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Deflection angle in the strong deflection limit for static and axisymmetric spacetimes: Local curvature, matter fields, and quasinormal modes

Takahisa Igata*

  • *Contact author: takahisa.igata@gakushuin.ac.jp
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Phys. Rev. D 113, 024036 – Published 20 January, 2026

DOI: https://doi.org/10.1103/ylrj-rm9j

Abstract

We investigate the deflection of photons in the strong deflection limit within static and axisymmetric spacetimes possessing reflection symmetry. As the impact parameter approaches its critical value, the deflection angle exhibits a logarithmic divergence. This divergence is characterized by a logarithmic coefficient and a constant offset, which we express in terms of the coordinate-invariant curvature quantities evaluated at the unstable circular photon orbit. The curvature contribution is encoded in the electric part of the Weyl tensor, reflecting tidal effects, and the matter contribution is encoded in the Einstein tensor, capturing the influence of local energy and pressure. We also express these coefficients using the Newman-Penrose scalars. By exploiting the relationship between the strong deflection limit and the quasinormal modes, we derive a new expression for the quasinormal mode frequency in the eikonal limit in terms of the curvature scalars. Our results provide a unified and coordinate-invariant framework that connects observable lensing features and quasinormal modes to the local geometry and matter distribution near compact objects.

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References (56)

  1. K. Akiyama et al. (Event Horizon Telescope Collaboration), Astrophys. J. Lett. 875, L1 (2019).
  2. K. Akiyama et al. (Event Horizon Telescope Collaboration), Astrophys. J. Lett. 930, L12 (2022).
  3. B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. Lett. 116, 061102 (2016).
  4. V. Perlick, Living Rev. Relativity 7, 9 (2004).
  5. V. Bozza, Phys. Rev. D 66, 103001 (2002).
  6. N. Tsukamoto, Phys. Rev. D 95, 064035 (2017).
  7. R. Shaikh, P. Banerjee, S. Paul, and T. Sarkar, Phys. Rev. D 99, 104040 (2019).
  8. V. Bozza, S. Capozziello, G. Iovane, and G. Scarpetta, Gen. Relativ. Gravit. 33, 1535 (2001).
  9. E. F. Eiroa, G. E. Romero, and D. F. Torres, Phys. Rev. D 66, 024010 (2002).
  10. N. Tsukamoto and Y. Gong, Phys. Rev. D 95, 064034 (2017).
  11. E. F. Eiroa and C. M. Sendra, Classical Quantum Gravity 28, 085008 (2011).
  12. D. Chen, Y. Chen, P. Wang, T. Wu, and H. Wu, Eur. Phys. J. C 84, 584 (2024).
  13. N. Tsukamoto, Phys. Rev. D 94, 124001 (2016).
  14. K. K. Nandi, R. N. Izmailov, A. A. Yanbekov, and A. A. Shayakhmetov, Phys. Rev. D 95, 104011 (2017).
  15. R. Shaikh, P. Banerjee, S. Paul, and T. Sarkar, J. Cosmol. Astropart. Phys. 07 (2019) 028; 12 (2023) E01(E).
  16. K. K. Nandi, Y. Z. Zhang, and A. V. Zakharov, Phys. Rev. D 74, 024020 (2006).
  17. J. M. Tejeiro and E. A. Larranaga R., Rom. J. Phys. 57, 736 (2012).
  18. A. Bhattacharya and A. A. Potapov, Mod. Phys. Lett. A 34, 1950040 (2019).
  19. R. N. Izmailov, E. R. Zhdanov, A. Bhattacharya, A. A. Potapov, and K. K. Nandi, Eur. Phys. J. Plus 134, 384 (2019).
  20. T. Kubo and N. Sakai, Phys. Rev. D 93, 084051 (2016).
  21. S. Chakraborty and S. SenGupta, J. Cosmol. Astropart. Phys. 07 (2017) 045.
  22. A. R. Soares, R. L. L. Vitória, and C. F. S. Pereira, Phys. Rev. D 110, 084004 (2024).
  23. J. R. Nascimento, A. Y. Petrov, P. J. Porfirio, and A. R. Soares, Phys. Rev. D 102, 044021 (2020).
  24. A. Ishihara, Y. Suzuki, T. Ono, and H. Asada, Phys. Rev. D 95, 044017 (2017).
  25. K. Takizawa and H. Asada, Phys. Rev. D 103, 104039 (2021).
  26. F. Feleppa, V. Bozza, and O. Y. Tsupko, Phys. Rev. D 111, 044018 (2025).
  27. Y. X. Gao, arXiv:2503.06895.
  28. T. Sasaki, arXiv:2504.00355.
  29. V. Bozza, Phys. Rev. D 67, 103006 (2003).
  30. S. R. Dolan, Phys. Rev. D 82, 104003 (2010).
  31. T. Hsieh, D. S. Lee, and C. Y. Lin, Phys. Rev. D 103, 104063 (2021).
  32. A. Chowdhuri, S. Ghosh, and A. Bhattacharyya, Front. Phys. 11, 1113909 (2023).
  33. M. Patil, P. Mishra, and D. Narasimha, Phys. Rev. D 95, 024026 (2017).
  34. H. Chakrabarty and Y. Tang, Phys. Rev. D 107, 084020 (2023).
  35. T. Igata, arXiv:2503.02320.
  36. V. Ferrari and B. Mashhoon, Phys. Rev. D 30, 295 (1984).
  37. V. Cardoso, A. S. Miranda, E. Berti, H. Witek, and V. T. Zanchin, Phys. Rev. D 79, 064016 (2009).
  38. I. Z. Stefanov, S. S. Yazadjiev, and G. G. Gyulchev, Phys. Rev. Lett. 104, 251103 (2010).
  39. B. Raffaelli, Gen. Relativ. Gravit. 48, 16 (2016).
  40. B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. Lett. 116, 221101 (2016).
  41. B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. D 103, 122002 (2021).
  42. S. V. Bolokhov and M. Skvortsova, arXiv:2504.05014.
  43. R. M. Wald, General Relativity (University of Chicago Press, Chicago, 1984).
  44. J. B. Griffiths and J. Podolský, Exact Space-Times in Einstein’s General Relativity (Cambridge University Press, Cambridge, England, 2012).
  45. H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselaers, and E. Herlt, Exact Solutions of Einstein’s Field Equations (Cambridge University Press, Cambridge, England, 2003), 10.1017/CBO9780511535185.
  46. Y. Koga and T. Harada, Phys. Rev. D 100, 064040 (2019).
  47. T. Assumpcao, V. Cardoso, A. Ishibashi, M. Richartz, and M. Zilhao, Phys. Rev. D 98, 064036 (2018).
  48. J. Shipley and S. R. Dolan, Classical Quantum Gravity 33, 175001 (2016).
  49. K. Nakashi and T. Igata, Phys. Rev. D 99, 124033 (2019).
  50. A. Allahyari, H. Firouzjahi, and B. Mashhoon, Phys. Rev. D 99, 044005 (2019).
  51. P. Pradhan, Pramana 87, 5 (2016).
  52. K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D 62, 084003 (2000).
  53. J. Maldacena, S. H. Shenker, and D. Stanford, J. High Energy Phys. 08 (2016) 106.
  54. K. Hashimoto and N. Tanahashi, Phys. Rev. D 95, 024007 (2017).
  55. D. Giataganas, A. Kehagias, and A. Riotto, J. High Energy Phys. 09 (2024) 168.
  56. E. Gallo and T. Mädler, Eur. Phys. J. C 85, 299 (2025).

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Phys. Rev. D 113, 024036– Published 20 January, 2026
  • Received 12 May 2025
  • Accepted 5 January 2026
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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.

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