• [1]

    C. M. Will, Living Rev. Relativ. 9, 3 (2006)

  • [2]

    C. M. Will, Living Rev. Relativ. 17, 4 (2014)

  • [3]

    P. Schneider, J. Ehlers, and E. E. Falco, Gravitational Lenses. 1992.

  • [4]

    M. Bartelmann, Class. Quantum Grav. 27, 233001 (2010)

  • [5]

    K. Akiyama and et al. (Event Horizon Telescope Collaboration), Astrophys. J. 910, L12 (2021)

  • [6]

    K. Akiyama and et al. (Event Horizon Telescope Collaboration), Astrophys. J. 910, L13 (2021)

  • [7]

    K. Akiyama and et al. (Event Horizon Telescope Collaboration), Astrophys. J. 875, L1 (2019)

  • [8]

    K. Akiyama and et al. (Event Horizon Telescope Collaboration), Astrophys. J. 875, L3 (2019)

  • [9]

    C. Goddi, H. Falcke, M. Kramer et al., Int. J. Mod. Phys. D 26, 1730001 (2017)

  • [10]

    X. Jiang, X. Ren, Z. Li et al., Mon. Not. R. Astron. Soc. 528, 1965 (2024)

  • [11]

    X.-M. Kuang, Z.-Y. Tang, B. Wang et al., Phys. Rev. D 106, 064012 (2022)

  • [12]

    S.-S. Zhao and Y. Xie, European Physical Journal C 77, 272 (2017)

  • [13]

    S. Chen and J. Jing, Class. Quantum Grav. 27, 225006 (2010)

  • [14]

    T. L. Smith, arXiv: 0907.4829

  • [15]

    A. Ditta, X. Tiecheng, F. Atamurotov et al., Chin. J. Phys. 83, 664 (2023)

  • [16]

    R. C. Pantig, L. Mastrototaro, G. Lambiase et al., Eur. Phys. J. C 82(12), 1155 (2022)

  • [17]

    S. I. Kruglov, Int. J. Mod. Phys. D 31(04), 2250025 (2022)

  • [18]

    Z. Turakhonov, F. Atamurotov, A. Ovgun et al., Commun. Theor. Phys. 76(11), 115401 (2024)

  • [19]

    B. Rahmatov, M. Zahid, S. U. Khan et al., Chin. Phys. C 49(7), 075105 (2025)

  • [20]

    A. Al-Badawi, Y. Sekhmani, and K. Boshkayev, Phys. Dark Univ. 48, 101865 (2025)

  • [21]

    H. P. Saikia, M. M. Gohain, and K. Bhuyan, arXiv: 2504.13466

  • [22]

    D. Flores-Alfonso, B. A. González-Morales, R. Linares et al., Phys. Lett. B 812, 136011 (2021)

  • [23]

    K. Karshiboev, F. Atamurotov, A. Övgün et al., New Astron. 109, 102200 (2024)

  • [24]

    B. Eslam Panah, PTEP 2024(2), 023E01 (2024)

  • [25]

    H. Kim, Phys. Rev. D 61, 085014 (2000)

  • [26]

    M. Born and L. Infeld, Proc. Roy. Soc. Lond. A 144(852), 425 (1934)

  • [27]

    G. W. Gibbons, AIP Conf. Proc. 589(1), 324 (2001)

  • [28]

    I. Bandos, K. Lechner, D. Sorokin et al., Phys. Rev. D 102, 121703 (2020)

  • [29]

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

  • [30]

    G. W. Gibbons and C. A. R. Herdeiro, Class. Quant. Grav. 18, 1677 (2001)

  • [31]

    R. Ruffini, Y.-B. Wu, and S.-S. Xue, Phys. Rev. D 88, 085004 (2013)

  • [32]

    B. P. Kosyakov, Phys. Lett. B 810, 135840 (2020)

  • [33]

    T. Clifton, P. G. Ferreira, A. Padilla et al., Phys. Rep. 513, 001 (2012)

  • [34]

    C. G. Böhmer and E. Jensko, Phys. Rev. D 104, 024010 (2021)

  • [35]

    S. Nojiri and S. D. Odintsov, Phys. Rep 505, 59 (2011)

  • [36]

    T. P. Sotiriou and S.-Y. Zhou, Phys. Rev. Lett. 112, 251102 (2014)

  • [37]

    S. Capozziello and M. de Laurentis, Phys. Rep. 509, 167 (2011)

  • [38]

    E. Berti and et al., Class. Quantum Grav. 32, 243001 (2015)

  • [39]

    A. De Felice and S. Tsujikawa, Living Rev. Relativ. 13, 3 (2010)

  • [40]

    K. Koyama, Reports on Progress in Physics 79, 046902 (2016)

  • [41]

    L. Heisenberg, Phys. Rep. 796, 001 (2019)

  • [42]

    J. Rayimbaev, R. C. Pantig, A. Övgün et al., Annals of Physics 454, 169335 (2023)

  • [43]

    B. Puliçe, R. C. Pantig, A. Övgün et al., Class. Quantum Grav. 40, 195003 (2023)

  • [44]

    S. Vagnozzi, R. Roy, Y.-D. Tsai et al., Class. Quantum Grav. 40, 165007 (2023)

  • [45]

    A. Uniyal, R. C. Pantig, and A. Övgün, Physics of the Dark Universe 40, 101178 (2023)

  • [46]

    R. C. Pantig and A. Övgün, Annals of Physics 448, 169197 (2023)

  • [47]

    F. Atamurotov, U. Papnoi, and K. Jusufi, Class. Quantum Grav. 39, 025014 (2022)

  • [48]

    E. Ghorani, B. Puliçe, F. Atamurotov et al., Eur. Phys. J. C 83, 318 (2023)

  • [49]

    H. Hoshimov, O. Yunusov, F. Atamurotov et al., Physics of the Dark Universe 43, 101392 (2024)

  • [50]

    R. Kumar, S. U. Islam, and S. G. Ghosh, European Physical Journal C 80, 1128 (2020)

  • [51]

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

  • [52]

    D. Psaltis, Living Rev. Relativ. 11, 9 (2008)

  • [53]

    E. Berti, V. Cardoso, and A. O. Starinets, Class. Quantum Grav. 26, 163001 (2009)

  • [54]

    D. Psaltis and et al., Phys. Rev. Lett. 125, 141104 (2020)

  • [55]

    N. K. Johnson-McDaniel, arXiv: 1905.05565

  • [56]

    S. W. Hawking and G. F. R. Ellis, The large-scale structure of space-time (Cambridge: Cambridge University Press, 1973)

  • [57]

    K. S. Thorne, Black holes and time warps: Einstein’s outrageous legacy (New York: W. W. Norton & Company, 1994).

  • [58]

    M. Volonteri, M. Habouzit, and M. Colpi, Nature Reviews Physics 3, 732 (2021)

  • [59]

    R. Tsuda, R. Suzuki, and S. Tomizawa, arXiv: 2308.02146

  • [60]

    J. Barrientos, A. Cisterna, M. Hassaine et al., Phys. Lett. B 860, 139214 (2025)

  • [61]

    B. Eslam Panah, A. Rincon, and N. Heidari, arXiv: 2411.02907

  • [62]

    J. F. Plebański and S. Hacyan, Journal of Mathematical Physics 20, 1004 (1979)

  • [63]

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

  • [64]

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

  • [65]

    K. S. Virbhadra, Phys. Rev. D 79, 083004 (2009)

  • [66]

    K. Akiyama and et al., ApJL 875, 44 (2019)

  • [67]

    V. Bozza and L. Mancini, General Relativity and Gravitation 36, 435 (2004)