• [1]

    H. C. Kim and W. Lee, The European Physical Journal C (2025), https://doi.org/10.1140/epjc/s10052-025-15008-w.

  • [2]

    R. Ruffini and J. A. Wheeler, Phys. Today 24, 30 (1971)

  • [3]

    J. D. Bekenstein, Lett. Nuovo Cim. 4, 737 (1972)

  • [4]

    J. D. Bekenstein, Phys. Rev. D 51, R6608 (1995)

  • [5]

    K. e. a. E. H. T. C. Akiyama, Astrophys. J. 875, L1 (2019), arXiv: 1906.11238[astro-ph.GA]

  • [6]

    K. e. a. E. H. T. C. Akiyama, Astrophys. J. 875, L4 (2019), arXiv: 1906.11241[astro-ph.GA]

  • [7]

    K. e. a. E. H. T. C. Akiyama, Astrophys. J. 930, L12 (2022), arXiv: 2311.08680[astro-ph.GA]

  • [8]

    C. M. Will, Living Rev. Rel. 17, 4 (2014), arXiv: 1403.7377[gr-qc]

  • [9]

    R. A. Hulse and J. H. Taylor, Astrophys. J. 195, L51 (1975)

  • [10]

    D. Psaltis, Living Rev. Rel. 11, 9 (2008), arXiv: 0806.1531[astro-ph]

  • [11]

    N. Yunes and X. Siemens, Living Rev. Rel. 16, 9 (2013), arXiv: 1304.3473[gr-qc]

  • [12]

    E. Berti et al., Class. Quant. Grav. 32, 243001 (2015), arXiv: 1501.07274[gr-qc]

  • [13]

    S. Chandrasekhar, The Mathematical Theory of Black Holes, Oxford University Press, New York, 1998.

  • [14]

    V. Bozza, Gen. Rel. Grav. 42, 2269 (2010), arXiv: 0911.2187[gr-qc]

  • [15]

    K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D. 62, 084003 (2000)

  • [16]

    P. V. P. Cunha and C. A. R. Herdeiro, Gen. Rel. Grav. 50, 42 (2018), arXiv: 1801.00860[gr-qc]

  • [17]

    S. U. Islam, R. Kumara, and S. G. Ghosha, J. Cosmol. A. P 2020, 030 (2020)

  • [18]

    V. Perlick, Living Rev. Rel. 7, 9 (2004)

  • [19]

    A. Abdujabbarov, B. Ahmedov, N. Dadhich, and F. Atamurotov, Phys. Rev. D 96, 084017 (2017)

  • [20]

    J. L. Synge, Mon. Not. R. Astron. Soc. 131, 463 (1966)

  • [21]

    H. Falcke, F. Melia, and E. Agol, Astrophys. J. Lett. 528, L13 (2000), arXiv: astro-ph/9912263

  • [22]

    K. Hioki and K.-I. Maeda, Phys. Rev. D 80, 024042 (2009)

  • [23]

    F. Atamurotov, A. Abdujabbarov, and B. Ahmedov, Phys. Rev. D 88, 064004 (2013)

  • [24]

    A. Abdujabbarov, F. Atamurotov, Y. Kucukakca, B. Ahmedov, and U. Camci, Astrophys. Space. Sci. 344, 429 (2013), arXiv: 1212.4949[physics.gen-ph]

  • [25]

    V. Perlick and O. Y. Tsupko, Phys. Rept. 947, 1 (2022), arXiv: 2105.07101[gr-qc]

  • [26]

    S. Vagnozzi, R. Roy, Y.-D. Tsai, L. Visinelli, M. Afrin, A. Allahyari, P. Bambhaniya, D. Dey, S. G. Ghosh, P. S. Joshi, K. Jusufi, M. Khodadi, R. K. Walia, A. Övgün, and C. Bambi, Classical and Quantum Gravity 40, 165007 (2023)

  • [27]

    G. Mustafa, F. Atamurotov, I. Hussain, S. Shaymatov, and A. Övgün, Chin. Phys. C 46, 125107 (2022), arXiv: 2207.07608[gr-qc]

  • [28]

    C. Bambi, Mod. Phys. Lett. A 28, 1340018 (2013), arXiv: 1309.2895[gr-qc]

  • [29]

    T. Johannsen, Phys. Rev. D 87, 124017 (2013), arXiv: 1304.0118[gr-qc]

  • [30]

    V. Cardoso and P. Pani, Living Rev. Rel. 22, 4 (2019), arXiv: 1904.05363[gr-qc]

  • [31]

    V. Perlick, O. Y. Tsupko, and G. S. Bisnovatyi-Kogan, Phys. Rev. D 92, 104031 (2015), arXiv: 1507.04217[grqc]

  • [32]

    V. Perlick and O. Y. Tsupko, Phys. Rev. D 95, 104003 (2017), arXiv: 1702.08768[gr-qc]

  • [33]

    A. Rogers, Mon. Not. R. Astron. Soc. 451, 17 (2015)

  • [34]

    F. Atamurotov, B. Ahmedov, and A. Abdujabbarov, Phys. Rev. D 92, 084005 (2015), arXiv: 1507.08131[grqc]

  • [35]

    F. Atamurotov, K. Jusufi, M. Jamil, A. Abdujabbarov, and M. Azreg-Aïnou, Phys. Rev. D 104, 064053 (2021), arXiv: 2109.08150[gr-qc]

  • [36]

    F. Atamurotov, M. Jamil, and K. Jusufi, Chin. Phys. C 47, 035106 (2023), arXiv: 2212.12949[gr-qc]

  • [37]

    F. Sarikulov, F. Atamurotov, A. Abdujabbarov, and B. Ahmedov, European Physical Journal C 82 (2022), 10.1140/epjc/s10052-022-10711-4, cited by: 39; All Open Access, Gold Open Access.

  • [38]

    G. Z. Babar, A. Z. Babar, and F. Atamurotov, Eur. Phys. J. C. 80, 761 (2020)

  • [39]

    G. Zaman Babar, F. Atamurotov, S. Ul Islam, and S. G. Ghosh, Phys. Rev. D 103, 084057 (2021), arXiv: 2104.00714[gr-qc]

  • [40]

    R. Ali, X. Tiecheng, R. Babar, and A. Övgün, Int. J. Theor. Phys. 64, 75 (2025), arXiv: 2402.07657[gr-qc]

  • [41]

    S. Kumar, A. Uniyal, and S. Chakrabarti, Phys. Rev. D 109, 104012 (2024), arXiv: 2308.05545[gr-qc]

  • [42]

    M. M. Gohain, K. Bhuyan, and P. Chutia, Phys. Dark Univ. 49, 102064 (2025), arXiv: 2504.00581[hep-th]

  • [43]

    A. Davlataliev, B. Narzilloev, I. Hussain, A. Abdujabbarov, and B. Ahmedov, Phys. Dark Univ. 42, 101340 (2023)

  • [44]

    K. Kobialko, I. Bogush, and D. Gal’tsov, Phys. Rev. D 109, 024060 (2024), arXiv: 2312.07498[gr-qc]

  • [45]

    A. Bakhodirov, B. Shodikulov, F. Atamurotov, A. Abdujabbarov, C. Zhou, and C. Yuan, Phys. Dark Univ. 49, 102008 (2025)

  • [46]

    Z. Turakhonov, F. Atamurotov, S. G. Ghosh, and A. Abdujabbarov, Phys. Dark Univ. 48, 101880 (2025)

  • [47]

    D. Umarov, O. Yunusov, F. Atamurotov, A. Abdujabbarov, and S. G. Ghosh, Chin. Phys. C 49, 055102 (2025)

  • [48]

    A. Alimova, Z. Turakhonov, F. Atamurotov, and A. Abdujabbarov, Phys. Dark Univ. 47, 101749 (2025)

  • [49]

    G. S. Bisnovatyi-Kogan and O. Y. Tsupko, Universe 3, 57 (2017), arXiv: 1905.06615[gr-qc]

  • [50]

    J. L. Synge, Relativity: The General Theory (New York, : Interscience Publishers, 1960).

  • [51]

    Y. Pahlavon, F. Atamurotov, K. Jusufi, M. Jamil, and A. Abdujabbarov, Physics of the Dark Universe 45, 101543 (2024)

  • [52]

    Y. Huang, Y.-P. Dong, and D.-J. Liu, International Journal of Modern Physics D 27, 1850114 (2018)

  • [53]

    Event Horizon Telescope Collaboration, Akiyama, and et al., Astrophys. J. Lett. 930, L17.

  • [54]

    W. Liu, Y. Liu, D. Wu, and Y.-X. Liu, “A universal framework for horizon-scale tests of gravity with black hole shadows, ” (2025), arXiv: 2511.06017[gr-qc].

  • [55]

    V. Bozza, General Relativity and Gravitation 42, 2269 (2010)

  • [56]

    G. S. Bisnovatyi-Kogan and O. Y. Tsupko, Universe 3 (2017), 10.3390/universe3030057.

  • [57]

    V. Bozza, Phys. Rev. D 66, 103001 (2002)