• [1]

    W. Israel, Phys. Rev 164, 1776 (1967)

  • [2]

    B. Carter, Phys. Rev. Lett. 26, 331 (1971)

  • [3]

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

  • [4]

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

  • [5]

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

  • [6]

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

  • [7]

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

  • [8]

    J. M. Bardeen, W. H. Press, and S. A. Teukolsky, Astrophys. J. 178, 347 (1972)

  • [9]

    X. J. Gao, J. M. Chen, H. Zhang et al., Phys. Lett. B 822, 136683 (2021)

  • [10]

    Y. Mizuno, Z. Younsi, C. M. Fromm et al., Nature Astron. 2, 585 (2018)

  • [11]

    V. Perlick and O.Y. Tsupko, Phys. Rep. 947, 39 (2022)

  • [12]

    X. J. Gao, X. K. Yan, Y. Yin et al., Eur. Phys. J. C 83, 281 (2023)

  • [13]

    D. Anish, A. Saha, and S. Gangopadhyay, Class. Quant. Grav. 38, 065015 (2021)

  • [14]

    C. K. Qiao and M. Zhou, JCAP 12, 005 (2023)

  • [15]

    R. Moderski and M. Rogatko, Phys. Rev. D 64, 044024 (2001)

  • [16]

    M. Bouhmadi-Lopez, S. Brahma, C. Y. Chen et al., arXiv: 1902.07874 [gr-qc]

  • [17]

    C. Y. Chen and P. Chen, Phys. Rev. D 99, 104003 (2019)

  • [18]

    C. V. Vishveshwara, Nature 227, 936 (1970)

  • [19]

    B. P. Abbott et al. (LIGO Scientific, Virgo Collaboration), Phys. Rev. Lett. 116, 061102 (2016)

  • [20]

    V. Boyanov, K. Destounis, R. Panosso Macedo et al., Phys. Rev. D 107, 064012 (2023)

  • [21]

    V. Cardoso, E. Franzin, and P. Pani, Phys. Rev. Lett. 116, 171101 (2016)

  • [22]

    R. A. Konoplya and A. Zhidenko, Rev. Mod. Phys. 83, 793 (2011)

  • [23]

    R. A. Konoplya, Phys. Rev. D 66, 044009 (2002)

  • [24]

    R. A. Konoplya, A. Zhidenko, and A F Zinhailo, Class. Quant. Grav. 36, 155002 (2019)

  • [25]

    R. A. Konoplya, Phys. Rev. D 68, 024018 (2003)

  • [26]

    R. A. Konoplya and A. Zinhailo, Phys. Rev. D 99, 104060 (2019)

  • [27]

    R. A. Konoplya, Phys. Lett. B 804, 135363 (2020)

  • [28]

    R. A. Konoplya and A. Zhidenko, JCAP 08, 008 (2023)

  • [29]

    R. A. Konoplya and A. Zhidenko, Phys. Rev. D 107, 044009 (2023)

  • [30]

    D. Kastor et al., Class. Quant. Grav. 27, 235014 (2010)

  • [31]

    Ė. Komatsu et al. (WMAP Collaboration), Astrophys. J. Suppl. 192, 18 (2011)

  • [32]

    R. Pantig and A. Övgün , Eur. Phys. J. C 82, 391 (2022)

  • [33]

    A. G. Riess, L. G. Strolger, J. Tonry et al., Astrophys. J. 607, 665 (2004)

  • [34]

    V. V. Kiselev, Class. Quantum Grav 20, 1187 (2003)

  • [35]

    S. B. Chen, B. Wang, and R. Su, Phys. Rev. D 77, 124011 (2008)

  • [36]

    G. Q. Li, Phys. Lett. B 735, 256 (2014)

  • [37]

    A. Younas, M. Jamil, and S. Hussain, Phys. Rev. D 92, 084042 (2015)

  • [38]

    H. Chen, B. C. Lütfüolu, H. Hassanabadi et al., Phys. Lett. B 827, 136994 (2022)

  • [39]

    B. Hamil and B.C. Lütfüoğlu , Nucl. Phys. B 990, 116191 (2023)

  • [40]

    G. Siddhartha, T. Matos, D. Nunez et al., arXiv: astro-ph/0003105

  • [41]

    H. X. Zhang, Y. Chen, T. C. Ma et al., Chin. Phys. C 45, 055103 (2021)

  • [42]

    S. Qi, Q. Li, Y. Zhang et al., Mod. Phys. Lett. A 38, 2350102 (2023)

  • [43]

    K. Saurabh and K. Jusufi, Eur. Phys. J. C 81, 490 (2021)

  • [44]

    S. Shaymatov, B. Ahmedov, and M. Jamil, Eur. Phys. J. C 81, 588 (2021)

  • [45]

    A. Das, A. R. Chowdhury, and S. Gangopadhyay, Class. Quant. Grav. 41, 015018 (2024)

  • [46]

    J. P.S. Lemos, Class. Quant. Grav. 12, 1081 (1995)

  • [47]

    C.-G. Huang and C.-B. Liang, Phys. Lett. A 201, 27 (1995)

  • [48]

    Jose P.S. Lemos and Vilson T. Zanchin, Phys. Rev. D 54, 3840 (1996)

  • [49]

