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Singular structures and causality of the Schwarzschild Green’s function in the frequency domain

Romeo Felice Rosato1,2,*, Marina De Amicis3,†, and Paolo Pani1,2,‡

  • *Contact author: romeofelice.rosato@uniroma1.it
  • †Contact author: mdeamicis@perimeterinstitute.ca
  • ‡Contact author: paolo.pani@uniroma1.it
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Phys. Rev. D 114, 024024 – Published 9 July, 2026

DOI: https://doi.org/10.1103/58mx-v2wz

Abstract

We study two singular spectral components of the Green’s function of a Schwarzschild black hole and their interpretation in the frequency domain: (i) the low-frequency branch cut, which yields corrections to Price’s law tails in the form of inverse power laws weighted by logarithmic terms; and (ii) the quasinormal-mode spectrum, which generates a redshifted response for sources extended toward the horizon. We show that the frequency-domain Green’s function admits a natural interpretation in terms of graybody factors, providing the first analytical justification of graybody-based ringdown models for sources at generic locations. For sources localized outside the peak of the potential barrier, we identify two tail contributions activated with a time delay, arising from backscattering of the prompt response and of the ringdown signal. We show that corrections to Price’s law can be relevant at intermediate times, when the ringdown still dominates the waveform. For sources localized inside the potential barrier peak, the tail is suppressed and the signal is instead dominated by quasinormal frequencies. In this regime, these spectral components produce both the ordinary quasinormal-mode ringdown and an infinite tower of exponentially decaying terms governed by the horizon surface gravity, the so-called redshift terms. We demonstrate that this component is not screened by geometric features of the background spacetime and persists up to late times, as supported by numerical investigations of perturbative waveforms. Our results provide a mathematical foundation for phenomenological modeling of the branch-cut contribution at intermediate times, which is relevant for prospective observations of tails, and strong evidence for the presence of redshifted components from intermediate to late times.

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References (70)

  1. B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), Phys. Rev. Lett. 116, 061102 (2016).
  2. R. Abbott et al. (LIGO Scientific, Virgo, and KAGRA Collaborations), Phys. Rev. D 112, 084080 (2025).
  3. A. G. Abac et al. (LIGO Scientific, KAGRA, and Virgo Collaborations), Astrophys. J. Lett. 995, L18 (2025).
  4. M. Saleem et al., Classical Quantum Gravity 39, 025004 (2022).
  5. A. Abac et al. (ET Collaboration), J. Cosmol. Astropart. Phys. 03 (2026) 081.
  6. M. Colpi et al. (LISA Collaboration), arXiv:2402.07571.
  7. E. Berti et al., Classical Quantum Gravity 32, 243001 (2015).
  8. E. Berti, K. Yagi, H. Yang, and N. Yunes, Gen. Relativ. Gravit. 50, 49 (2018).
  9. V. Cardoso and P. Pani, Living Rev. Relativity 22, 4 (2019).
  10. J. Abedi et al., arXiv:2505.23895.
  11. E. W. Leaver, Phys. Rev. D 34, 384 (1986).
  12. N. Andersson, Phys. Rev. D 55, 468 (1997).
  13. R. H. Price, Phys. Rev. D 5, 2419 (1972).
  14. M. De Amicis, E. Cannizzaro, G. Carullo, and L. Sberna, Phys. Rev. D 113, 024048 (2026).
  15. A. Kuntz, SciPost Phys. 20, 120 (2026).
  16. P. Arnaudo, J. Carballo, and B. Withers, arXiv:2510.18956.
  17. M. De Amicis and E. Cannizzaro, arXiv:2601.11706.
  18. J. Su, N. Khera, M. Casals, S. Ma, A. Chowdhuri, and H. Yang, Phys. Rev. D 113, 104013 (2026).
  19. E. W. Leaver, Proc. R. Soc. A 402, 285 (1985).
  20. R. H. Price, Phys. Rev. D 5, 2439 (1972).
  21. C. T. Cunningham, R. H. Price, and V. Moncrief, Astrophys. J. 224, 643 (1978).
  22. C. T. Cunningham, R. H. Price, and V. Moncrief, Astrophys. J. 230, 870 (1979).
  23. E. W. Leaver, J. Math. Phys. (N.Y.) 27, 1238 (1986).
  24. R. Gómez and J. Winicour, J. Math. Phys. (N.Y.) 33, 1445 (1992).
  25. C. Gundlach, R. H. Price, and J. Pullin, Phys. Rev. D 49, 883 (1994).
  26. C. Gundlach, R. H. Price, and J. Pullin, Phys. Rev. D 49, 890 (1994).
  27. E. S. C. Ching, P. T. Leung, W. M. Suen, and K. Young, Phys. Rev. Lett. 74, 2414 (1995).
  28. L. M. Burko and A. Ori, Phys. Rev. D 56, 7820 (1997).
  29. L. Barack, Phys. Rev. D 59, 044017 (1999).
  30. S. Bernuzzi, A. Nagar, and R. De Pietri, Phys. Rev. D 77, 044042 (2008).
  31. S. Hod, Classical Quantum Gravity 26, 028001 (2009).
  32. E. Poisson, Phys. Rev. D 66, 044008 (2002).
  33. R. F. Rosato and P. Pani, Phys. Rev. D 112, 024080 (2025).
  34. M. Casals and A. C. Ottewill, Phys. Rev. D 92, 124055 (2015).
  35. S. Hod, Phys. Rev. D 60, 104053 (1999).
  36. A. Z. Smith and L. M. Burko, Phys. Rev. D 74, 028501 (2006).
  37. M. Casals and A. C. Ottewill, Phys. Rev. Lett. 109, 111101 (2012).
  38. N. Oshita, J. Cosmol. Astropart. Phys. 04 (2023) 013.
  39. N. Oshita, Phys. Rev. D 109, 104028 (2024).
  40. R. F. Rosato, K. Destounis, and P. Pani, Phys. Rev. D 110, L121501 (2024).
  41. N. Oshita, K. Takahashi, and S. Mukohyama, Phys. Rev. D 110, 084070 (2024).
  42. R. F. Rosato, S. Yi, E. Berti, and P. Pani, Phys. Rev. D 113, 064060 (2026).
  43. K. Okabayashi and N. Oshita, Phys. Rev. D 110, 064086 (2024).
  44. M. Dafermos and I. Rodnianski, Commun. Pure Appl. Math. 62, 859 (2009).
  45. Y. Mino and J. Brink, Phys. Rev. D 78, 124015 (2008).
  46. A. Zimmerman and Y. Chen, Phys. Rev. D 84, 084012 (2011).
  47. A. Laeuger, C. Weller, D. Li, and Y. Chen, Phys. Rev. D 112, 084042 (2025).
  48. N. Oshita, S. Ma, Y. Chen, and H. Yang, arXiv:2509.09165.
  49. A. Kankani and S. T. McWilliams, arXiv:2603.15474.
  50. F. J. Zerilli, Phys. Rev. D 2, 2141 (1970).
  51. K. Martel, Phys. Rev. D 69, 044025 (2004).
  52. M. Sasaki and H. Tagoshi, Living Rev. Relativity 6, 6 (2003).
  53. K. Martel and E. Poisson, Phys. Rev. D 71, 104003 (2005).
  54. A. Nagar, T. Damour, and A. Tartaglia, Classical Quantum Gravity 24, S109 (2007).
  55. S. Bernuzzi and A. Nagar, Phys. Rev. D 81, 084056 (2010).
  56. S. Bernuzzi, A. Nagar, and A. Zenginoglu, Phys. Rev. D 84, 084026 (2011).
  57. M. De Amicis, S. Albanesi, and G. Carullo, Phys. Rev. D 110, 104005 (2024).
  58. D. Chiaramello and A. Nagar, Phys. Rev. D 101, 101501 (2020).
  59. S. Albanesi, A. Nagar, and S. Bernuzzi, Phys. Rev. D 104, 024067 (2021).
  60. F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge University Press, Cambridge, 2010).
  61. S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, Oxford, 1985).
  62. S. Ma, K. Mitman, L. Sun, N. Deppe, F. Hébert, L. E. Kidder, J. Moxon, W. Throwe, N. L. Vu, and Y. Chen, Phys. Rev. D 106, 084036 (2022).
  63. R. F. Rosato, S. Biswas, S. Chakraborty, and P. Pani, Phys. Rev. D 111, 084051 (2025).
  64. R. A. Konoplya and A. Zhidenko, J. Cosmol. Astropart. Phys. 09 (2024) 068.
  65. R. A. Konoplya and A. Zhidenko, Phys. Lett. B 861, 139288 (2025).
  66. A. A. Starobinskij and S. M. Churilov, Zh. Eksp. Teor. Fiz. 65, 3 (1973).
  67. R. Brito, V. Cardoso, and P. Pani, Lect. Notes Phys. 906, 1 (2015).
  68. S. L. Detweiler, Proc. R. Soc. A 352, 381 (1977).
  69. E. Maggio, V. Cardoso, S. R. Dolan, and P. Pani, Phys. Rev. D 99, 064007 (2019).
  70. S. A. Teukolsky, Astrophys. J. 185, 635 (1973).

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Phys. Rev. D 114, 024024– Published 9 July, 2026
  • Received 6 April 2026
  • Accepted 28 May 2026
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