The relativistic restricted three-body problem: geometry and motion around tidally perturbed black holes
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2026
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| author | Katagiri, Takuya Cardoso, Vitor |
| author_facet | Katagiri, Takuya Cardoso, Vitor |
| contents | 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, 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 LISA 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 quasi-periodic oscillations in X-ray light curves from accreting black holes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_14979 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The relativistic restricted three-body problem: geometry and motion around tidally perturbed black holes Katagiri, Takuya Cardoso, Vitor General Relativity and Quantum Cosmology Astrophysics of Galaxies High Energy Astrophysical Phenomena 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, 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 LISA 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 quasi-periodic oscillations in X-ray light curves from accreting black holes. |
| title | The relativistic restricted three-body problem: geometry and motion around tidally perturbed black holes |
| topic | General Relativity and Quantum Cosmology Astrophysics of Galaxies High Energy Astrophysical Phenomena |
| url | https://arxiv.org/abs/2601.14979 |