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.
Categories: gr-qc astro-ph.GA astro-ph.HE