $s_\parallel \neq s$.
• Figure 4: The plots of the gravitational waveform polarization $h_+$ from (\ref{['pressexp']}). The observer is located at $(\Theta,\Phi)=(\pi/2,0)$. The waveforms are generated by the orbit of a spinning particle with $\gamma_m = 0.97$ and $|\lambda_m| = 4.2$ with (without) spin $s_\parallel$ around a Kerr-Newman black hole with $a/M = 0.5$ and a small or large charge $Q$, respectively. The parameters $(Q/M, s/m, s_\parallel/m)$ are chosen to be: (a) $( 0.4, 0, 0 )$, (b) $( 0.4, 0.3, 0.1 )$, (c) $( 0.8, 0, 0 )$, (d) $( 0.8, 0.3, 0.1 )$. The red and orange (blue and purple) lines represent the waveform polarization $h_+$ for the direct (retrograde) orbits with $\lambda_m > 0$ ($\lambda_m < 0$).
• Figure 5: The modes of the waveform in the frequency domain $|h_{lmn}|$ with the parameters in Fig. \ref{['GWaveform']}(d) for the direct orbit are shown in (a) the mode $l$ of $\Omega_r^B$ with $(m,n) = (0,0)$, (b) the mode $m$ of $\Omega_{\delta \theta}^B$ with $(l,n) = (0,0)$, and (c) the mode $n$ of $\Omega_\phi^B$ with $(l,m) = (0,0)$. The red (green) line represents $|h_{lmn}|$ in the basis $( \Omega_r^B, \Omega_{\delta \theta}^B, \Omega_\phi^B )$ with (without) the particle spin $s$ and $s_\parallel$. Note that, as $s_\parallel =s$, the motion is confined to the equatorial plane, hence there is no frequency $\Omega_{\delta \theta}^B$.