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

Takuya Katagiri1 and Vitor Cardoso2,3

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Phys. Rev. D 113, 104036 – Published 13 May, 2026

DOI: https://doi.org/10.1103/83tn-ljl3

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 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.

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References (140)

  1. B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), Phys. Rev. Lett. 116, 221101 (2016); 121, 129902(E) (2018).
  2. V. Cardoso and P. Pani, Living Rev. Relativity 22, 4 (2019).
  3. R. Abbott et al. (LIGO Scientific, VIRGO, and KAGRA Collaborations), Phys. Rev. D 112, 084080 (2025).
  4. A. G. Abac et al. (LIGO Scientific, VIRGO, and KAGRA Collaborations), Phys. Rev. Lett. 135, 111403 (2025).
  5. E. Berti et al., arXiv:2505.23895.
  6. V. Cardoso, S. Biswas, and S. Sarkar, arXiv:2511.14841.
  7. P. Amaro-Seoane et al. (LISA Collaboration), arXiv:1702.00786.
  8. M. Branchesi et al., J. Cosmol. Astropart. Phys. 07 (2023) 068.
  9. A. J. Dittmann, A. M. Dempsey, and H. Li, Astrophys. J. 964, 61 (2024).
  10. S.-C. Yang, W.-B. Han, H. Tagawa, S. Li, Y. Jiang, P. Shen, Q. Yun, C. Zhang, and X.-Y. Zhong, Astrophys. J. Lett. 988, L41 (2025).
  11. A. G. Abac et al. (LIGO Scientific, VIRGO, and KAGRA Collaborations), Astrophys. J. Lett. 993, L21 (2025).
  12. J. S. Santos, V. Cardoso, J. Natário, and M. van de Meent, Phys. Rev. Lett. 135, 211402 (2025).
  13. S. Naoz, C. M. Will, E. Ramirez-Ruiz, A. Hees, A. M. Ghez, and T. Do, Astrophys. J. Lett. 888, L8 (2020).
  14. C. M. Will, S. Naoz, A. Hees, A. Tucker, E. Zhang, T. Do, and A. Ghez, Astrophys. J. 959, 58 (2023).
  15. O. Straub et al. (GRAVITY Collaboration), Astron. Astrophys. 672, A63 (2023); 677, C2(E) (2023).
  16. V. Cardoso and A. Foschi, Phys. Rev. D 104, 024004 (2021).
  17. F. Camilloni, G. Grignani, T. Harmark, R. Oliveri, M. Orselli, and D. Pica, Phys. Rev. D 107, 084011 (2023).
  18. E. Poisson, Phys. Rev. D 91, 044004 (2015).
  19. P. Landry and E. Poisson, Phys. Rev. D 91, 104018 (2015).
  20. P. Pani, L. Gualtieri, A. Maselli, and V. Ferrari, Phys. Rev. D 92, 024010 (2015).
  21. Y. Mino, M. Sasaki, and T. Tanaka, Phys. Rev. D 55, 3457 (1997).
  22. T. C. Quinn and R. M. Wald, Phys. Rev. D 56, 3381 (1997).
  23. E. Poisson, Living Rev. Relativity 7, 6 (2004).
  24. E. Poisson, A. Pound, and I. Vega, Living Rev. Relativity 14, 7 (2011).
  25. S. A. Balbus and J. F. Hawley, Astrophys. J. 376, 214 (1991).
  26. J. F. Hawley and S. A. Balbus, Astrophys. J. 376, 223 (1991).
  27. J. F. Hawley and S. A. Balbus, Astrophys. J. 400, 595 (1992).
  28. S. A. Balbus and J. F. Hawley, Astrophys. J. 400, 610 (1992).
  29. S. A. Balbus and J. F. Hawley, Rev. Mod. Phys. 70, 1 (1998).
  30. M. A. Abramowicz and P. C. Fragile, Living Rev. Relativity 16, 1 (2013).
  31. D. C. Robinson, Phys. Rev. Lett. 34, 905 (1975).
  32. B. Carter, Phys. Rev. Lett. 26, 331 (1971).
  33. S. W. Hawking, Commun. Math. Phys. 25, 152 (1972).
  34. J. D. Bekenstein, in 2nd International Sakharov Conference on Physics (1996), pp. 216–219, arXiv:gr-qc/9605059.
  35. B. Carter, in 8th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Gravitation and Relativistic Field Theories (MG 8) (1997), pp. 136–155, arXiv:gr-qc/9712038.
  36. P. T. Chrusciel, J. Lopes Costa, and M. Heusler, Living Rev. Relativity 15, 7 (2012).
  37. V. Cardoso and L. Gualtieri, Classical Quantum Gravity 33, 174001 (2016).
  38. E. Poisson, Phys. Rev. D 70, 084044 (2004).
  39. S. Taylor and E. Poisson, Phys. Rev. D 78, 084016 (2008).
  40. K. Chatziioannou, E. Poisson, and N. Yunes, Phys. Rev. D 87, 044022 (2013).
  41. T. Katagiri, https://github.com/TakuyaKatagiri/Tidally_Deformed_metric (2026).
  42. T. Katagiri and V. Cardoso, https://the-center-of-gravity.com/data-and-routines/ (2026).
  43. W. M. Kinnersley, Type D gravitational fields, Ph.D. thesis, Caltech, 1968.
  44. E. Poisson and I. Vlasov, Phys. Rev. D 81, 024029 (2010).
  45. T. Binnington and E. Poisson, Phys. Rev. D 80, 084018 (2009).
  46. H. S. Chia, Phys. Rev. D 104, 024013 (2021).
  47. P. Charalambous, S. Dubovsky, and M. M. Ivanov, J. High Energy Phys. 05 (2021) 038.
  48. E. Poisson, Phys. Rev. D 104, 104062 (2021).
  49. S. E. Gralla, Classical Quantum Gravity 35, 085002 (2018).
  50. E. Poisson, Phys. Rev. D 103, 064023 (2021).
  51. T. Katagiri, K. Yagi, and V. Cardoso, Phys. Rev. D 111, 084080 (2025).
  52. T. Katagiri, V. Cardoso, T. Ikeda, and K. Yagi, Phys. Rev. D 111, 084081 (2025).
  53. J. M. Bardeen, W. H. Press, and S. A. Teukolsky, Astrophys. J. 178, 347 (1972).
  54. G. Contopoulos, Order and Chaos in Dynamical Astronomy (Springer, Berlin, 2002).
  55. J. R. Gair, C. Li, and I. Mandel, Phys. Rev. D 77, 024035 (2008).
  56. G. Lukes-Gerakopoulos, Phys. Rev. D 86, 044013 (2012).
  57. K. Destounis and P. G. S. Fernandes, Phys. Rev. D 113, 044040 (2026).
  58. J. Brink, M. Geyer, and T. Hinderer, Phys. Rev. D 91, 083001 (2015).
  59. W. Schmidt, Classical Quantum Gravity 19, 2743 (2002).
  60. J. Brink, Phys. Rev. D 78, 102002 (2008).
  61. V. I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics (Springer, New York, 1989).
  62. J. Laskar, Celest. Mech. Dyn. Astron. 56, 191 (1993).
  63. H. Poincaré, Rendiconti del Circolo Matematico di Palermo (1884-1940) 33, 375 (1912).
  64. G. D. Birkhoff, Trans. Am. Math. Soc. 14, 14 (1913).
  65. G. Lukes-Gerakopoulos, T. A. Apostolatos, and G. Contopoulos, Phys. Rev. D 81, 124005 (2010).
  66. T. A. Apostolatos, G. Lukes-Gerakopoulos, and G. Contopoulos, Phys. Rev. Lett. 103, 111101 (2009).
  67. J. Brink, M. Geyer, and T. Hinderer, Phys. Rev. Lett. 114, 081102 (2015).
  68. J. Levin and G. Perez-Giz, Phys. Rev. D 77, 103005 (2008).
  69. E. Gueron and P. S. Letelier, arXiv:astro-ph/0108042.
