- Physical Review Journals
All JournalsPhysics Magazine

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

Black-hole scattering with numerical relativity: Self-force extraction and post-Minkowskian validation

Oliver Long1,*, Harald P. Pfeiffer1, Lawrence E. Kidder2, and Mark A. Scheel3

  • *Contact author: oliver.long@aei.mpg.de
PDF

Phys. Rev. D 112, 124038 – Published 9 December, 2025

DOI: https://doi.org/10.1103/bdsb-sp9c

Abstract

The asymptotic nature of unbound binary-black-hole encounters provides a clean method for comparing different approaches for modeling the two-body problem in general relativity. In this work, we use numerical relativity simulations of black-hole scattering, generated using the Spectral Einstein Code, to explore the self-force and post-Minkowskian expansions of the scattering angle. First, we use a set of unequal-mass simulations to extract the self-force contributions to the scattering angle. Our main result is that using information up to second-order in the symmetric mass ratio (2SF) reproduces numerical relativity within the error bars across the full range of mass-ratios, including equal mass. Next, we compare our numerical relativity results to state-of-the-art post-Minkowskian predictions at larger impact parameters than previously explored. We find good agreement in the weak-field regime and discuss the relative importance of higher order terms.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (65)

  1. Z. Bern, J. Parra-Martinez, R. Roiban, M. S. Ruf, C.-H. Shen, M. P. Solon, and M. Zeng, Phys. Rev. Lett. 126, 171601 (2021).
  2. Z. Bern, J. Parra-Martinez, R. Roiban, M. S. Ruf, C.-H. Shen, M. P. Solon, and M. Zeng, Phys. Rev. Lett. 128, 161103 (2022).
  3. P. H. Damgaard, E. R. Hansen, L. Planté, and P. Vanhove, J. High Energy Phys. 09 (2023) 183.
  4. C. Dlapa, G. Kälin, Z. Liu, and R. A. Porto, Phys. Rev. Lett. 128, 161104 (2022).
  5. C. Dlapa, G. Kälin, Z. Liu, and R. A. Porto, J. High Energy Phys. 08 (2023) 109.
  6. G. U. Jakobsen, G. Mogull, J. Plefka, B. Sauer, and Y. Xu, Phys. Rev. Lett. 131, 151401 (2023).
  7. G. U. Jakobsen, G. Mogull, J. Plefka, and B. Sauer, Phys. Rev. D 109, L041504 (2024).
  8. M. Driesse, G. U. Jakobsen, A. Klemm, G. Mogull, C. Nega, J. Plefka, B. Sauer, and J. Usovitsch, Nature (London) 641, 603 (2025).
  9. M. Driesse, G. U. Jakobsen, G. Mogull, J. Plefka, B. Sauer, and J. Usovitsch, Phys. Rev. Lett. 132, 241402 (2024).
  10. C. Dlapa, G. Kälin, Z. Liu, J. Neef, and R. A. Porto, Phys. Rev. Lett. 130, 101401 (2023).
  11. A. Buonanno, G. Mogull, R. Patil, and L. Pompili, Phys. Rev. Lett. 133, 211402 (2024).
  12. T. Damour, A. Nagar, A. Placidi, and P. Rettegno, arXiv:2503.05487.
  13. T. Damour and P. Rettegno, Phys. Rev. D 107, 064051 (2023).
  14. A. Buonanno, G. U. Jakobsen, and G. Mogull, Phys. Rev. D 110, 044038 (2024).
  15. O. Long, H. P. Pfeiffer, A. Buonanno, G. U. Jakobsen, G. Mogull, A. Ramos-Buades, H. R. Rüter, L. E. Kidder, and M. A. Scheel, arXiv:2507.08071.
  16. A. Clark and G. Pratten, arXiv:2508.12126.
  17. G. Kälin and R. A. Porto, J. High Energy Phys. 01 (2020) 072.
  18. O. Long, C. Whittall, and L. Barack, Phys. Rev. D 110, 044039 (2024).
  19. B. Wardell, A. Pound, N. Warburton, J. Miller, L. Durkan, and A. Le Tiec, Phys. Rev. Lett. 130, 241402 (2023).
  20. N. Warburton, A. Pound, B. Wardell, J. Miller, and L. Durkan, Phys. Rev. Lett. 127, 151102 (2021).
  21. J. Mathews, B. Wardell, A. Pound, and N. Warburton, arXiv:2510.16113.
  22. L. Barack and O. Long, Phys. Rev. D 106, 104031 (2022).
  23. C. Whittall and L. Barack, Phys. Rev. D 108, 064017 (2023).
  24. C. Whittall, L. Barack, and O. Long, arXiv:2509.19439.
  25. L. Barack et al., Phys. Rev. D 108, 024025 (2023).
  26. T. Damour, F. Guercilena, I. Hinder, S. Hopper, A. Nagar, and L. Rezzolla, Phys. Rev. D 89, 081503 (2014).
