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Trinity of black hole correspondences: Shadows, quasinormal modes, graybody factors, and cautionary remarks

Davide Pedrotti1,2,3,* and Marco Calzà1,2,†

  • *Contact author: davide.pedrotti-1@unitn.it
  • †Contact author: marco.calza@unitn.it
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Phys. Rev. D 111, 124056 – Published 30 June, 2025

DOI: https://doi.org/10.1103/1q35-mjjz

Abstract

Correspondences between apparently distant concepts are ubiquitous in theoretical physics. In the context of black holes (BHs), quasinormal modes (QNMs) were shown to be linked to both shadows, in the so-called eikonal limit, and graybody factors (GBFs), using the WKB approximation. We test the accuracy of the QNM-GBF correspondence in the context of the Hawking radiation for static and rotating black hole configurations, with particular attention to the superradiant regime. Our analysis reveals the correspondence failure to accurately reproduce the Hawking spectrum due to divergences. Furthermore, we bridge the gap between BH shadows and GBFs by drawing a correspondence between such quantities in the case of generic static and spherically symmetric spacetime configurations. The shadow-GBF correspondence is tested for some case studies, including regular BHs, and its limitations and applicability are discussed. This study opens new perspectives by introducing a new correspondence and remarking on the caution needed when considering these connections.

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References (178)

  1. J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998).
  2. E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998).
  3. O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri, and Y. Oz, Phys. Rep. 323, 183 (2000).
  4. S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96, 181602 (2006).
  5. C. Barcelo, S. Liberati, and M. Visser, Living Rev. Relativity 8, 12 (2005).
  6. C. Barcelo, S. Liberati, and M. Visser, Classical Quantum Gravity 18, 1137 (2001).
  7. M. Baker and S. L. Glashow, Phys. Rev. 128, 2462 (1962).
  8. J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 106, 162 (1957).
  9. Y. Nambu, Phys. Rev. 117, 648 (1960).
  10. Y. Nambu, Phys. Rev. Lett. 4, 380 (1960).
  11. Y. Nambu and G. Jona-Lasinio, Phys. Rev. 124, 246 (1961).
  12. Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961).
  13. S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967).
  14. P. W. Anderson, Phys. Rev. 130, 439 (1963).
  15. F. Englert and R. Brout, Phys. Rev. Lett. 13, 321 (1964).
  16. P. W. Higgs, Phys. Rev. Lett. 13, 508 (1964).
  17. A. Pelissetto and E. Vicari, Phys. Rep. 368, 549 (2002).
  18. J. Berges, N. Tetradis, and C. Wetterich, Phys. Rep. 363, 223 (2002).
  19. K. Huang, Int. J. Mod. Phys. A 28, 1330050 (2013).
  20. G. Jona-Lasinio, Prog. Theor. Phys. Suppl. 184, 1 (2010).
  21. S. W. Hawking, Nature (London) 248, 30 (1974).
  22. S. W. Hawking, Commun. Math. Phys. 43, 199 (1975); 46, 206(E) (1976).
  23. G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2738 (1977).
  24. E. Witten, Eur. Phys. J. Plus 140, 430 (2025).
  25. K. Jusufi, Phys. Rev. D 101, 084055 (2020).
  26. I. Z. Stefanov, S. S. Yazadjiev, and G. G. Gyulchev, Phys. Rev. Lett. 104, 251103 (2010).
  27. H. Yang, Phys. Rev. D 103, 084010 (2021).
  28. D. Pedrotti and S. Vagnozzi, Phys. Rev. D 110, 084075 (2024).
  29. H.-P. Nollert, Classical Quantum Gravity 16, R159 (1999).
  30. K. D. Kokkotas and B. G. Schmidt, Living Rev. Relativity 2, 2 (1999).
  31. E. Berti, V. Cardoso, and A. O. Starinets, Classical Quantum Gravity 26, 163001 (2009).
  32. R. A. Konoplya and A. Zhidenko, Rev. Mod. Phys. 83, 793 (2011).
  33. S. Chen, J. Jing, W.-L. Qian, and B. Wang, Sci. China Phys. Mech. Astron. 66, 260401 (2023).
  34. J. L. Synge, Mon. Not. R. Astron. Soc. 131, 463 (1966).
  35. J. P. Luminet, Astron. Astrophys. 75, 228 (1979).
  36. K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D 62, 084003 (2000).
  37. H. Falcke, F. Melia, and E. Agol, Astrophys. J. Lett. 528, L13 (2000).
  38. R. Narayan, M. D. Johnson, and C. F. Gammie, Astrophys. J. Lett. 885, L33 (2019).
  39. P. V. P. Cunha and C. A. R. Herdeiro, Gen. Relativ. Gravit. 50, 42 (2018).
  40. V. I. Dokuchaev and N. O. Nazarova, Usp. Fiz. Nauk 190, 627 (2020).
  41. V. Perlick and O. Y. Tsupko, Phys. Rep. 947, 1 (2022).
  42. M. Wang, S. Chen, and J. Jing, Commun. Theor. Phys. 74, 097401 (2022).
  43. A. Chowdhuri, S. Ghosh, and A. Bhattacharyya, Front. Phys. 11, 1113909 (2023).
  44. V. Cardoso, A. S. Miranda, E. Berti, H. Witek, and V. T. Zanchin, Phys. Rev. D 79, 064016 (2009).
  45. R. Abbott et al. (KAGRA, Virgo, and LIGO Scientific Collaborations), Phys. Rev. X 13, 041039 (2023).
  46. K. Akiyama et al. (Event Horizon Telescope Collaboration), Astrophys. J. Lett. 875, L1 (2019).
  47. K. Akiyama et al. (Event Horizon Telescope Collaboration), Astrophys. J. Lett. 930, L12 (2022).
  48. C.-Y. Chen, Y.-J. Chen, M.-Y. Ho, and Y.-H. Tseng, Phys. Lett. B 845, 138153 (2023).
  49. E. Berti, Gen. Relativ. Gravit. 56, 145 (2024).
  50. S. H. Völkel, N. Franchini, E. Barausse, and E. Berti, Phys. Rev. D 106, 124036 (2022).
  51. S. H. Völkel, E. Barausse, N. Franchini, and A. E. Broderick, Classical Quantum Gravity 38, 21LT01 (2021).
  52. R. A. Konoplya and A. Zhidenko, J. Cosmol. Astropart. Phys. 09 (2024) 068.
  53. R. A. Konoplya and A. Zhidenko, Phys. Lett. B 861, 139288 (2025).
  54. N. Oshita, Phys. Rev. D 109, 104028 (2024).
  55. N. Oshita, K. Takahashi, and S. Mukohyama, Phys. Rev. D 110, 084070 (2024).
  56. V. Ferrari and B. Mashhoon, Phys. Rev. D 30, 295 (1984).
  57. K. Jusufi, Phys. Rev. D 101, 124063 (2020).
  58. W. Kinnersley, J. Math. Phys. (N.Y.) 10, 1195 (1969).
  59. A. Z. Petrov, Gen. Relativ. Gravit. 32, 1661 (2000).
  60. A. Coley, R. Milson, V. Pravda, and A. Pravdova, Classical Quantum Gravity 21, L35 (2004).
  61. E. Newman and R. Penrose, J. Math. Phys. (N.Y.) 3, 566 (1962).
  62. S. A. Teukolsky, Phys. Rev. Lett. 29, 1114 (1972).
  63. S. A. Teukolsky, Astrophys. J. 185, 635 (1973).
  64. A. Arbey, J. Auffinger, M. Geiller, E. R. Livine, and F. Sartini, Phys. Rev. D 103, 104010 (2021).
  65. E. Berti, V. Cardoso, and M. Casals, Phys. Rev. D 73, 024013 (2006); 73, 109902(E) (2006).
  66. M. Isi, M. Giesler, W. M. Farr, M. A. Scheel, and S. A. Teukolsky, Phys. Rev. Lett. 123, 111102 (2019).
  67. O. Dreyer, B. J. Kelly, B. Krishnan, L. S. Finn, D. Garrison, and R. Lopez-Aleman, Classical Quantum Gravity 21, 787 (2004).
  68. E. Berti, V. Cardoso, and C. M. Will, Phys. Rev. D 73, 064030 (2006).
  69. P. Kocherlakota et al. (Event Horizon Telescope Collaboration), Phys. Rev. D 103, 104047 (2021).
  70. K. Akiyama et al. (Event Horizon Telescope Collaboration), Astrophys. J. Lett. 930, L17 (2022).
  71. S. Vagnozzi and L. Visinelli, Phys. Rev. D 100, 024020 (2019).
  72. S. Vagnozzi et al., Classical Quantum Gravity 40, 165007 (2023).
  73. S. Vagnozzi and L. Visinelli, Res. Notes AAS 6, 106 (2022).
  74. J.-Z. Qi and X. Zhang, Chin. Phys. C 44, 055101 (2020).
  75. E. González, K. Jusufi, G. Leon, and E. N. Saridakis, Phys. Dark Universe 42, 101304 (2023).
  76. I. Banerjee, S. Sau, and S. SenGupta, arXiv:2207.06034.
  77. K. Jusufi, M. Azreg-A𝚤nou, M. Jamil, and T. Zhu, Int. J. Geom. Methods Mod. Phys. 19, 2250068 (2022).
  78. O. Y. Tsupko, Z. Fan, and G. S. Bisnovatyi-Kogan, Classical Quantum Gravity 37, 065016 (2020).
  79. C. Bambi, K. Freese, S. Vagnozzi, and L. Visinelli, Phys. Rev. D 100, 044057 (2019).
  80. M. Afrin, S. Vagnozzi, and S. G. Ghosh, Astrophys. J. 944, 149 (2023).
  81. M. Khodadi, S. Vagnozzi, and J. T. Firouzjaee, Sci. Rep. 14, 26932 (2024).
  82. S. Vagnozzi, C. Bambi, and L. Visinelli, Classical Quantum Gravity 37, 087001 (2020).
  83. R. Ghosh, S. Sk, and S. Sarkar, Phys. Rev. D 108, L041501 (2023).
  84. R. Ghosh and C. Singha, Phys. Rev. D 111, 044008 (2025).
  85. K. S. Virbhadra, Phys. Rev. D 79, 083004 (2009).
  86. K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D 65, 103004 (2002).
  87. C.-M. Claudel, K. S. Virbhadra, and G. F. R. Ellis, J. Math. Phys. (N.Y.) 42, 818 (2001).
  88. A. B. Joshi, D. Dey, P. S. Joshi, and P. Bambhaniya, Phys. Rev. D 102, 024022 (2020).
  89. S. E. Gralla, D. E. Holz, and R. M. Wald, Phys. Rev. D 100, 024018 (2019).
  90. Q. Gan, P. Wang, H. Wu, and H. Yang, Phys. Rev. D 104, 044049 (2021).
  91. L. F. D. da Silva, F. S. N. Lobo, G. J. Olmo, and D. Rubiera-Garcia, Phys. Rev. D 108, 084055 (2023).
  92. G. J. Olmo, J. L. Rosa, D. Rubiera-Garcia, and D. Saez-Chillon Gomez, Classical Quantum Gravity 40, 174002 (2023).
  93. K. Jusufi, Gen. Relativ. Gravit. 53, 87 (2021).
  94. B. F. Schutz and C. M. Will, Astrophys. J. Lett. 291, L33 (1985).
  95. R. A. Konoplya and Z. Stuchlík, Phys. Lett. B 771, 597 (2017).
  96. R. A. Konoplya, A. F. Zinhailo, and Z. Stuchlík, Phys. Rev. D 99, 124042 (2019).
  97. K. Glampedakis and H. O. Silva, Phys. Rev. D 100, 044040 (2019).
  98. C.-Y. Chen and P. Chen, Phys. Rev. D 101, 064021 (2020).
  99. H. O. Silva and K. Glampedakis, Phys. Rev. D 101, 044051 (2020).
  100. C.-Y. Chen, M. Bouhmadi-López, and P. Chen, Eur. Phys. J. Plus 136, 253 (2021).
  101. S. Li, A. A. Abdujabbarov, and W.-B. Han, Eur. Phys. J. C 81, 649 (2021).
  102. F. Moura and J. a. Rodrigues, Phys. Lett. B 819, 136407 (2021).
  103. A. Bryant, H. O. Silva, K. Yagi, and K. Glampedakis, Phys. Rev. D 104, 044051 (2021).
  104. K. Nomura and D. Yoshida, Phys. Rev. D 105, 044006 (2022).
  105. G. Guo, P. Wang, H. Wu, and H. Yang, J. High Energy Phys. 06 (2022) 060.
  106. R. A. Konoplya, Phys. Lett. B 838, 137674 (2023).
  107. S.-W. Wei and Y.-X. Liu, Chin. Phys. C 44, 115103 (2020).
  108. M. Cvetic, G. W. Gibbons, and C. N. Pope, Phys. Rev. D 94, 106005 (2016).
  109. G. Khanna and R. H. Price, Phys. Rev. D 95, 081501 (2017).
  110. R. F. Rosato, K. Destounis, and P. Pani, Phys. Rev. D 110, L121501 (2024).
  111. R. A. Konoplya, A. Zhidenko, and A. F. Zinhailo, Classical Quantum Gravity 36, 155002 (2019).
  112. S. Iyer and C. M. Will, Phys. Rev. D 35, 3621 (1987).
  113. R. Brito, V. Cardoso, and P. Pani, Lect. Notes Phys. 906, 1 (2015).
  114. M. Calzá, Phys. Dark Universe 48, 101900 (2025).
  115. https://pages.jh.edu/eberti2/ringdown/
  116. M. Calzá, Phys. Rev. D 107, 044067 (2023).
  117. M. Calzà, J. a. G. Rosa, and F. Serrano, J. High Energy Phys. 05 (2024) 140.
  118. M. Calzà, D. Pedrotti, and S. Vagnozzi, Phys. Rev. D 111, 024009 (2025).
  119. M. Calzà, D. Pedrotti, and S. Vagnozzi, Phys. Rev. D 111, 024010 (2025).
  120. A. Arbey, M. Calzà, and Y. F. Perez-Gonzalez, Phys. Dark Universe 48, 101903 (2025).
  121. B. Carter, Commun. Math. Phys. 10, 280 (1968).
  122. B. Carter, Phys. Rev. 174, 1559 (1968).
  123. S. Chandrasekhar, The mathematical theory of black holes (1985), https://ui.adsabs.harvard.edu/abs/1983mtbh.book.....C.
  124. J. Bardeen, in Proceedings, 5th International Conference on Gravitation and the Theory of Relativity: Tbilisi, USSR, 1968 (1968), p. 174, https://ui.adsabs.harvard.edu/abs/1968qtr..conf...87B.
  125. S. A. Hayward, Phys. Rev. Lett. 96, 031103 (2006).
  126. A. Borde, Phys. Rev. D 55, 7615 (1997).
  127. E. Ayon-Beato and A. Garcia, Phys. Rev. Lett. 80, 5056 (1998).
  128. E. Ayon-Beato and A. Garcia, Phys. Lett. B 464, 25 (1999).
  129. K. A. Bronnikov and J. C. Fabris, Phys. Rev. Lett. 96, 251101 (2006).
  130. W. Berej, J. Matyjasek, D. Tryniecki, and M. Woronowicz, Gen. Relativ. Gravit. 38, 885 (2006).
  131. K. A. Bronnikov, R. A. Konoplya, and A. Zhidenko, Phys. Rev. D 86, 024028 (2012).
  132. M. Rinaldi, Phys. Rev. D 86, 084048 (2012).
  133. Z. Stuchlík and J. Schee, Int. J. Mod. Phys. D 24, 1550020 (2014).
  134. J. Schee and Z. Stuchlik, J. Cosmol. Astropart. Phys. 06 (2015) 048.
  135. T. Johannsen, Phys. Rev. D 88, 044002 (2013).
  136. R. Myrzakulov, L. Sebastiani, S. Vagnozzi, and S. Zerbini, Classical Quantum Gravity 33, 125005 (2016).
  137. Z.-Y. Fan and X. Wang, Phys. Rev. D 94, 124027 (2016).
  138. L. Sebastiani, S. Vagnozzi, and R. Myrzakulov, Adv. High Energy Phys. 2017, 3156915 (2017).
  139. B. Toshmatov, Z. Stuchlík, and B. Ahmedov, Phys. Rev. D 95, 084037 (2017).
  140. S. Chinaglia and S. Zerbini, Gen. Relativ. Gravit. 49, 75 (2017).
  141. V. P. Frolov, EPJ Web Conf. 168, 01001 (2018).
  142. M. Bertipagani, M. Rinaldi, L. Sebastiani, and S. Zerbini, Phys. Dark Universe 33, 100853 (2021).
  143. G. G. L. Nashed and E. N. Saridakis, J. Cosmol. Astropart. Phys. 05 (2022) 017.
  144. A. Simpson and M. Visser, J. Cosmol. Astropart. Phys. 03 (2022) 011.
  145. E. Franzin, S. Liberati, J. Mazza, R. Dey, and S. Chakraborty, Phys. Rev. D 105, 124051 (2022).
  146. L. Chataignier, A. Y. Kamenshchik, A. Tronconi, and G. Venturi, Phys. Rev. D 107, 023508 (2023).
  147. M. Khodadi and R. Pourkhodabakhshi, Phys. Rev. D 106, 084047 (2022).
  148. S. N. Sajadi, M. Khodadi, O. Luongo, and H. Quevedo, Phys. Dark Universe 45, 101525 (2024).
  149. F. Javed, S. Mumtaz, G. Mustafa, F. Atamurotov, and S. G. Ghosh, Chin. J. Phys. (Taipei) 88, 55 (2024).
  150. A. Ditta, T. Xia, R. Ali, G. Mustafa, G. Mustafa, and A. Mahmood, Phys. Dark Universe 43, 101418 (2024).
  151. A. Al-Badawi, S. Shaymatov, M. Alloqulov, and A. Wang, Commun. Theor. Phys. 76, 085401 (2024).
  152. D. Corona, R. Giambò, and O. Luongo, Int. J. Geom. Methods Mod. Phys. 21, 2440019 (2024).
  153. P. Bueno, P. A. Cano, and R. A. Hennigar, Phys. Lett. B 861, 139260 (2025).
  154. S. Ansoldi, in Conference on Black Holes and Naked Singularities (2008), arXiv:0802.0330.
  155. P. Nicolini, Int. J. Mod. Phys. A 24, 1229 (2009).
  156. L. Sebastiani and S. Zerbini, Astronomy 1, 99 (2022).
  157. R. Torres, Regular Black Holes (Springer Singapore, 2022).
  158. C. Lan, H. Yang, Y. Guo, and Y.-G. Miao, Int. J. Theor. Phys. 62, 202 (2023).
  159. E. Ayon-Beato and A. Garcia, Phys. Lett. B 493, 149 (2000).
  160. E. Ayon-Beato and A. Garcia, Gen. Relativ. Gravit. 37, 635 (2005).
  161. R. V. Maluf and J. C. S. Neves, Int. J. Mod. Phys. D 28, 1950048 (2018).
  162. A. D. Sakharov, Sov. Phys. JETP 22, 241 (1966).
  163. E. B. Gliner, Sov. J. Exp. Theor. Phys. 22, 378 (1966).
  164. M. A. Markov, Sov. J. Exp. Theor. Phys. Lett. 36, 265 (1982), http://jetpletters.ru/ps/1334/article_20160.pdf.
  165. M. A. Markov, Sov. J. Exp. Theor. Phys. Lett. 46, 431 (1987), http://jetpletters.ru/ps/1232/article_18605.pdf.
  166. V. F. Mukhanov and R. H. Brandenberger, Phys. Rev. Lett. 68, 1969 (1992).
  167. A. Kumar, D. V. Singh, and S. G. Ghosh, Ann. Phys. (Amsterdam) 419, 168214 (2020).
  168. S. I. Kruglov, Gravitation Cosmol. 27, 78 (2021).
  169. A. Addazi et al., Prog. Part. Nucl. Phys. 125, 103948 (2022).
  170. R. Alves Batista et al., Classical Quantum Gravity 42, 032001 (2025).
  171. B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), Phys. Rev. Lett. 123, 011102 (2019).
  172. E. Berti, K. Yagi, H. Yang, and N. Yunes, Gen. Relativ. Gravit. 50, 49 (2018).
  173. R. C. Pantig, A. Övgün, and D. Demir, Eur. Phys. J. C 83, 250 (2023).
  174. S.-W. Wei and Y.-X. Liu, Eur. Phys. J. Plus 136, 436 (2021).
  175. S. C. Gandhi and C. J. Efthimiou, Am. J. Phys. 74, 638 (2006).
  176. J. C. Lazenby and D. J. Griffiths, Am. J. Phys. 48, 432 (1980).
  177. E. H. Lieb, Studies in Mathematical Physics. Essays in Honor of Valentine Bargmann (1976), https://ui.adsabs.harvard.edu/abs/1976smpe.book.....L.
  178. S. H. Völkel, R. Konoplya, and K. D. Kokkotas, Phys. Rev. D 99, 104025 (2019).

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Phys. Rev. D 111, 124056– Published 30 June, 2025
  • Received 7 May 2025
  • Accepted 2 June 2025
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