• [1]

    B. P. Abbott et al. (LIGO Scienti c and Virgo), Phys. Rev. Lett. 116(24), 241103 (2016)

  • [2]

    B. P. Abbott et al. (LIGO Scienti c and Virgo), Phys. Rev. Lett. 116(24), 241102 (2016)

  • [3]

    B. P. Abbott et al. (LIGO Scienti c and Virgo), Phys. Rev. Lett. 116(6), 061102 (2016)

  • [4]

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

  • [5]

    K. Akiyama et al. (Event Horizon Telescope), Astrophys. J. Lett. 875(1), L2 (2019)

  • [6]

    K. Akiyama et al. (Event Horizon Telescope), Astrophys. J. Lett. 875(1), L3 (2019)

  • [7]

    K. Akiyama et al. (Event Horizon Telescope), Astrophys. J. Lett. 875(1), L4 (2019)

  • [8]

    K. Akiyama et al. (Event Horizon Telescope), Astrophys. J. Lett. 875(1), L5 (2019)

  • [9]

    K. Akiyama et al. (Event Horizon Telescope), Astrophys. J. Lett. 875(1), L6 (2019)

  • [10]

    K. Akiyama et al. (Event Horizon Telescope), Astrophys. J. Lett. 930(2), L12 (2022)

  • [11]

    K. Akiyama et al. (Event Horizon Telescope), Astrophys. J. Lett. 930(2), L13 (2022)

  • [12]

    K. Akiyama et al. (Event Horizon Telescope), Astrophys. J. Lett. 930(2), L14 (2022)

  • [13]

    K. Akiyama et al. (Event Horizon Telescope), Astrophys. J. Lett. 930(2), L15 (2022)

  • [14]

    K. Akiyama et al. (Event Horizon Telescope), Astrophys. J. Lett. 930(2), L16 (2022)

  • [15]

    K. Akiyama et al. (Event Horizon Telescope), Astrophys. J. Lett. 930(2), L17 (2022)

  • [16]

    S. E. Gralla, D. E. Holz, and R. M. Wald, Phys. Rev. D 100(2), 024018 (2019)

  • [17]

    V. Perlick and O. Y. Tsupko, Phys. Rept. 947, 1 (2022)

  • [18]

    J. L. Synge, Mon. Not. Roy. Astron. Soc. 131(3), 463 (1966)

  • [19]

    J. P. Luminet, Astron. Astrophys. 75, 228 (1979)

  • [20]

    J. M. Bardeen, Proceedings, Ecole d'Eté de Physique Théorique: Les Astres Occlus: Les Houches, France, August, 1972, 215-240, (1973)

  • [21]

    E. F. Eiroa and C. M. Sendra, Eur. Phys. J. C 78(2), 91 (2018)

  • [22]

    K. Hioki and U. Miyamoto, Phys. Rev. D 78, 044007 (2008)

  • [23]

    P. V. P. Cunha and C. A. R. Herdeiro, Gen. Rel. Grav. 50(4), 42 (2018)

  • [24]

    L. Amarilla, E. F. Eiroa, and G. Giribet, Phys. Rev. D 81, 124045 (2010)

  • [25]

    L. Amarilla and E. F. Eiroa, Phys. Rev. D 85, 064019 (2012)

  • [26]

    M. Amir, B. P. Singh, and S. G. Ghosh, Eur. Phys. J. C 78(5), 399 (2018)

  • [27]

    B. P. Singh and S. G. Ghosh, Annals Phys. 395, 127 (2018)

  • [28]

    S. Vagnozzi and L. Visinelli, Phys. Rev. D 100(2), 024020 (2019)

  • [29]

    I. Banerjee, S. Chakraborty, and S. SenGupta, Phys. Rev. D 101(4), 041301 (2020)

  • [30]

    C. Bambi and K. Freese, Phys. Rev. D 79, 043002 (2009)

  • [31]

    F. Atamurotov, A. Abdujabbarov, and B. Ahmedov, Phys. Rev. D 88(6), 064004 (2013)

  • [32]

    F. Atamurotov and B. Ahmedov, Phys. Rev. D 92, 084005 (2015)

  • [33]

    M. Wang, S. Chen, and J. Jing, Phys. Rev. D 97(6), 064029 (2018)

  • [34]

    M. Guo, N. A. Obers, and H. Yan, Phys. Rev. D 98(8), 084063 (2018)

  • [35]

    R. A. Konoplya, Phys. Lett. B 795, 1 (2019)

  • [36]

    C. Bambi, K. Freese, S. Vagnozzi et al., Phys. Rev. D 100(4), 044057 (2019)

  • [37]

    A. Allahyari, M. Khodadi, S. Vagnozzi et al., JCAP 02, 003 (2020)

  • [38]

    S. Vagnozzi, C. Bambi, and L. Visinelli, Class. Quant. Grav. 37(8), 087001 (2020)

  • [39]

    M. Khodadi, A. Allahyari, S. Vagnozzi and D. F. Mota, JCAP 09, 026 (2020)

  • [40]

    N. Tsukamoto, Z. Li, and C. Bambi, JCAP 06, 043 (2014)

  • [41]

    N. Tsukamoto, Phys. Rev. D 97(6), 064021 (2018)

  • [42]

    Z. Hu, Z. Zhong, P. C. Li et al., Phys. Rev. D 103(4), 044057 (2021)

  • [43]

    J. Peng, M. Guo, and X. H. Feng, Chin. Phys. C 45(8), 085103 (2021)

  • [44]

    Z. Zhong, Z. Hu, H. Yan et al., Phys. Rev. D 104(10), 104028 (2021)

  • [45]

    Y. Hou, Z. Zhang, H. Yan et al., Phys. Rev. D 106(6), 064058 (2022)

  • [46]

    X. X. Zeng, G. P. Li, and K. J. He, Nucl. Phys. B 974, 115639 (2022)

  • [47]

    G. P. Li and K. J. He, JCAP 06, 037 (2021)

  • [48]

    X. X. Zeng, K. J. He, and G. P. Li, Sci. China Phys. Mech. Astron. 65(9), 290411 (2022)

  • [49]

    K. J. He, X. Zhang, and X. Li, Chin. Phys. C 46(7), 075103 (2022)

  • [50]

    K. J. He, S. C. Tan, and G. P. Li, Eur. Phys. J. C 82(1), 81 (2022)

  • [51]

    S. W. Wei and Y. X. Liu, Eur. Phys. J. Plus 136(4), 436 (2021)

  • [52]

    G. P. Li and K. J. He, Eur. Phys. J. C 81(11), 1018 (2021)

  • [53]

    P. V. P. Cunha, E. Berti, and C. A. R. Herdeiro, Phys. Rev. Lett. 119(25), 251102 (2017)

  • [54]

    M. Guo and S. Gao, Phys. Rev. D 103(10), 104031 (2021)

  • [55]

    A. B. Abdikamalov, A. A. Abdujabbarov, D. Ayzenberg et al., Phys. Rev. D 100(2), 024014 (2019)

  • [56]

    B. Narzilloev, J. Rayimbaev, S. Shaymatov et al., Phys. Rev. D 102(4), 044013 (2020)

  • [57]

    C. A. R. Herdeiro, A. M. Pombo, E. Radu et al., JCAP 04, 051 (2021)

  • [58]

    J. L. Rosa, Phys. Rev. D 107(8), 084048 (2023)

  • [59]

    X. Wang, Y. Hou, and M. Guo, JCAP 05, 036 (2023)

  • [60]

    X. Wang, P. C. Li, C. Y. Zhang et al., Phys. Lett. B 811, 135930 (2020)

  • [61]

    M. Wielgus, J. Horak, F. Vincent et al., Phys. Rev. D 102(8), 084044 (2020)

  • [62]

    M. Guerrero, G. J. Olmo, and D. Rubiera-Garcia, JCAP 04, 066 (2021)

  • [63]

    N. Tsukamoto, Phys. Rev. D 103(6), 064031 (2021)

  • [64]

    J. Peng, M. Guo, and X. H. Feng, Phys. Rev. D 104(12), 124010 (2021)

  • [65]

    S. Guo, G. R. Li, and E. W. Liang, Eur. Phys. J. C 83(7), 663 (2023)

  • [66]

    Y. Chen, P. Wang, H. Wu et al., Eur. Phys. J. C 83(5), 361 (2023)

  • [67]

    P. G. Nedkova, V. K. Tinchev, and S. S. Yazadjiev, Phys. Rev. D 88(12), 124019 (2013)

  • [68]

    M. A. Bugaev, I. D. Novikov, S. V. Repin et al., Astron. Rep. 65(12), 1185 (2021)

  • [69]

    S. Kasuya and M. Kobayashi, Phys. Rev. D 103(10), 104050 (2021)

  • [70]

    G. J. Olmo, J. L. Rosa, D. Rubiera-Garcia et al., Class. Quant. Grav. 40(17), 174002 (2023)

  • [71]

    K. A. Bronnikov, R. A. Konoplya, and T. D. Pappas, Phys. Rev. D 103(12), 124062 (2021)

  • [72]

    N. Tsukamoto, Phys. Rev. D 104(6), 064022 (2021)

  • [73]

    N. Tsukamoto, Phys. Rev. D 105(8), 084036 (2022)

  • [74]

    P. V. P. Cunha, J. Grover, C. Herdeiro et al., Phys. Rev. D 94(10), 104023 (2016)

  • [75]

    J. L. Rosa and D. Rubiera-Garcia, Phys. Rev. D 106(8), 084004 (2022)

  • [76]

    J. L. Rosa, C. F. B. Macedo, and D. Rubiera-Garcia, Phys. Rev. D 108(4), 044021 (2023)

  • [77]

    J. L. Rosa, P. Garcia, F. H. Vincent et al., Phys. Rev. D 106(4), 044031 (2022)

  • [78]

    S. W. Hawking and R. Penrose, Proc. Roy. Soc. Lond. A 314 , 529-548 (1970)

  • [79]

    S. W. Hawking and G. F. R. Ellis, Cambridge University Press, (2023).

  • [80]

    J. M. Bardeen, Proc. Int. Conf. GR5, Tbilisi. 174 (1968)

  • [81]

    E. Ayon-Beato and A. Garcia, Phys. Rev. Lett. 80, 5056 (1998)

  • [82]

    E. Ayon-Beato and A. Garcia, Gen. Rel. Grav. 31, 629 (1999)

  • [83]

    E. Ayon-Beato and A. Garcia, Phys. Lett. B 464, 25 (1999)

  • [84]

    E. Ayon-Beato and A. Garcia, Phys. Lett. B 493, 149 (2000)

  • [85]

    S. A. Hayward, Phys. Rev. Lett. 96, 031103 (2006)

  • [86]

    K. J. He, S. Guo, S. C. Tan et al., Chin. Phys. C 46(8), 085106 (2022)

  • [87]

    M. Visser, Nucl. Phys. B 328, 203 (1989)