  27. S. Hopper, A. Nagar, and P. Rettegno, Phys. Rev. D 107, 124034 (2023).
  28. P. Rettegno, G. Pratten, L. M. Thomas, P. Schmidt, and T. Damour, Phys. Rev. D 108, 124016 (2023).
  29. S. Albanesi, A. Rashti, F. Zappa, R. Gamba, W. Cook, B. Daszuta, S. Bernuzzi, A. Nagar, and D. Radice, Phys. Rev. D 111, 024069 (2025).
  30. S. Swain, G. Pratten, and P. Schmidt, Phys. Rev. D 111, 064048 (2025).
  31. J. Fontbuté, S. Bernuzzi, P. Rettegno, S. Albanesi, and W. Tichy, arXiv:2506.11204.
  32. F. Pretorius and D. Khurana, Classical Quantum Gravity 24, S83 (2007).
  33. U. Sperhake, V. Cardoso, F. Pretorius, E. Berti, and J. A. Gonzalez, Phys. Rev. Lett. 101, 161101 (2008).
  34. H. Witek, M. Zilhao, L. Gualtieri, V. Cardoso, C. Herdeiro, A. Nerozzi, and U. Sperhake, Phys. Rev. D 82, 104014 (2010).
  35. U. Sperhake, E. Berti, V. Cardoso, F. Pretorius, and N. Yunes, Phys. Rev. D 83, 024037 (2011).
  36. U. Sperhake, E. Berti, V. Cardoso, and F. Pretorius, Phys. Rev. Lett. 111, 041101 (2013).
  37. U. Sperhake, E. Berti, V. Cardoso, and F. Pretorius, Phys. Rev. D 93, 044012 (2016).
  38. A. Kankani and S. T. McWilliams, Phys. Rev. D 110, 064033 (2024).
  39. P. E. Nelson, Z. B. Etienne, S. T. McWilliams, and V. Nguyen, Phys. Rev. D 100, 124045 (2019).
  40. S. Jaraba and J. Garcia-Bellido, Phys. Dark Universe 34, 100882 (2021).
  41. J. L. Rodríguez-Monteverde, S. Jaraba, and J. García-Bellido, Phys. Dark Universe 47, 101776 (2025).
  42. Y.-B. Bae, Y.-H. Hyun, and G. Kang, Phys. Rev. Lett. 132, 261401 (2024).
  43. J. Fontbuté, T. Andrade, R. Luna, J. Calderón Bustillo, G. Morrás, S. Jaraba, J. García-Bellido, and G. L. Izquierdo, Phys. Rev. D 111, 044024 (2025).
  44. H. Kogan, F. C. L. Pardoe, and H. Witek, arXiv:2511.00307.
  45. https://www.black-holes.org/SpEC.html.
  46. A. H. Mroue et al., Phys. Rev. Lett. 111, 241104 (2013).
  47. M. Boyle et al., Classical Quantum Gravity 36, 195006 (2019).
  48. M. A. Scheel et al., Classical Quantum Gravity 42, 195017 (2025).
  49. L. Lindblom, M. A. Scheel, L. E. Kidder, R. Owen, and O. Rinne, Classical Quantum Gravity 23, S447 (2006).
  50. L. E. Kidder and L. S. Finn, Phys. Rev. D 62, 084026 (2000).
  51. M. A. Scheel, M. Boyle, T. Chu, L. E. Kidder, K. D. Matthews, and H. P. Pfeiffer, Phys. Rev. D 79, 024003 (2009).
  52. B. Szilagyi, L. Lindblom, and M. A. Scheel, Phys. Rev. D 80, 124010 (2009).
  53. D. A. Hemberger, M. A. Scheel, L. E. Kidder, B. Szilágyi, G. Lovelace, N. W. Taylor, and S. A. Teukolsky, Classical Quantum Gravity 30, 115001 (2013).
  54. O. Rinne, Classical Quantum Gravity 23, 6275 (2006).
  55. O. Rinne, L. Lindblom, and M. A. Scheel, Classical Quantum Gravity 24, 4053 (2007).
  56. S. Ossokine, L. E. Kidder, and H. P. Pfeiffer, Phys. Rev. D 88, 084031 (2013).
  57. H. P. Pfeiffer, L. E. Kidder, M. A. Scheel, and S. A. Teukolsky, Comput. Phys. Commun. 152, 253 (2003).
  58. S. Ossokine, F. Foucart, H. P. Pfeiffer, M. Boyle, and B. Szilágyi, Classical Quantum Gravity 32, 245010 (2015).
  59. J. W. York, Jr., Phys. Rev. Lett. 82, 1350 (1999).
  60. H. P. Pfeiffer and J. W. York, Jr., Phys. Rev. D 67, 044022 (2003).
  61. G. B. Cook and H. P. Pfeiffer, Phys. Rev. D 70, 104016 (2004).
  62. I. B. Mendes, N. L. Vu, O. Long, H. P. Pfeiffer, and R. Owen, arXiv:2509.07291.
  63. T. Damour, Phys. Rev. D 102, 024060 (2020).
  64. Black Hole Perturbation Toolkit (bhptoolkit.org).
  65. O. Long, H. P. Pfeiffer, L. E. Kidder, and M. A. Scheel, , v1 https://arxiv.org/src/2511.10196v1/anc/ScatteringAngles.dat (2025).

Outline

Information

Phys. Rev. D 112, 124038– Published 9 December, 2025
  • Received 18 November 2025
  • Accepted 20 November 2025
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. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation