- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review D

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Horizon multipole moments of a Kerr black hole

Eric Gourgoulhon1,2,*, Alexandre Le Tiec1,3,†, and Marc Casals4,5,6,‡

  • *Contact author: eric.gourgoulhon@obspm.fr
  • †Contact author: alexandre.letiec@obspm.fr
  • ‡Contact author: marc.casals@uni-leipzig.de
PDF

Phys. Rev. D 114, 024014 – Published 6 July, 2026

DOI: https://doi.org/10.1103/rgbl-yk9n

Abstract

The horizon multipole moments of a Kerr black hole are computed from two distinct definitions that have been proposed in the literature. The first one [Ashtekar et al., Classical Quantum Gravity 21, 2549 (2004)] regards axisymmetric isolated horizons, while the second one [Ashtekar et al., J. High Energy Phys. 01 (2022) 028] applies to generic (i.e., not necessarily axisymmetric) nonexpanding horizons. We review these definitions in a common frame and perform a detailed study of the resulting multipole moments for the Kerr event horizon. The horizon multipoles are found to share several properties with the (Hansen) field multipoles, including parity constraints and the leading scaling behavior with respect to the Kerr spin parameter a in the regime of small a. For the axisymmetry-based definition, we have obtained a closed-form expression of the multipole moments in terms of a and the spherical harmonic degree ℓ. For the generic definition, we have established closed-form expressions for the conformal unit round metric, the “electric” and “magnetic” potentials related to the multipoles, and the values of the multipoles in the small a limit. We show that the two definitions lead to different values of the Kerr horizon multipoles as soon as ℓ⩾1 (generic nonzero value of a) or ℓ⩾2 (small a limit).

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (84)

