- 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

Gravitational radiation from hyperbolic orbits: Comparison between self-force, post-Minkowskian, post-Newtonian, and numerical relativity results

Niels Warburton

PDF

Phys. Rev. D 113, 084059 – Published 28 April, 2026

DOI: https://doi.org/10.1103/v6rz-4f6k

Abstract

In this work I use a frequency-domain Regge-Wheeler-Zerilli approach to compute the gravitational-wave energy radiated by a compact body moving along a hyperbolic or parabolic geodesic of a Schwarzschild black hole. I compare my results with the latest post-Minkowskian (PM) calculations for the radiated energy and find agreement for hyperbolic orbits with large impact parameters and characterized by a velocity at infinity, v∞, as large as v∞/c=0.7. I also find agreement between my results and the leading-order PM expansion for the radiation absorbed by the black hole. I make further comparisons with post-Newtonian (PN) theory and show the effectiveness of a simple PN-PM hybrid model. Finally, I make a first comparison of the radiated energy between self-force and numerical relativity.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (124)

  1. A. G. Abac et al. (LIGO Scientific Collaboration, VIRGO Collaboration, and KAGRA Collaboration), GWTC-4.0: Updating the gravitational-wave transient catalog with observations from the first part of the fourth LIGO-Virgo-KAGRA observing run, arXiv:2508.18082.
  2. C. P. L. Berry and J. R. Gair, Observing the galaxy’s massive black hole with gravitational wave bursts, Mon. Not. R. Astron. Soc. 429, 589 (2013).
  3. D. J. Oliver, A. D. Johnson, J. Berrier, K. Glampedakis, and D. Kennefick, Gravitational wave peeps from EMRIs and their implication for LISA signal confusion noise, Classical Quantum Gravity 41, 115004 (2024).
  4. C. Henshaw, J. Lange, P. Lott, R. O’Shaughnessy, and L. Cadonati, Parameter estimation of gravitational waves from hyperbolic black hole encounters, arXiv:2507.01156.
  5. A. Buonanno and T. Damour, Effective one-body approach to general relativistic two-body dynamics, Phys. Rev. D 59, 084006 (1999).
  6. L. Pompili et al., Laying the foundation of the effective-one-body waveform models SEOBNRv5: Improved accuracy and efficiency for spinning nonprecessing binary black holes, Phys. Rev. D 108, 124035 (2023).
  7. G. Riemenschneider, P. Rettegno, M. Breschi, A. Albertini, R. Gamba, S. Bernuzzi, and A. Nagar, Assessment of consistent next-to-quasicircular corrections and postadiabatic approximation in effective-one-body multipolar waveforms for binary black hole coalescences, Phys. Rev. D 104, 104045 (2021).
  8. M. Boyle, D. A. Brown, L. E. Kidder, A. H. Mroue, H. P. Pfeiffer, M. A. Scheel, G. B. Cook, and S. A. Teukolsky, High-accuracy comparison of numerical relativity simulations with post-Newtonian expansions, Phys. Rev. D 76, 124038 (2007).
  9. M. Hannam, S. Husa, B. Bruegmann, and A. Gopakumar, Comparison between numerical-relativity and post-Newtonian waveforms from spinning binaries: The orbital hang-up case, Phys. Rev. D 78, 104007 (2008).
  10. S. L. Detweiler, A Consequence of the gravitational self-force for circular orbits of the Schwarzschild geometry, Phys. Rev. D 77, 124026 (2008).
  11. L. Blanchet, S. L. Detweiler, A. Le Tiec, and B. F. Whiting, Post-Newtonian and numerical calculations of the gravitational self-force for circular orbits in the Schwarzschild geometry, Phys. Rev. D 81, 064004 (2010).
  12. M. Favata, Conservative self-force correction to the innermost stable circular orbit: Comparison with multiple post-Newtonian-based methods, Phys. Rev. D 83, 024027 (2011).
  13. A. Le Tiec, A. H. Mroue, L. Barack, A. Buonanno, H. P. Pfeiffer, N. Sago, and A. Taracchini, Periastron advance in black hole binaries, Phys. Rev. Lett. 107, 141101 (2011).
  14. S. R. Dolan, N. Warburton, A. I. Harte, A. Le Tiec, B. Wardell, and L. Barack, Gravitational self-torque and spin precession in compact binaries, Phys. Rev. D 89, 064011 (2014).
  15. A. Le Tiec, The overlap of numerical relativity, perturbation theory and post-Newtonian theory in the binary black hole problem, Int. J. Mod. Phys. D 23, 1430022 (2014).
  16. S. R. Dolan, P. Nolan, A. C. Ottewill, N. Warburton, and B. Wardell, Tidal invariants for compact binaries on quasicircular orbits, Phys. Rev. D 91, 023009 (2015).
  17. S. Akcay, A. Le Tiec, L. Barack, N. Sago, and N. Warburton, Comparison between self-force and post-Newtonian dynamics: Beyond circular orbits, Phys. Rev. D 91, 124014 (2015).
  18. A. Antonelli, M. van de Meent, A. Buonanno, J. Steinhoff, and J. Vines, Quasicircular inspirals and plunges from nonspinning effective-one-body Hamiltonians with gravitational self-force information, Phys. Rev. D 101, 024024 (2020).
  19. A. Antonelli, A. Buonanno, J. Steinhoff, M. van de Meent, and J. Vines, Energetics of two-body Hamiltonians in post-Minkowskian gravity, Phys. Rev. D 99, 104004 (2019).
  20. S. Borhanian, K. G. Arun, H. P. Pfeiffer, and B. S. Sathyaprakash, Comparison of post-Newtonian mode amplitudes with numerical relativity simulations of binary black holes, Classical Quantum Gravity 37, 065006 (2020).
  21. M. van de Meent, A. Buonanno, D. P. Mihaylov, S. Ossokine, L. Pompili, N. Warburton, A. Pound, B. Wardell, L. Durkan, and J. Miller, Enhancing the SEOBNRv5 effective-one-body waveform model with second-order gravitational self-force fluxes, Phys. Rev. D 108, 124038 (2023).
  22. B. Leather, A. Buonanno, and M. van de Meent, Inspiral-merger-ringdown waveforms with gravitational self-force results within the effective-one-body formalism, Phys. Rev. D 112, 044012 (2025).
  23. T. Damour, Gravitational scattering, post-Minkowskian approximation and effective one-body theory, Phys. Rev. D 94, 104015 (2016).
  24. T. Damour, High-energy gravitational scattering and the general relativistic two-body problem, Phys. Rev. D 97, 044038 (2018).
  25. N. E. J. Bjerrum-Bohr, P. H. Damgaard, L. Plante, and P. Vanhove, The SAGEX review on scattering amplitudes Chapter 13: Post-Minkowskian expansion from scattering amplitudes, J. Phys. A 55, 443014 (2022).
  26. D. A. Kosower, R. Monteiro, and D. O’Connell, The SAGEX review on scattering amplitudes Chapter 14: Classical gravity from scattering amplitudes, J. Phys. A 55, 443015 (2022).
  27. A. Buonanno, M. Khalil, D. O’Connell, R. Roiban, M. P. Solon, and M. Zeng, Snowmass White paper: Gravitational waves and scattering amplitudes, in Snowmass 2021 (2022).
  28. M. Driesse, G. U. Jakobsen, A. Klemm, G. Mogull, C. Nega, J. Plefka, B. Sauer, and J. Usovitsch, Emergence of Calabi–Yau manifolds in high-precision black-hole scattering, Nature (London) 641, 603 (2025).
  29. L. Blanchet and G. Schaefer, Higher order gravitational radiation losses in binary systems, Mon. Not. R. Astron. Soc. 239, 845 (1989); 242, 704(E) (1990).
  30. D. Bini, T. Damour, and A. Geralico, Sixth post-Newtonian nonlocal-in-time dynamics of binary systems, Phys. Rev. D 102, 084047 (2020).
  31. G. Cho, Third post-Newtonian gravitational radiation from two-body scattering. II. Hereditary energy radiation, Phys. Rev. D 105, 104035 (2022).
  32. E. Herrmann, J. Parra-Martinez, M. S. Ruf, and M. Zeng, Gravitational bremsstrahlung from reverse unitarity, Phys. Rev. Lett. 126, 201602 (2021).
  33. C. Dlapa, G. Kälin, Z. Liu, J. Neef, and R. A. Porto, Radiation reaction and gravitational waves at fourth post-Minkowskian order, Phys. Rev. Lett. 130, 101401 (2023).
  34. P. H. Damgaard, E. R. Hansen, L. Planté, and P. Vanhove, Classical observables from the exponential representation of the gravitational S-matrix, J. High Energy Phys. 09 (2023) 183.