  70. E. Teo, Gen. Relativ. Gravit. 53, 10 (2021).
  71. B. Carter, Phys. Rev. 174, 1559 (1968).
  72. K. Destounis, G. Huez, and K. D. Kokkotas, Gen. Relativ. Gravit. 55, 71 (2023).
  73. R. Fujita and W. Hikida, Classical Quantum Gravity 26, 135002 (2009).
  74. R. V. Wagoner, Phys. Rep. 311, 259 (1999).
  75. M. A. Abramowicz and W. Kluzniak, Astron. Astrophys. 374, L19 (2001).
  76. W. Kluzniak and M. A. Abramowicz, arXiv:astro-ph/0105057.
  77. A. Ingram and S. Motta, New Astron. Rev. 85, 101524 (2019).
  78. J. Horak, in Workshop on Processes in the Vicinity of Black Holes and Neutron Stars (2004), arXiv:astro-ph/0408092.
  79. J. Horák, M. A. Abramowicz, W. Kluźniak, P. Rebusco, and G. Török, Astron. Astrophys. 499, 535 (2009).
  80. P. Rebusco, Publ. Astron. Soc. Jpn. 56, 553 (2004).
  81. W. H. Lee, M. A. Abramowicz, and W. Kluzniak, Astrophys. J. Lett. 603, L93 (2004).
  82. S. Kato, Publ. Astron. Soc. Jpn. 55, 801 (2003).
  83. S. Kato, Publ. Astron. Soc. Jpn. 56, 905 (2004).
  84. D. Barret, J.-F. Olive, and M. C. Miller, Mon. Not. R. Astron. Soc. 361, 855 (2005).
  85. D. Barret, J.-F. Olive, and M. C. Miller, Mon. Not. R. Astron. Soc. 376, 1139 (2007).
  86. J. F. Steiner, J. E. McClintock, R. A. Remillard, L. Gou, S. Yamada, and R. Narayan, Astrophys. J. Lett. 718, L117 (2010).
  87. J. A. Orosz, J. F. Steiner, J. E. McClintock, M. M. Buxton, C. D. Bailyn, D. Steeghs, A. Guberman, and M. A. P. Torres, Astrophys. J. 794, 154 (2014).
  88. K. Sawada, T. Matsuda, and I. Hachisu, Mon. Not. R. Astron. Soc. 219, 75 (1986).
  89. P. P. Eggleton, Astrophys. J. 268, 368 (1983).
  90. G. P. Kuiper, Astrophys. J. 93, 133 (1941).
  91. F. H. Shu, S. H. Lubow, and L. Anderson, Astrophys. J. 229, 223 (1979).
  92. K. Nariai and D. Sugimoto, Publ. Astron. Soc. Jpn. 28, 593 (1976).
  93. R. Gold, V. Paschalidis, M. Ruiz, S. L. Shapiro, Z. B. Etienne, and H. P. Pfeiffer, Phys. Rev. D 90, 104030 (2014).
  94. V. Paschalidis, J. Bright, M. Ruiz, and R. Gold, Astrophys. J. Lett. 910, L26 (2021).
  95. D. B. Bowen, M. Campanelli, J. H. Krolik, V. Mewes, and S. C. Noble, Astrophys. J. 838, 42 (2017).
  96. R. Gold, Galaxies 7, 63 (2019).
  97. B. Bonga, H. Yang, and S. A. Hughes, Phys. Rev. Lett. 123, 101103 (2019).
  98. P. Gupta, B. Bonga, A. J. K. Chua, and T. Tanaka, Phys. Rev. D 104, 044056 (2021).
  99. P. Gupta, L. Speri, B. Bonga, A. J. K. Chua, and T. Tanaka, Phys. Rev. D 106, 104001 (2022).
  100. A. Ori and K. S. Thorne, Phys. Rev. D 62, 124022 (2000).
  101. M. Cocco, G. Grignani, T. Harmark, M. Orselli, D. Pereñiguez, and M. van de Meent, arXiv:2601.00954.
  102. E. Grilli, M. Orselli, D. Pereñiguez, and D. Pica, J. Cosmol. Astropart. Phys. 02 (2025) 028.
  103. J. Aasi et al. (LIGO Scientific Collaboration), Classical Quantum Gravity 32, 074001 (2015).
  104. F. Acernese et al. (VIRGO Collaboration), Classical Quantum Gravity 32, 024001 (2015).
