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

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access
  • Access by Xinjiang University

Analytic black hole perturbation theory package for gravitational scattering amplitudes

Jovan Markovic1,2,* and Mikhail M. Ivanov3,†

  • *Contact author: jm2725@cam.ac.uk
  • †Contact author: ivanov99@mit.edu
PDF

Phys. Rev. D 114, 024002 – Published 1 July, 2026

DOI: https://doi.org/10.1103/pjrx-p7j6

Abstract

Applications of effective field theory (EFT) and scattering amplitudes to gravitational problems have recently produced many unique results that advanced our understanding of the dynamics of compact binaries. Many of these results were made possible by comparing gravitational scattering amplitudes in EFT with exact expressions from general relativity. However, the latter expressions are not easily available as they require intricate solution techniques for the Teukolsky master equation, such as the Mano-Suzuki-Takasugi (MST) method. In this paper, we develop and present the first public package that enables computations of gravitational scattering amplitudes in black hole perturbation theory within the MST framework. Our package supports both analytic computations to a given post-Minkowskian (PM) order in the low-frequency limit and exact numerical computations for an arbitrary frequency of the perturbing field. As an application, we compute scattering phase shifts and inelasticity parameters for massless spin—0, 1, and 2 fields resulting from scattering off a rotating Kerr black hole through the third PM order and compare these results with the exact numerical solutions.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (79)

