Relativistic restricted three-body problem: Geometry and motion around tidally perturbed black holes

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 semianalytic estimates for the critical tidal amplitudes associated with each transition. Our estimates, within the idealized test-particle description, indicate that, within the frequency band of ground-based gravitational-wave detectors, the matter flow around black holes may already be depleted, whereas the Laser Interferometer Space Antenna and (B-)DECIGO could probe the earlier stages. Our results suggest that an object orbiting a tidally deformed massive black hole may remain near resonances in a long term, indicating an accumulated, non-negligible impact on the gravitational-wave phase. Another finding is that tidal perturbations can modulate nonlinear couplings among epicyclic oscillations of geodesics, and could therefore, in principle, affect resonant excitation mechanism potentially relevant to quasiperiodic oscillations in x-ray light curves from accreting black holes.