The relativistic restricted three-body problem: geometry and motion around tidally perturbed black holes

TL;DR

This work analyzes the motion of test particles around a tidally deformed, rotating black hole in an adiabatically evolving binary, introducing a small tidal strength parameter $\epsilon$ to explore strong-field three-body dynamics.Using linear tidal perturbation theory and metric reconstruction, it constructs a tidally perturbed Kerr geometry and demonstrates that tidal coupling generically destroys integrability, producing resonances and chaotic layers in the geodesic flow.The paper identifies a robust four-stage evolution of bound geodesics as $\epsilon$ increases: bound, chaotic, collapsing, and depleted, with semi-analytic estimates for the critical amplitudes $\epsilon_c^-,\epsilon_c^+,\epsilon_c^0$ that match numerical results and connect to Roche-lobe and disk depletion scales.Astrophysical implications include resonant locking and disk-oscillation modulation, tidal triggering of inward accretion and disk depletion, and potentially detectable imprints on gravitational waves from EMRIs, with LISA/DECIGO capable of probing the earlier stages while ground-based detectors may observe depleted-phase effects.Overall, the work provides a framework to model GW phase corrections and EM signatures in binaries involving tidally perturbed Kerr black holes, guiding future waveform modeling and multi-messenger analyses.

Abstract

We investigate the geometry of a tidally deformed, rotating black hole and timelike geodesics in its vicinity. Our framework provides a local picture of the structural evolution of a relativistic restricted three-body problem around a deformed black hole in an adiabatically evolving binary, motivated by various astrophysical settings including disk dynamics and extreme mass-ratio inspirals. As the tidal-field strength is increased, initially regular, bound geodesics undergo four stages: (i) weak chaos emerges within the bound motion; (ii) a subset of trajectories plunges into the black hole; (iii) a fraction of the remaining trajectories becomes unbound; and (iv) no bound trajectories persist. We provide semi-analytic estimates for the critical tidal amplitudes associated with each transition. Our estimates indicate that, within the frequency band of ground-based gravitational-wave detectors, the matter flow around black holes may already be depleted, whereas LISA and (B-)DECIGO could probe the earlier stages. Our results suggest that an object orbiting a tidally deformed massive BH may remain near resonances over a wide range of separations, indicating an accumulated, non-negligible impact on the gravitational-wave phase. Tidal perturbations can also introduce nonlinear couplings among epicyclic oscillations of geodesics, offering a potential avenue to resonant excitation of quasi-periodic oscillations in X-ray light curves from accreting black holes.

Figures (6)

• Figure 1: Magnitude of $\epsilon$ as a function of the orbital separation $b/M$, $b/M_{\rm ext}$, for various mass ratios.
• Figure 2: $\epsilon$-independent deviation of the radius of the static-limit surface from $r_{\rm ergo}^{(0)}:=M+\sqrt{M^2-a^2\cos^2\theta}$ for the BH with $a/M= 4/5$, c.f. Eq. \ref{['eq:rergo_shift']}. Colors denote $\theta=0,\pi$ (black solid), $\theta=0.1\pi,0.9\pi$ (green dashed), $\theta=0.25\pi,0.75\pi$ (blue dotted), and $\theta=0.5\pi$ (red dotted-dashed), which reflects the symmetry with respect to the equatorial plane.
• Figure 3: Prograde bound orbits with initial parameters $(\hat{E}, \hat{L}|_{\lambda=0})=(0.98,1.8r_+)$ which starts its motion at various discrete values of $r_0$, around a tidally deformed Kerr BH with $a/M=4/5$. In the left panel, the sequences in green and magenta correspond to the examples of the $(2,3)$ and $(5,7)$ resonant islands, respectively. The dot at $(d\lambda/dr,r/r_+)\simeq (0,15.176)$ in the left panel corresponds to an originally spherical orbit at $r=r_{\rm sph}$ in Eq. \ref{['eq:rsph']}.
• Figure 4: Frequency ratio, $\omega_r/\omega_\theta$, associated with the $(q,p)=(2,3)$ (green) and $(5,7)$ (magenta) resonant islands, highlighted in the corresponding color in the left panel of Fig. \ref{['fig:PoincareMap']}. The red dot-dashed smooth curve denotes the unperturbed Kerr case. The inset presents the zoomed-out figure, showing that no clear plateaus are visible outside the zoomed-in region.
• Figure 5: Evolution of originally bound orbits with initial parameters $(\hat{E}, \hat{L}|_{\lambda=0})=(0.98,1.8r_+)$, evolved from $r_0/r_+=2.8$ (magenta dashed), $r_0/r_+=10$ (green dotted), $r_0/r_+=15.176$ (blue solid), $r_0/r_+=21$ (red dot-dashed), in the tidally deformed Kerr spacetime with $a/M=4/5$ for various values of $\epsilon$. Note that the blue solid trajectories correspond to an originally spherical orbit. Top-left, top-right, bottom-left, and bottom-right panels represent bound phase, chaotic phase, collapsing phase, and depleted phase, defined in the main text, respectively.