    R.-G. Cai and Y.-Z. Zhang, Phys. Rev. D 54, 4891 (1996)

  • [50]

    R.-G. Cai, J.-Y. Ji, and K.-S. Soh, Phys. Rev. D 57, 6547 (1998)

  • [51]

    S. Aminneborg, I. Bengtsson, S. Holst et al., Class. Quant. Grav. 13, 2707 (1996)

  • [52]

    R. B. Mann, Class. Quant. Grav., 14, L109 (1997)

  • [53]

    L. Vanzo, Phys. Rev. D 56, 6475 (1997)

  • [54]

    D. R. Brill, J. Louko, and P. Peldan, Phys. Rev. D 56, 3600 (1997)

  • [55]

    D. Klemm, V. Moretti, and L. Vanzo, Phys. Rev. D 57, 6127 (1998)

  • [56]

    R.-G. Cai and K.-S. Soh, Phys. Rev. D 59, 044013 (1999)

  • [57]

    D. Birmingham, Class. Quant. Grav. 16, 1197 (1999)

  • [58]

    R.-G. Cai, Phys. Rev. D 65, 084014 (2002)

  • [59]

    M.-Y. Zhang, H. Chen, H. Hassanabadi et al., Eur. Phys. J. C 83, 773 (2023)

  • [60]

    G. Abbas and R. H. Ali, Eur. Phys. J. C 83, 407 (2023)

  • [61]

    M. H. Li and K. C. Yang, Phys. Rev. D 86, 123015 (2012)

  • [62]

    V. Ferrari and B. Mashhoon, Phys. Rev. D 30, 295 (1984)

  • [63]

    E. W. Leaver, Proc. Roy. Soc. Lond. A 402, 285 (1985)

  • [64]

    N. Heidari and H. Hassanabadi, Phys. Lett. B 839, 137814 (2023)

  • [65]

    L. Dong, Y. Yang, A. Övgün et al., Eur. Phys. J. C 83, 565 (2023)

  • [66]

    A. Baruah, A. Övgün, and A. Deshamukhya , Ann. Phys. 455, 169393 (2023)

  • [67]

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

  • [68]

    B. F. Schutz and C. M. Will, Astrophys. J. 291, L33 (1985)

  • [69]

    i. Sakalli and S. Kanzi, Turk. J. Phys. 46, 51 (2022)

  • [70]

    L. C. Crispino, A. Higuchi, E. S. Oliveira et al., Phys. Rev. D 87, 104034 (2013)

  • [71]

    V. Cardoso, M. Cavaglia, and L. Gualtieri, Phys. Rev. Lett. 96, 071301 (2006)

  • [72]

    J. A. V Campos, M. A. Anacleto, F.A. Brito et al., Sci. Rep. 12, 8516 (2022)

  • [73]

    S. Iyer and C. M. Will, Phys. Rev. D 35, 3621 (1987)

  • [74]

    S. Dey and S. Chakrabarti, Eur. Phys. J. C 79, 504 (2019)

  • [75]

    H. Chen, H. Hassanabadi, B. C. Lütfüolu et al., Gen. Rel. Grav. 54, 143 (2022)

  • [76]

    S. Hawking, Commun. Math. Phys. 43, 199 (1975)

  • [77]

    K. Jusufi, Phys. Rev. D 101, 124063 (2020)

  • [78]

    H. Falcke, F. Melia, and E. Agol, Astrophys. J. 528, L13 (1999)

  • [79]

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

  • [80]

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

  • [81]

    P. C. Li, M. Y. Guo, and B. Chen, Phys. Rev. D 101, 084041 (2020)

  • [82]

    I. Cimdiker, D. Demir, and A. Övgün , Phys. Dark Univ. 34, 100900 (2021)

  • [83]

    A. F. Zakharov, F. Paolis, G. Ingrosso et al., New Astron. Rev. 56, 64 (2012)

  • [84]

    S. Vagnozzi, R. Roy, Y. D. Tsai et al., Class. Quant. Grav. 40, 16 (2023)

  • [85]

    R. Kumar and S. G. Ghosh, Astrophys. J. 892, 78 (2020)

  • [86]

    B. Cuadros-Melgar, R. Fontana, and J. Oliveira, Phys. Lett. B 811, 135966 (2020)

  • [87]

    R. Pantig, P. K. Yu, E. T. Rodulfo et al., Ann. Phys. 436, 168722 (2022)

  • [88]

    G. Mustafa, F. Atamurotov, I. Hussain et al., Chin. Phys. C 46, 125107 (2022)

  • [89]

    B. Hamil and B. C. Lütfüoğlu, Phys. Dark Univ. 42, 101293 (2023)

  • [90]

    Y. Yang, D. Liu, A. Övgün et al., Eur. Phys. J. C 84, 63 (2024)

  • [91]

    Y. Yang, D. Liu, A. Övgün et al., Phys. Rev. D 109, 024002 (2024)

  • [92]

    K. Jusufi, Phys. Rev. D 101, 084055 (2020)

  • [93]

    B. Carter, Phys. Rev. 174, 1559 (1968)

  • [94]

    E. González, K. Jusufi, G. Leon et al., Phys. Dark Univ. 42, 101304 (2023)

  • [95]

    S. Capozziello, S. Zare, D. F. Mota et al., J. Cosmol. Astropart. Phys. 05, 027 (2023)