  35. W. D. Goldberger and I. Z. Rothstein, Horizon radiation reaction forces, J. High Energy Phys. 10 (2020) 026.
  36. C. R. T. Jones and M. S. Ruf, Absorptive effects and classical black hole scattering, J. High Energy Phys. 03 (2024) 015.
  37. G. Cho, S. Dandapat, and A. Gopakumar, Third order post-Newtonian gravitational radiation from two-body scattering: Instantaneous energy and angular momentum radiation, Phys. Rev. D 105, 084018 (2022).
  38. D. Bini, A. Geralico, C. Kavanagh, A. Pound, and D. Usseglio, Post-Minkowskian self-force in the low-velocity limit: Scalar field scattering, Phys. Rev. D 110, 064050 (2024).
  39. A. Geralico, Scattering of a point mass by a Schwarzschild black hole: Radiated energy and angular momentum, Phys. Rev. D 112, 064088 (2025).
  40. G. Kälin and R. A. Porto, From boundary data to bound states, J. High Energy Phys. 01 (2020) 072.
  41. G. Kälin and R. A. Porto, From boundary data to bound states. Part II. Scattering angle to dynamical invariants (with twist), J. High Energy Phys. 02 (2020) 120.
  42. G. Cho, G. Kälin, and R. A. Porto, From boundary data to bound states. Part III. Radiative effects, J. High Energy Phys. 04 (2022) 154; 07 (2022) 002(E).
  43. T. Adamo, R. Gonzo, and A. Ilderton, Gravitational bound waveforms from amplitudes, J. High Energy Phys. 05 (2024) 034.
  44. M. Khalil, A. Buonanno, J. Steinhoff, and J. Vines, Energetics and scattering of gravitational two-body systems at fourth post-Minkowskian order, Phys. Rev. D 106, 024042 (2022).
  45. A. Buonanno, G. U. Jakobsen, and G. Mogull, Post-Minkowskian theory meets the spinning effective-one-body approach for two-body scattering, Phys. Rev. D 110, 044038 (2024).
  46. T. Damour, F. Guercilena, I. Hinder, S. Hopper, A. Nagar, and L. Rezzolla, Strong-field scattering of two black holes: Numerics versus analytics, Phys. Rev. D 89, 081503 (2014).
  47. P. Rettegno, G. Pratten, L. M. Thomas, P. Schmidt, and T. Damour, Strong-field scattering of two spinning black holes: Numerical relativity versus post-Minkowskian gravity, Phys. Rev. D 108, 124016 (2023).
  48. O. Long, H. P. Pfeiffer, L. E. Kidder, and M. A. Scheel, Black-hole scattering with numerical relativity: Self-force extraction and post-Minkowskian validation, Phys. Rev. D 112, 124038 (2025).
  49. 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, Highly accurate simulations of asymmetric black-hole scattering and cross validation of effective-one-body models, Phys. Rev. D 112, 124039 (2025).
  50. A. Clark and G. Pratten, Strong field scattering of two black holes: Exploring gauge flexibility, Phys. Rev. D 112, 104074 (2025).
  51. H. Kogan, F. C. L. Pardoe, and H. Witek, Spin-up and mass-gain in hyperbolic encounters of spinning black holes, Phys. Rev. D 113, 064016 (2026).
  52. A. Kankani and S. T. McWilliams, Testing the boundary-to-bound correspondence with numerical relativity, Phys. Rev. D 110, 064033 (2024).
  53. S. Swain, G. Pratten, and P. Schmidt, Strong field scattering of black holes: Assessing resummation strategies, Phys. Rev. D 111, 064048 (2025).
  54. J. Fontbuté, T. Andrade, R. Luna, J. Calderón Bustillo, G. Morrás, S. Jaraba, J. García-Bellido, and G. L. Izquierdo, Numerical-relativity surrogate model for hyperbolic encounters of black holes: Challenges in parameter estimation, Phys. Rev. D 111, 044024 (2025).
  55. P. A. Sundararajan, G. Khanna, and S. A. Hughes, Towards adiabatic waveforms for inspiral into Kerr black holes. I. A New model of the source for the time domain perturbation equation, Phys. Rev. D 76, 104005 (2007).
  56. P. A. Sundararajan, G. Khanna, S. A. Hughes, and S. Drasco, Towards adiabatic waveforms for inspiral into Kerr black holes: II. Dynamical sources and generic orbits, Phys. Rev. D 78, 024022 (2008).
  57. S. E. Field, S. Gottlieb, Z. J. Grant, L. F. Isherwood, and G. Khanna, A GPU-accelerated mixed-precision WENO method for extremal black hole and gravitational wave physics computations, Appl. Math. Comput. 5, 97 (2023).
  58. E. Harms, S. Bernuzzi, A. Nagar, and A. Zenginoglu, A new gravitational wave generation algorithm for particle perturbations of the Kerr spacetime, Classical Quantum Gravity 31, 245004 (2014).
  59. E. Harms, S. Bernuzzi, and B. Brügmann, Numerical solution of the 2+1 Teukolsky equation on a hyperboloidal and horizon penetrating foliation of Kerr and application to late-time decays, Classical Quantum Gravity 30, 115013 (2013).
  60. A. Placidi, S. Albanesi, A. Nagar, M. Orselli, S. Bernuzzi, and G. Grignani, Exploiting Newton-factorized, 2PN-accurate waveform multipoles in effective-one-body models for spin-aligned noncircularized binaries, Phys. Rev. D 105, 104030 (2022).
  61. S. Albanesi, A. Nagar, and S. Bernuzzi, Effective one-body model for extreme-mass-ratio spinning binaries on eccentric equatorial orbits: Testing radiation reaction and waveform, Phys. Rev. D 104, 024067 (2021).
  62. S. Albanesi, A. Nagar, S. Bernuzzi, A. Placidi, and M. Orselli, Assessment of effective-one-body radiation reactions for generic planar orbits, Phys. Rev. D 105, 104031 (2022).
  63. G. Faggioli, M. van de Meent, A. Buonanno, A. Gamboa, M. Khalil, and G. Khanna, Testing eccentric corrections to the radiation-reaction force in the test-mass limit of effective-one-body models, Phys. Rev. D 111, 044036 (2025).
  64. K. Martel, Gravitational wave forms from a point particle orbiting a Schwarzschild black hole, Phys. Rev. D 69, 044025 (2004).
  65. K. I. Oohara and T. Nakamura, Gravitational waves from a particle scattered by a Schwarzschild black hole, Prog. Theor. Phys. 71, 91 (1984).
  66. K.-I. Oohara and T. Nakamura, Gravitational radiation from a particle scattered by a non-rotating black hole, Phys. Lett. A 98, 407 (1983).
  67. Y. Kojima and T. Nakamura, Gravitational radiation from a particle scattered by a Kerr black hole, Prog. Theor. Phys. 72, 494 (1984).
  68. S. Hopper, Unbound motion on a Schwarzschild background: Practical approaches to frequency domain computations, Phys. Rev. D 97, 064007 (2018).
  69. S. Hopper and V. Cardoso, Scattering of point particles by black holes: gravitational radiation, Phys. Rev. D 97, 044031 (2018).
  70. K. I. Oohara and T. Nakamura, Energy, momentum and angular momentum of gravitational waves induced by a particle plunging into a Schwarzschild black hole, Prog. Theor. Phys. 70, 757 (1983).
  71. Y. Kojima and T. Nakamura, Gravitational radiation from a particle with orbital angular momentum plunging into a Kerr black hole, Prog. Theor. Phys. 71, 79 (1984).
  72. E. Berti, V. Cardoso, T. Hinderer, M. Lemos, F. Pretorius, U. Sperhake, and N. Yunes, Semianalytical estimates of scattering thresholds and gravitational radiation in ultrarelativistic black hole encounters, Phys. Rev. D 81, 104048 (2010).
  73. Y. Yin, R. K. L. Lo, and X. Chen, Gravitational radiation from Kerr black holes using the Sasaki-Nakamura formalism: waveforms and fluxes at infinity, arXiv:2511.08673.
  74. A. Pound and B. Wardell, Black Hole Perturbation Theory and Gravitational Self-Force (Springer, Singapore, 2021).
  75. L. Barack and O. Long, Self-force correction to the deflection angle in black-hole scattering: A scalar charge toy model, Phys. Rev. D 106, 104031 (2022).
  76. C. Whittall and L. Barack, Frequency-domain approach to self-force in hyperbolic scattering, Phys. Rev. D 108, 064017 (2023).
  77. C. Whittall, L. Barack, and O. Long, Frequency-domain self-force calculations using Gegenbauer reconstruction, Phys. Rev. D 112, 124045 (2025).
  78. O. Long, C. Whittall, and L. Barack, Black hole scattering near the transition to plunge: Self-force and resummation of post-Minkowskian theory, Phys. Rev. D 110, 044039 (2024).
  79. L. Barack et al., Comparison of post-Minkowskian and self-force expansions: Scattering in a scalar charge toy model, Phys. Rev. D 108, 024025 (2023).
  80. R. O. Hansen, Postnewtonian gravitational radiation from point masses in a hyperbolic kepler orbit, Phys. Rev. D 5, 1021 (1972).
  81. M. Turner, Gravitational radiation from point-masses in unbound orbits: Newtonian results, Astrophys. J. 216, 610 (1977).
  82. D. Bini, T. Damour, and A. Geralico, Radiative contributions to gravitational scattering, Phys. Rev. D 104, 084031 (2021).
  83. K. Alvi, Energy and angular momentum flow into a black hole in a binary, Phys. Rev. D 64, 104020 (2001).
  84. W. Junker and G. Schäfer, Binary systems: higher order gravitational radiation damping and wave emission, Mon. Not. R. Astron. Soc. 254, 146 (1992).
  85. Y. F. Bautista, Y.-T. Huang, and J.-W. Kim, Absorptive effects in black hole scattering, Phys. Rev. D 111, 044043 (2025).
  86. T. Damour, Radiative contribution to classical gravitational scattering at the third order in G, Phys. Rev. D 102, 124008 (2020).
  87. A. V. Manohar, A. K. Ridgway, and C.-H. Shen, Radiated angular momentum and dissipative effects in classical scattering, Phys. Rev. Lett. 129, 121601 (2022).
  88. P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, Classical gravitational observables from the Eikonal operator, Phys. Lett. B 843, 138049 (2023).
  89. C. Heissenberg, Radiation-reaction and angular momentum loss at O(G4), Phys. Rev. D 111, 126012 (2025).
  90. C. Heissenberg and R. Russo, Reconstructing the gravitational waveform from its probe limit, arXiv:2511.13835.
  91. A. Cipriani, F. Fucito, C. Heissenberg, J. F. Morales, and R. Russo, “Waveforms” at the horizon, arXiv:2602.05766.
  92. G. U. Jakobsen, G. Mogull, J. Plefka, and B. Sauer, Dissipative scattering of spinning black holes at fourth post-Minkowskian order, Phys. Rev. Lett. 131, 241402 (2023).
  93. C. Cutler, D. Kennefick, and E. Poisson, Gravitational radiation reaction for bound motion around a Schwarzschild black hole, Phys. Rev. D 50, 3816 (1994).
  94. M. Van De Meent and N. Warburton, Fast self-forced inspirals, Classical Quantum Gravity 35, 144003 (2018).
  95. T. Regge and J. A. Wheeler, Stability of a Schwarzschild singularity, Phys. Rev. 108, 1063 (1957).
  96. F. J. Zerilli, Effective potential for even parity Regge-Wheeler gravitational perturbation equations, Phys. Rev. Lett. 24, 737 (1970).
  97. K. Martel and E. Poisson, Gravitational perturbations of the Schwarzschild spacetime: A Practical covariant and gauge-invariant formalism, Phys. Rev. D 71, 104003 (2005).
  98. V. Moncrief, Gravitational perturbations of spherically symmetric systems. I. The exterior problem, Ann. Phys. (Amsterdam) 88, 323 (1974).
  99. C. T. Cunningham, R. H. Price, and V. Moncrief, Radiation from collapsing relativistic stars. I—Linearized odd-parity radiation, Astrophys. J. 224, 643 (1978).
  100. S. Hopper and C. R. Evans, Gravitational perturbations and metric reconstruction: Method of extended homogeneous solutions applied to eccentric orbits on a Schwarzschild black hole, Phys. Rev. D 82, 084010 (2010).
  101. R. A. Isaacson, Gravitational radiation in the limit of high frequency. II. Nonlinear terms and the effective stress tensor, Phys. Rev. 166, 1272 (1968).
  102. S. E. Field, J. S. Hesthaven, and S. R. Lau, Discontinuous Galerkin method for computing gravitational waveforms from extreme mass ratio binaries, Classical Quantum Gravity 26, 165010 (2009).