  1. J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley, New York, 1998).
  2. E. Poisson and C. M. Will, Gravity: Newtonian, Post-Newtonian, Relativistic (Cambridge University Press, Cambridge, England, 2014).
  3. R. P. Geroch, Multipole moments. I. Flat space, J. Math. Phys. (N.Y.) 11, 1955 (1970).
  4. R. P. Geroch, Multipole moments. II. Curved space, J. Math. Phys. (N.Y.) 11, 2580 (1970).
  5. R. O. Hansen, Multipole moments of stationary space-times, J. Math. Phys. (N.Y.) 15, 46 (1974).
  6. K. S. Thorne, Multipole expansions of gravitational radiation, Rev. Mod. Phys. 52, 299 (1980).
  7. R. Beig and W. Simon, On the multipole expansion for stationary space-times, Proc. R. Soc. A 376, 333 (1981).
  8. W. Simon and R. Beig, The multipole structure of stationary space-times, J. Math. Phys. (N.Y.) 24, 1163 (1983).
  9. Y. Gürsel, Multipole moments for stationary systems: The equivalence of the Geroch-Hansen formulation and the Thorne formulation, Gen. Relativ. Gravit. 15, 737 (1983).
  10. G. Fodor, C. Hoenselaers, and Z. Perjés, Multipole moments of axisymmetric systems in relativity, J. Math. Phys. (N.Y.) 30, 2252 (1989).
  11. W. G. Dixon, A covariant multipole formalism for extended test bodies in general relativity, Nuovo Cimento (1955–1965) 34, 317 (1964).
  12. W. G. Dixon, Dynamics of extended bodies in general relativity I. Momentum and angular momentum, Proc. R. Soc. A 314, 499 (1970).
  13. W. G. Dixon, The definition of multipole moments for extended bodies, Gen. Relativ. Gravit. 4, 199 (1973).
  14. W. G. Dixon, Dynamics of extended bodies in general relativity III. Equations of motion, Phil. Trans. R. Soc. A 277, 59 (1974).
  15. W. Dixon, Extended bodies in general relativity: Their description and motion, in Isolated Gravitating Systems in General Relativity, Proceedings of the International School of Physics Enrico Fermi, Vol. 67, edited by J. Ehlers (North-Holland, Amsterdam, 1979), p. 156.
  16. A. I. Harte, Mechanics of extended masses in general relativity, Classical Quantum Gravity 29, 055012 (2012).
  17. A. I. Harte, Motion in classical field theories and the foundations of the self-force problem, Fund. Theor. Phys. 179, 327 (2015).
  18. A. Ashtekar, J. Engle, T. Pawlowski, and C. Van Den Broeck, Multipole moments of isolated horizons, Classical Quantum Gravity 21, 2549 (2004).
  19. E. Schnetter, B. Krishnan, and F. Beyer, Introduction to dynamical horizons in numerical relativity, Phys. Rev. D 74, 024028 (2006).
  20. R. Owen, The final remnant of binary black hole mergers: Multipolar analysis, Phys. Rev. D 80, 084012 (2009).
  21. A. Ashtekar, M. Campiglia, and S. Shah, Dynamical black holes: Approach to the final state, Phys. Rev. D 88, 064045 (2013).
  22. A. Ashtekar, N. Khera, M. Kolanowski, and J. Lewandowski, Non-expanding horizons: Multipoles and the symmetry group, J. High Energy Phys. 01 (2022) 028.
  23. N. Vasset, J. Novak, and J. L. Jaramillo, Excised black hole spacetimes: Quasilocal horizon formalism applied to the Kerr example, Phys. Rev. D 79, 124010 (2009).
  24. V. Prasad, Generalized source multipole moments of dynamical horizons in binary black hole mergers, arXiv:2109.01193.
  25. V. Prasad, A. Gupta, S. Bose, and B. Krishnan, Tidal deformation of dynamical horizons in binary black hole mergers, Phys. Rev. D 105, 044019 (2022).
  26. V. Prasad, Tidal deformation of dynamical horizons in binary black hole mergers and its imprint on gravitational radiation, Phys. Rev. D 109, 044033 (2024).
  27. A. Ribes Metidieri, B. Bonga, and B. Krishnan, Tidal deformations of slowly spinning isolated horizons, Phys. Rev. D 110, 024069 (2024).
  28. L. Rezzolla, R. P. Macedo, and J. L. Jaramillo, Understanding the “Antikick” in the Merger of Binary Black Holes, Phys. Rev. Lett. 104, 221101 (2010).
  29. J. L. Jaramillo, R. P. Macedo, P. Moesta, and L. Rezzolla, Black-hole horizons as probes of black-hole dynamics. I. Post-merger recoil in head-on collisions, Phys. Rev. D 85, 084030 (2012), arXiv:1108.0060 [gr-qc].
  30. D. Pook-Kolb, O. Birnholtz, J. L. Jaramillo, B. Krishnan, and E. Schnetter, Horizons in a binary black hole merger II: Fluxes, multipole moments and stability, arXiv:2006.03940.
  31. P. Mourier, X. Jiménez Forteza, D. Pook-Kolb, B. Krishnan, and E. Schnetter, Quasinormal modes and their overtones at the common horizon in a binary black hole merger, Phys. Rev. D 103, 044054 (2021).
  32. A. Gupta, B. Krishnan, A. B. Nielsen, and E. Schnetter, Dynamics of marginally trapped surfaces in a binary black hole merger: Growth and approach to equilibrium, Phys. Rev. D 97, 084028 (2018).
  33. Y. Chen et al., Multipole moments on the common horizon in a binary-black-hole simulation, Phys. Rev. D 106, 124045 (2022).
  34. A. Ribes Metidieri, B. Bonga, and B. Krishnan, Black hole tomography: Unveiling black hole horizon dynamics via ringdown observations, Phys. Rev. D 111, 104075 (2025).
  35. A. Ashtekar and B. Krishnan, Quasi-local black hole horizons: recent advances, Living Rev. Relativity 28, 8 (2025).
  36. L. Blanchet, Post-Newtonian theory for gravitational waves, Living Rev. Relativity 27, 4 (2024).
  37. P. T. Chruściel, J. L. Costa, and M. Heusler, Stationary black holes: Uniqueness and beyond, Living Rev. Relativity 15, 7 (2012).
  38. R. M. Wald, General Relativity (University of Chicago Press, Chicago, 1984).
  39. S. W. Hawking, Black holes in general relativity, Commun. Math. Phys. 25, 152 (1972).
  40. P. T. Chrusciel and R. M. Wald, On the topology of stationary black holes, Classical Quantum Gravity 11, L147 (1994).
  41. T. Damour, Quelques propriétés mécaniques, électromagnétiques, thermodynamiques et quantiques des trous noirs, Thèse de Doctorat d’État, Université Paris 6 (1979), available at https://www.ihes.fr/damour/Articles/.
  42. T. Damour, Surface effects in black-hole physics, in Proceedings of the Second Marcel Grossmann Meeting on General Relativity, edited by R. Ruffini (North Holland, Amsterdam, 1982), p. 587.
  43. E. Gourgoulhon and J. L. Jaramillo, A 3+1 perspective on null hypersurfaces and isolated horizons, Phys. Rep. 423, 159 (2006).
  44. P. Hájíček, Exact models of charged black holes. I. Geometry of totally geodesic null hypersurface, Commun. Math. Phys. 34, 37 (1973).
  45. P.Hájíček, Can outside fields destroy black holes?, J. Math. Phys. (N.Y.) 15, 1554 (1974).
  46. A. Ashtekar, S. Fairhurst, and B. Krishnan, Isolated horizons: Hamiltonian evolution and the first law, Phys. Rev. D 62, 104025 (2000).
  47. A. Ashtekar, C. Beetle, and J. Lewandowski, Geometry of generic isolated horizons, Classical Quantum Gravity 19, 1195 (2002).
  48. P. Hájiček, Exact models of charged black holes: I. Geometry of totally geodesic null hypersurface, Commun. Math. Phys. 34, 37 (1973).
  49. P. Szekeres, The gravitational compass, J. Math. Phys. (N.Y.) 6, 1387 (1965).
  50. S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, Oxford, 1983).
  51. A. Ashtekar and S. Bahrami, Asymptotics with a positive cosmological constant. IV. The no-incoming radiation condition, Phys. Rev. D 100, 024042 (2019).
  52. A. Ashtekar, C. Beetle, and J. Lewandowski, Mechanics of rotating isolated horizons, Phys. Rev. D 64, 044016 (2001).
  53. E. Gourgoulhon, Generalized Damour-Navier-Stokes equation applied to trapping horizons, Phys. Rev. D 72, 104007 (2005).
  54. S. A. Hayward, Angular momentum conservation for dynamical black holes, Phys. Rev. D 74, 104013 (2006).
  55. J. L. Jaramillo and E. Gourgoulhon, Mass and angular momentum in general relativity, in Mass and Motion in General Relativity, edited by L. Blanchet, A. Spallicci, and B. Whiting (Springer, Dordrecht, 2011), pp. 87–124.
  56. A. Ashtekar and B. Krishnan, Isolated and dynamical horizons and their applications, Living Rev. Relativity 7, 10 (2004).
  57. M. Korzyński, Quasi-local angular momentum of non-symmetric isolated and dynamical horizons from the conformal decomposition of the metric, Classical Quantum Gravity 24, 5935 (2007).
  58. B. Chow, The Ricci flow on the 2-sphere, J. Diff. Geom. 33, 325 (1991).
  59. H.-P. Gittel, J. Jezierski, J. Kijowski, and S. Łeski, Rigid spheres in Riemannian spaces, Classical Quantum Gravity 30, 175010 (2013).
  60. L. Blanchet and T. Damour, Radiative gravitational fields in general relativity i. General structure of the field outside the source, Phil. Trans. R. Soc. A 320, 379 (1986).
  61. S. A. Teukolsky and W. H. Press, Perturbations of a rotating black hole. III. Interaction of the hole with gravitational and electromagnetic radiation, Astrophys. J. 193, 443 (1974).
  62. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, New York, 1973).
  63. R. H. Boyer and R. W. Lindquist, Maximal analytic extension of the Kerr metric, J. Math. Phys. (N.Y.) 8, 265 (1967).
  64. S. W. Hawking and J. B. Hartle, Energy and angular momentum flow into a black hole, Commun. Math. Phys. 27, 283 (1972).
  65. J. B. Hartle, Tidal shapes and shifts on rotating black holes, Phys. Rev. D 9, 2749 (1974).
  66. E. Poisson, Absorption of mass and angular momentum by a black hole: Time-domain formalisms for gravitational perturbations, and the small-hole or slow-motion approximation, Phys. Rev. D 70, 084044 (2004).
  67. W. Kinnersley, Type D vacuum metrics, J. Math. Phys. (N.Y.) 10, 1195 (1969).
  68. H. Fang and G. Lovelace, Tidal coupling of a Schwarzschild black hole and circularly orbiting moon, Phys. Rev. D 72, 124016 (2005).
  69. L. Smarr, Surface geometry of charged rotating black holes, Phys. Rev. D 7, 289 (1973).
  70. A. Le Tiec, M. Casals, and E. Franzin, Tidal Love numbers of Kerr black holes, Phys. Rev. D 103, 084021 (2021).
  71. A. Le Tiec and M. Casals, Spinning black holes fall in Love, Phys. Rev. Lett. 126, 131102 (2021).
  72. T. Damour and O. M. Lecian, Gravitational polarizability of black holes, Phys. Rev. D 80, 044017 (2009).
  73. P. Landry and E. Poisson, Relativistic theory of surficial Love numbers, Phys. Rev. D 89, 124011 (2014).
  74. https://zenodo.org/records/18511585.
  75. DLMF, NIST Digital Library of Mathematical Functions, https://dlmf.nist.gov, Release 1.2.5 (2025), edited by F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain.
  76. The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org (2026), sequence A032443.
  77. The SageMath Developers, SageMath (2026), https://www.sagemath.org.
  78. E. Gourgoulhon and M. Mancini, Symbolic tensor calculus on manifolds: A SageMath implementation, Les cours du CIRM 6, 1 (2018).
  79. The SageManifolds Developers, SageManifolds (2026), https://sagemanifolds.obspm.fr.
  80. The FLINT team, FLINT: Fast Library for Number Theory (2025), https://flintlib.org.
  81. https://nbviewer.org/url/zenodo.org/records/18511585/files/Kerr_cross_section.ipynb.
  82. https://nbviewer.org/url/zenodo.org/records/18511585/files/Kerr_Hajicek_form_B.ipynb.
  83. https://nbviewer.org/url/zenodo.org/records/18511585/files/Kerr_multipoles.ipynb.
  84. https://nbviewer.org/url/zenodo.org/records/18511585/files/hypergeom_arctan.ipynb.

Outline

Information

Phys. Rev. D 114, 024014– Published 6 July, 2026
  • Received 7 February 2026
  • Accepted 6 April 2026
Crossmark - Check for updates button

© 2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation