• [1]

    K. Akiyama et al. (Event Horizon Telescope Collaboration), Astrophys. J. Lett. 875, L1 (2019), arXiv: 1906.11238 [astro-ph.GA]

  • [2]

    K. Akiyama et al. (Event Horizon Telescope Collaboration), Astrophys. J. Lett. 875, L4 (2019), arXiv: 1906.11241 [astro-ph.GA]

  • [3]

    K. Akiyama et al. (Event Horizon Telescope Collaboration), Astrophys. J. Lett. 930, L12 (2022), arXiv: 2311.08680 [astro-ph.HE]

  • [4]

    B. P. Abbott and others (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. Lett. 116, 061102 (2016), arXiv: 1602.03837[gr-qc]

  • [5]

    A. Duran-Cabacxes, D. Rubiera-Garcia, and D. SxaezChillxon Gxomez, Phys. Rev. D 112, 044016 (2025), arXiv: 2506.10814[gr-qc]

  • [6]

    M. Guerrero, G. J. Olmo, D. Rubiera-Garcia et al., JCAP (08), 036 (2021), arXiv: 2105.15073[grqc]

  • [7]

    G. J. Olmo, J. L. Rosa, D. Rubiera-Garcia et al., Class. Quant. Grav. 40, 174002 (2023), arXiv: 2302.12064[gr-qc]

  • [8]

    S. E. Gralla, D. E. Holz, and R. M. Wald, Phys. Rev. D 100, 024018 (2019), arXiv: 1906.00873[astroph.HE]

  • [9]

    R. Penrose, Phys. Rev. Lett. 14, 57 (1965)

  • [10]

    M. Reuter, Phys. Rev. D 57, 971 (1998), arXiv: hepth/9605030

  • [11]

    M. Niedermaier and M. Reuter, Living Reviews in Relativity 9, 5 (2006)

  • [12]

    M. Reuter and F. Saueressig, Quantum Gravity and the Functional Renormalization Group: The Road towards Asymptotic Safety, (Cambridge University Press, 2019)

  • [13]

    A. Bonanno and M. Reuter, Phys. Rev. D 62, 043008 (2000), arXiv: hep-th/0002196

  • [14]

    A. Bonanno and M. Reuter, Phys. Rev. D 73, 083005 (2006), arXiv: hep-th/0602159

  • [15]

    K. Falls, C. R. King, D. F. Litim et al., Phys. Rev. D 97, 086006 (2018), arXiv: 1801.00162[hep-th]

  • [16]

    G. Alencar, T. M. Crispim, C. R. Muniz et al., (2026), arXiv: 2603.05130v1[gr-qc]

  • [17]

    J. L. Synge, Relativity: The General Theory (NorthHolland, Amsterdam, 1960)

  • [18]

    V. Perlick, Living Reviews in Relativity 7, 9 (2004)

  • [19]

    O. Y. Tsupko and G. S. Bisnovatyi-Kogan, MNRAS 427, 2017 (2012)

  • [20]

    G. S. Bisnovatyi-Kogan and O. Y. Tsupko, MNRAS 404, 1790 (2010), arXiv: 1006.2321[astro-ph.CO]

  • [21]

    F. Feleppa, V. Bozza, and O. Y. Tsupko, Phys. Rev. D 110, 064031 (2024)

  • [22]

    F. Feleppa, V. Bozza, and O. Y. Tsupko, Phys. Rev. D 111, 044018 (2025)

  • [23]

    F. Feleppa, F. Aratore, and V. Bozza, Phys. Rev. D 112, 1 (2025)

  • [24]

    V. Perlick, O. Y. Tsupko, and G. S. Bisnovatyi-Kogan, Phys. Rev. D 92, 104031 (2015)

  • [25]

    J. L. Synge, MNRAS131, 463 (1966)

  • [26]

    V. Bozza, Phys. Rev. D 66, 103001 (2002), arXiv: gr-qc/0208075

  • [27]

    A. Rogers, MNRAS 451, 17 (2015)

  • [28]

    V. S. Morozova, B. J. Ahmedov, and A. A. Tursunov, Astrophys. Space Sci. 346, 513 (2013)

  • [29]

    A. Abdujabbarov, B. Toshmatov, Z. Stuchlxık et al., Int. J. Mod. Phys. D 25, 1650023 (2015)

  • [30]

    M. Alloqulov, S. Shaymatov, A. Jawad et al., Commun. Theor. Phys. 77, 015402 (2025)

  • [31]

    A. Al-Badawi, M. Alloqulov, S. Shaymatov et al., Chin. Phys. C 48, 095105 (2024)

  • [32]

    M. Alloqulov, F. Atamurotov, A. Abdujabbarov et al., Chin. Phys. C 47, 075103 (2023)

  • [33]

    H. Jiang, M. Alloqulov, Q. Wu et al., Phys. Dark Univ. 46, 101627 (2024)

  • [34]

    M. Alloqulov, F. Atamurotov, A. Abdujabbarov et al., Chin. Phys. C 48, 115116 (2024)

  • [35]

    A. Al-Badawi, S. Shaymatov, M. Alloqulov et al., Commun. Theor. Phys. 76, 085401 (2024)

  • [36]

    M. Alloqulov, F. Atamurotov, A. Abdujabbarov et al., Chin. Phys. C 48, 025104 (2024)

  • [37]

    F. Atamurotov, M. Alloqulov, A. Abdujabbarov et al., Eur. Phys. J. Plus 137, 634 (2022)

  • [38]

    J. Sharipov, M. Alloqulov, P. Sheoran et al., Chin. J Phys. 99, 461 (2026)

  • [39]

    M. Alloqulov, Y. Isaqjonov, S. Shaymatov et al., Chin. Phys. C 49, 045104 (2025)

  • [40]

    J. Khasanov, M. Alloqulov, P. Sheoran et al., JCAP 12, 023 (2025)

  • [41]

    W. Javed, I. Hussain, and A. Övgün, Eur. Phys. J. Plus 137, 148 (2022), arXiv: 2201.09879[gr-qc]

  • [42]

    F. Atamurotov, I. Hussain, G. Mustafa et al., Chin. Phys. C 47, 025102 (2023)

  • [43]

    B. Turimov, O. Rahimov, A. Davlataliev et al., Phys. Dark Univ. 48, 101855 (2025), arXiv: 2502.11210[gr-qc]

  • [44]

    A. Davlataliev, B. Narzilloev, I. Hussain et al., Phys. Dark Univ. 42, 101340 (2023)

  • [45]

    A. Chowdhuri and A. Bhattacharyya, Phys. Rev. D 104, 064039 (2021)

  • [46]

    E. Battista, S. Capozziello, and C. Y. Chen, (2026), arXiv: 2601.10806[gr-qc]

  • [47]

    K. S. Virbhadra, Phys. Rev. D 109, 124004 (2024), arXiv: 2402.17190[gr-qc]

  • [48]

    K. S. Virbhadra, Can. J. Phys. 102, 523 (2024), arXiv: 2204.01792[gr-qc]

  • [49]

    S. L. Adler and K. S. Virbhadra, Gen. Rel. Grav. 54, 93 (2022), arXiv: 2205.04628[gr-qc]

  • [50]

    K. S. Virbhadra, Phys. Rev. D 106, 064038 (2022), arXiv: 2204.01879[gr-qc]

  • [51]

    K. S. Virbhadra and C. R. Keeton, Phys. Rev. D 77, 124014 (2008), arXiv: 0710.2333[gr-qc]

  • [52]

    K. S. Virbhadra, Phys. Rev. D 79, 083004 (2009), arXiv: 0810.2109[gr-qc]

  • [53]

    K. S. Virbhadra, D. Narasimha, and S. M. Chitre, Astron. Astrophys. 337, 1 (1998), arXiv: astro-ph/9801174

  • [54]

    C. M. Claudel, K. S. Virbhadra, and G. F. R. Ellis, J. Math. Phys. 42, 818 (2001), arXiv: gr-qc/0005050

  • [55]

    K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D 65, 103004 (2002)

  • [56]

    K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D 62, 084003 (2000), arXiv: astro-ph/9904193

  • [57]

    Y. Mizuno, Z. Younsi, C. M. Fromm et al., Nature Astron. 2, 585 (2018), arXiv: 1804.05812[astro-ph.GA]

  • [58]

    M. Zhang, Y. Mizuno, C. M. Fromm et al., Astron. Astrophys. 687, A88 (2024), arXiv: 2404.04033[astro-ph.HE]

  • [59]

    Y. Song, Y. Hou, L. Huang, and B. Chen, Phys. Rev. D 113, 043054 (2026), arXiv: 2507.23281[astro-ph.HE]

  • [60]

    K. Moriyama, A. Cruz-Osorio, Y. Mizuno et al., Astron. Astrophys. 694, A135 (2025), arXiv: 2501.08720[astro-ph.HE]

  • [61]

    A. Chael, Mon. Not. Roy. Astron. Soc. 532, 3198 (2024), arXiv: 2404.01471[astro-ph.HE]

  • [62]

    P. Kocherlakota and L. Rezzolla, Mon. Not. Roy. Astron. Soc. 513, 1229 (2022), arXiv: 2201.05641[gr-qc]

  • [63]

    C. M. Fromm et al., Astron. Astrophys. 660, A107 (2022), arXiv: 2111.02518[astro-ph.HE]

  • [64]

    D. Bardiev, Y. Wang, M. Kološ et al., Astrophys. J. 972, 55 (2024)

  • [65]

    D. Bardiev, J. Schee, and Z. Stuchlxık, Astrophys. J. 993, 62 (2025)

  • [66]

    O. Y. Tsupko and G. S. Bisnovatyi-Kogan, Gravit. Cosmol. 15, 184 (2009)

  • [67]

    S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, 1983)

  • [68]

    G. Z. Babar, F. Atamurotov, and A. Z. Babar, Physics of the Dark Universe 32, 100798 (2021)

  • [69]

    S. Vxazquez and E. Esteban, Il Nuovo Cimento B 119, 489 (2004)

  • [70]

    C. Bambi, K. Freese, S. Vagnozzi et al., Phys. Rev. D 100, 044057 (2019)

  • [71]

    K. Akiyama et al. (Event Horizon Telescope Collaboration), Astron. Astrophys. 681, A79 (2024)

  • [72]

    M. De Laurentis and P. Salucci, Astrophys. J. 929, 17 (2022), arXiv: 2206.01997[astro-ph.CO]

  • [73]

    R. Abuter et al. (GRAVITY), Astron. Astrophys. 636, L5 (2020), arXiv: 2004.07187[astro-ph.GA]

  • [74]

    A. Rogers, Mon. Not. Roy. Astron. Soc. 451, 17 (2015), arXiv: 1505.06790[gr-qc]

  • [75]

    X. Er and A. Rogers, Mon. Not. Roy. Astron. Soc. 475, 867 (2017), arXiv: 1712.06900[astro-ph.GA]

  • [76]

    P. Goldreich and W. H. Julian, Astrophys. J. 157, 869 (1969)

  • [77]

    F. Atamurotov, A. Abdujabbarov, and J. Rayimbaev, Eur. Phys. J. C 81, 118 (2021)

  • [78]

    O. Y. Tsupko and G. S. Bisnovatyi-Kogan, Plasma Physics Reports 41, 562 (2015)

  • [79]

    P. Schneider, J. Ehlers, and E. E. Falco, Gravitational Lenses, Astronomy and Astrophysics Library (Heidelberg: Springer Berlin,1992)

  • [80]

    F. Atamurotov, K. Jusufi, M. Jamil et al., Phys. Rev. D 104, 064053 (2021), arXiv: 2109.08150[gr-qc]