  105. K. Somiya (KAGRA Collaboration), Classical Quantum Gravity 29, 124007 (2012).
  106. M. Gröbner, W. Ishibashi, S. Tiwari, M. Haney, and P. Jetzer, Astron. Astrophys. 638, A119 (2020).
  107. J. D. Schnittman, Classical Quantum Gravity 28, 094021 (2011).
  108. S. Kawamura et al., Prog. Theor. Exp. Phys. 2021, 05A105 (2021).
  109. M. Punturo et al., Classical Quantum Gravity 27, 194002 (2010).
  110. B. Sathyaprakash et al., Classical Quantum Gravity 29, 124013 (2012); 30, 079501(E) (2013).
  111. A. Abac et al. (ET Collaboration), J. Cosmol. Astropart. Phys. 03 (2026) 081.
  112. D. Reitze et al., Bull. Am. Astron. Soc. 51, 035 (2019).
  113. P. Amaro-Seoane et al., arXiv:1702.00786.
  114. M. Dotti et al., arXiv:2512.21359.
  115. R. P. Geroch, A. Held, and R. Penrose, J. Math. Phys. (N.Y.) 14, 874 (1973).
  116. R. Berens, T. Gravely, and A. Lupsasca, Classical Quantum Gravity 41, 195004 (2024).
  117. J. M. Cohen and L. S. Kegeles, Phys. Rev. D 10, 1070 (1974).
  118. P. L. Chrzanowski, Phys. Rev. D 11, 2042 (1975).
  119. L. S. Kegeles and J. M. Cohen, Phys. Rev. D 19, 1641 (1979).
  120. R. M. Wald, Phys. Rev. Lett. 41, 203 (1978).
  121. A. Le Tiec, M. Casals, and E. Franzin, Phys. Rev. D 103, 084021 (2021).
  122. SFI, https://www.sfi.org.bm/.
  123. SF, https://www.simonsfoundation.org/.
  124. T. Katagiri and V. Cardoso, Generator of tidally distorted rotating black hole solutions (2026), https://the-center-of-gravity.com/data-and-routines/.
  125. E. Newman and R. Penrose, J. Math. Phys. (N.Y.) 3, 566 (1962).
  126. S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford Univ. Press, Oxford, 1985).
  127. W. Kinnersley, J. Math. Phys. (N.Y.) 10, 1195 (1969).
  128. K. S. Thorne and J. B. Hartle, Phys. Rev. D 31, 1815 (1985).
  129. E. Berti and K. D. Kokkotas, Phys. Rev. D 71, 124008 (2005).
  130. J. N. Goldberg, A. J. MacFarlane, E. T. Newman, F. Rohrlich, and E. C. G. Sudarshan, J. Math. Phys. (N.Y.) 8, 2155 (1967).
  131. S. A. Teukolsky, Astrophys. J. 185, 635 (1973).
  132. S. A. Teukolsky and W. H. Press, Astrophys. J. 193, 443 (1974).
  133. N. Yunes and J. Gonzalez, Phys. Rev. D 73, 024010 (2006); 89, 089902(E) (2014).
  134. J. M. Stewart, Proc. R. Soc. A 367, 527 (1979).
  135. B. F. Whiting and L. R. Price, Classical Quantum Gravity 22, S589 (2005).
  136. A. Ori, Phys. Rev. D 67, 124010 (2003).
  137. T. S. Keidl, A. G. Shah, J. L. Friedman, D.-H. Kim, and L. R. Price, Phys. Rev. D 82, 124012 (2010); 90, 109902(E) (2014).
  138. C. O. Lousto and B. F. Whiting, Phys. Rev. D 66, 024026 (2002).
  139. T. S. Keidl, J. L. Friedman, and A. G. Wiseman, Phys. Rev. D 75, 124009 (2007).
  140. T. Regge and J. A. Wheeler, Phys. Rev. 108, 1063 (1957).

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Phys. Rev. D 113, 104036– Published 13 May, 2026
  • Received 25 January 2026
  • Accepted 20 April 2026
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