  103. L. J. G. Da Silva, R. Panosso Macedo, J. E. Thompson, J. A. V. Kroon, L. Durkan, and O. Long, Hyperboloidal discontinuous time-symmetric numerical algorithm with higher order jumps for gravitational self-force computations in the time domain, arXiv:2306.13153.
  104. O. Long and L. Barack, Time-domain metric reconstruction for hyperbolic scattering, Phys. Rev. D 104, 024014 (2021).
  105. R. Panosso Macedo, B. Leather, N. Warburton, B. Wardell, and A. Zenginoğlu, Hyperboloidal method for frequency-domain self-force calculations, Phys. Rev. D 105, 104033 (2022).
  106. B. Leather, Gravitational self-force with hyperboloidal slicing and spectral methods, Gen. Relativ. Gravit. 57, 112 (2025).
  107. Black Hole Perturbation Toolkit, (https://bhptoolkit.org).
  108. S. J. Kovacs and K. S. Thorne, The generation of gravitational waves. 4. Bremsstrahlung, Astrophys. J. 224, 62 (1978).
  109. K. Glampedakis and D. Kennefick, Zoom and whirl: Eccentric equatorial orbits around spinning black holes and their evolution under gravitational radiation reaction, Phys. Rev. D 66, 044002 (2002).
  110. S. Akcay, N. Warburton, and L. Barack, Frequency-domain algorithm for the Lorenz-gauge gravitational self-force, Phys. Rev. D 88, 104009 (2013).
  111. L. Barack, R. Gonzo, B. Leather, O. Long, and N. Warburton, Resummed energy loss in extreme-mass-ratio scattering using critical orbits, arXiv:2602.10089.
  112. K. Cunningham, C. Kavanagh, A. Pound, D. Trestini, N. Warburton, and J. Neef, Gravitational memory: New results from post-Newtonian and self-force theory, Classical Quantum Gravity 42, 135009 (2025); 42, 199401(A) (2025).
  113. A. Georgoudis, V. Goncalves, C. Heissenberg, and J. Parra-Martinez, Nonlinear gravitational memory in the post-Minkowskian expansion, arXiv:2506.20733.
  114. M. Colpi et al. (LISA Collaboration), LISA definition study report, arXiv:2402.07571.
  115. I. Qunbar and N. C. Stone, Enhanced extreme mass ratio inspiral rates and intermediate mass black holes, Phys. Rev. Lett. 133, 141401 (2024).
  116. D. Mancieri, L. Broggi, M. Bonetti, and A. Sesana, Hanging on the cliff: Extreme mass ratio inspiral formation with local two-body relaxation and post-Newtonian dynamics, Astron. Astrophys. 694, A272 (2025).
  117. C. E. A. Chapman-Bird et al., Efficient waveforms for asymmetric-mass eccentric equatorial inspirals into rapidly spinning black holes, Phys. Rev. D 112, 104023 (2025).
  118. D. Mancieri, L. Broggi, M. Vinciguerra, A. Sesana, and M. Bonetti, Eccentricity distribution of extreme mass ratio inspirals, Phys. Rev. D 113, 043062 (2026).
  119. M. Casals, S. Dolan, A. C. Ottewill, and B. Wardell, Self-force and green function in Schwarzschild spacetime via quasinormal modes and branch cut, Phys. Rev. D 88, 044022 (2013).
  120. B. Wardell, C. R. Galley, A. Zenginoğlu, M. Casals, S. R. Dolan, and A. C. Ottewill, Self-force via Green functions and worldline integration, Phys. Rev. D 89, 084021 (2014).
  121. C. O’Toole, A. Ottewill, and B. Wardell, Characteristic formulation of the Regge-Wheeler and Zerilli Green functions, Phys. Rev. D 103, 124022 (2021).
  122. L. Barack, A. Ori, and N. Sago, Frequency-domain calculation of the self force: The high-frequency problem and its resolution, Phys. Rev. D 78, 084021 (2008).
  123. S. A. Teukolsky, Perturbations of a rotating black hole. 1. Fundamental equations for gravitational electromagnetic and neutrino field perturbations, Astrophys. J. 185, 635 (1973).
  124. H. O. Silva, G. Tambalo, K. Glampedakis, and K. Yagi, Gravitational radiation from a particle plunging into a Schwarzschild black hole: Frequency-domain and semirelativistic analyses, Phys. Rev. D 109, 024036 (2024).

Outline

Information

Phys. Rev. D 113, 084059– Published 28 April, 2026
  • Received 7 March 2026
  • Accepted 30 March 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