  1. J. Aasi et al. (LIGO Scientific Collaboration), Classical Quantum Gravity 32, 074001 (2015).
  2. F. Acernese et al. (Virgo Collaboration), Classical Quantum Gravity 32, 024001 (2015).
  3. B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), Phys. Rev. Lett. 116, 061102 (2016).
  4. B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), Phys. Rev. X 9, 031040 (2019).
  5. R. Abbott et al. (LIGO Scientific and Virgo Collaborations), Phys. Rev. X 11, 021053 (2021).
  6. R. Abbott et al. (LIGO Scientific and Virgo Collaborations), Phys. Rev. D 109, 022001 (2024).
  7. R. Abbott et al. (LIGO Scientific, Virgo, and KAGRA Collaborations), Phys. Rev. X 13, 041039 (2023).
  8. T. Akutsu et al. (KAGRA Collaboration), Prog. Theor. Exp. Phys. 2021, 05A102 (2021).
  9. A. G. Abac et al. (LIGO Scientific, Virgo, and KAGRA Collaborations), arXiv:2508.18079.
  10. W. D. Goldberger and I. Z. Rothstein, Phys. Rev. D 73, 104029 (2006).
  11. W. D. Goldberger and I. Z. Rothstein, Phys. Rev. D 73, 104030 (2006).
  12. R. A. Porto, Phys. Rev. D 77, 064026 (2008).
  13. I. Z. Rothstein, Gen. Relativ. Gravit. 46, 1726 (2014).
  14. W. D. Goldberger, arXiv:2212.06677.
  15. W. D. Goldberger, in 2022 Snowmass Summer Study (2022), arXiv:2206.14249.
  16. R. A. Porto, Phys. Rep. 633, 1 (2016).
  17. G. Kälin and R. A. Porto, J. High Energy Phys. 11 (2020) 106.
  18. G. Mogull, J. Plefka, and J. Steinhoff, J. High Energy Phys. 02 (2021) 048.
  19. C. Cheung, I. Z. Rothstein, and M. P. Solon, Phys. Rev. Lett. 121, 251101 (2018).
  20. D. A. Kosower, B. Maybee, and D. O’Connell, J. High Energy Phys. 02 (2019) 137.
  21. Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M. P. Solon, and M. Zeng, J. High Energy Phys. 10 (2019) 206.
  22. Z. Bern, A. Luna, R. Roiban, C.-H. Shen, and M. Zeng, Phys. Rev. D 104, 065014 (2021).
  23. A. Buonanno, M. Khalil, D. O’Connell, R. Roiban, M. P. Solon, and M. Zeng, in Snowmass 2021 (2022), arXiv:2204.05194.
  24. C. Cheung, J. Parra-Martinez, I. Z. Rothstein, N. Shah, and J. Wilson-Gerow, Phys. Rev. Lett. 132, 091402 (2024).
  25. D. Kosmopoulos and M. P. Solon, J. High Energy Phys. 03 (2024) 125.
  26. N. E. J. Bjerrum-Bohr, P. H. Damgaard, L. Plante, and P. Vanhove, J. Phys. A 55, 443014 (2022).
  27. Y. F. Bautista, G. Bonelli, C. Iossa, A. Tanzini, and Z. Zhou, Phys. Rev. D 109, 084071 (2024).
  28. A. Brandhuber, G. R. Brown, G. Chen, S. De Angelis, J. Gowdy, and G. Travaglini, J. High Energy Phys. 06 (2023) 048.
  29. A. Herderschee, R. Roiban, and F. Teng, J. High Energy Phys. 06 (2023) 004.
  30. A. Elkhidir, D. O’Connell, M. Sergola, and I. A. Vazquez-Holm, J. High Energy Phys. 07 (2024) 272.
  31. A. Georgoudis, C. Heissenberg, and I. Vazquez-Holm, J. High Energy Phys. 06 (2023) 126.
  32. S. Caron-Huot, M. Giroux, H. S. Hannesdottir, and S. Mizera, J. High Energy Phys. 01 (2024) 139.
  33. W. D. Goldberger, in Les Houches Summer School—Session 86: Particle Physics and Cosmology: The Fabric of Spacetime (2007), arXiv:hep-ph/0701129.
  34. P. Charalambous, S. Dubovsky, and M. M. Ivanov, J. High Energy Phys. 05 (2021) 038.
  35. P. Charalambous, S. Dubovsky, and M. M. Ivanov, Phys. Rev. Lett. 127, 101101 (2021).
  36. P. Charalambous, S. Dubovsky, and M. M. Ivanov, J. High Energy Phys. 10 (2022) 175.
  37. P. Charalambous and M. M. Ivanov, J. High Energy Phys. 07 (2023) 222.
  38. P. Charalambous, S. Dubovsky, and M. M. Ivanov, J. High Energy Phys. 06 (2025) 180.
  39. L. Hui, A. Joyce, R. Penco, L. Santoni, and A. R. Solomon, J. Cosmol. Astropart. Phys. 04 (2021) 052.
  40. L. Hui, A. Joyce, R. Penco, L. Santoni, and A. R. Solomon, J. Cosmol. Astropart. Phys. 01 (2022) 032.
  41. L. Hui, A. Joyce, R. Penco, L. Santoni, and A. R. Solomon, J. High Energy Phys. 09 (2022) 049.
  42. M. M. Ivanov and Z. Zhou, arXiv:2208.08459.
  43. M. M. Ivanov and Z. Zhou, Phys. Rev. D 107, 084030 (2023).
  44. M. M. Ivanov and Z. Zhou, Phys. Rev. Lett. 130, 091403 (2023).
  45. M. V. S. Saketh, Z. Zhou, and M. M. Ivanov, Phys. Rev. D 109, 064058 (2024).
  46. M. M. Ivanov, Y.-Z. Li, J. Parra-Martinez, and Z. Zhou, Phys. Rev. Lett. 132, 131401 (2024).
  47. M. M. Ivanov, Y.-Z. Li, J. Parra-Martinez, and Z. Zhou, Phys. Rev. Lett. 135, 141401 (2025).
  48. R. A. Matzner, J. Math. Phys. (N.Y.) 9, 163 (1968).
  49. R. A. Matzner and M. P. Ryan Jr., Astrophys. J. 36, 451 (1978).
  50. R. A. Matzner and M. P. Ryan, Phys. Rev. D 16, 1636 (1977).
  51. S. Teukolsky, Phys. Rev. Lett. 29, 1114 (1972).
  52. S. A. Teukolsky, Astrophys. J. 185, 635 (1973).
  53. S. A. Teukolsky and W. Press, Astrophys. J. 193, 443 (1974).
  54. S. Mano and E. Takasugi, Prog. Theor. Phys. 97, 213 (1997).
  55. S. Mano, H. Suzuki, and E. Takasugi, Prog. Theor. Phys. 96, 549 (1996).
  56. S. Mano, H. Suzuki, and E. Takasugi, Prog. Theor. Phys. 95, 1079 (1996).
  57. M. Sasaki and H. Tagoshi, Living Rev. Relativity 6, 6 (2003).
  58. G. Bonelli, C. Iossa, D. P. Lichtig, and A. Tanzini, Phys. Rev. D 105, 044047 (2022).
  59. G. Bonelli, C. Iossa, D. Panea Lichtig, and A. Tanzini, Commun. Math. Phys. 397, 635 (2023).
  60. A. Castro, J. M. Lapan, A. Maloney, and M. J. Rodriguez, Phys. Rev. D 88, 044003 (2013).
  61. A. Castro, J. M. Lapan, A. Maloney, and M. J. Rodriguez, Classical Quantum Gravity 30, 165005 (2013).
  62. Z. Nasipak, Classical Quantum Gravity 42, 165001 (2025).
  63. S. R. Dolan, Classical Quantum Gravity 25, 235002 (2008).
  64. Y. F. Bautista, A. Guevara, C. Kavanagh, and J. Vines, J. High Energy Phys. 05 (2023) 211.
  65. S. Caron-Huot, M. Correia, G. Isabella, and M. Solon, Phys. Rev. Lett. 135, 191601 (2025).
  66. Black Hole Perturbation Toolkit, https://bhptoolkit.org/.
  67. E. Newman and R. Penrose, J. Math. Phys. (N.Y.) 3, 566 (1962).
  68. E. Newman and R. Penrose, J. Math. Phys. (N.Y.) 4, 998 (1963).
  69. E. Berti, V. Cardoso, and M. Casals, Phys. Rev. D 73, 024013 (2006).
  70. http://bhptoolkit.org/bhptoolkit.org.
  71. M. Casals and R. Teixeira Da Costa, Classical Quantum Gravity 38, 165016 (2021).
  72. R. Fujita and H. Tagoshi, Prog. Theor. Phys. 113, 1165 (2005).
  73. M. Casals, C. Kavanagh, and A. C. Ottewill, Phys. Rev. D 94, 124053 (2016).
  74. M. M. Ivanov, Y.-Z. Li, J. Parra-Martinez, and Z. Zhou, arXiv:2602.06951.
  75. K. Cunningham, C. Kavanagh, A. Pound, D. Trestini, N. Warburton, and J. Neef, Classical Quantum Gravity 42, 135009 (2025); 42, 199401(A) (2025).
  76. J. C. Castillo, C. R. Evans, C. Kavanagh, J. Neef, A. Ottewill, and B. Wardell, Phys. Rev. D 111, 084004 (2025).
  77. J. C. Castillo, C. R. Evans, C. Kavanagh, J. Neef, B. Wardell, and A. Ottewill, Phys. Rev. D 112, 104001 (2025).
  78. J. Markovic and M. Ivanov, Jovan888777/bh-scattering-phase: Analytic black hole perturbation theory package for gravitational scattering amplitudes (2026).
  79. https://github.com/Jovan888777/bh-scattering-phase/tree/main.

Outline

Information

Phys. Rev. D 114, 024002– Published 1 July, 2026
  • Received 16 December 2025
  • Accepted 28 May 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation