• [1]

    A. Abdujabbarov, B. Toshmatov, Z. Stuchlík, and B. Ahmedov, Int. J. Mod. Phys. D 26, 1750051 (2017)

  • [2]

    Event Horizon Telescope Collaboration, Astrophysical Journal Letters 875, L1 (2019), arXiv: 1906.11238[astro-ph.GA]

  • [3]

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

  • [4]

    C. Bambi, Black Holes: A Laboratory for Testing Strong Gravity (Springer, Singapore, 2017).

  • [5]

    T. Johannsen, Classical and Quantum Gravity 33, 124001 (2016)

  • [6]

    R. A. Konoplya and A. Zhidenko, Reviews of Modern Physics 83, 793 (2011)

  • [7]

    N. Heidari, A. A. A. Filho, R. C. Pantig, and A. Övgün, arXiv preprint 2024, arXiv:2410.08246 (2025), arXiv:2410.08246[gr-qc].

  • [8]

    D. Psaltis, Living Reviews in Relativity 11, 9 (2008)

  • [9]

    V. Cardoso, E. Franzin, and P. Pani, Physical Review Letters 116, 171101 (2016)

  • [10]

    K. A. Bronnikov and O. B. Zaslavskii, Phys. Rev. D 84, 084013 (2011), arXiv: 1107.4701[gr-qc]

  • [11]

    L. Herrera and N. O. Santos, Phys. Rep. 286, 53 (1997)

  • [12]

    I. Cho and H.-C. Kim, Chinese Physics C 43, 025101 (2019), arXiv: 1703.01103[gr-qc]

  • [13]

    M. S. Morris and K. S. Thorne, Am. J. Phys. 56, 395 (1988)

  • [14]

    H.-C. Kim, B.-H. Lee, W. Lee, and Y. S. Myung, Phys. Rev. D 101, 064067 (2020), arXiv: 1912.09709[gr-qc]

  • [15]

    B.-H. Lee, W. Lee, and Y. S. Myung, Phys. Rev. D 103, 064026 (2021), arXiv: 2101.04862[gr-qc]

  • [16]

    A. Ghosh and U. Debnath, arXiv preprint 2025, arXiv:2506.15240 (2025), arXiv:2506.15240[gr-qc].

  • [17]

    J. Rayimbaev, A. Abdujabbarov, M. Jamil, and W.-B. Han, Nuclear Physics B 966, 115364 (2021), arXiv: 2009.04898[gr-qc]

  • [18]

    Z. Stuchlík and J. Vrba, arXiv preprint 2021, arXiv:2110.07411 (2021), arXiv:2110.07411[astro-ph.HE].

  • [19]

    Y. Feng, A. Ashraf, S. Mumtaz, S. K. Maurya, G. Mustafa, and F. Atamurotov, Journal of High Energy Astrophysics 43, 158 (2024)

  • [20]

    A. Çalışkan, G. Mustafa, T. Naseer, S. K. Maurya, E. Güdekli, S. Murodov, and F. Atamurotov, Journal of High Energy Astrophysics 44, 99 (2024)

  • [21]

    M. Zajaček, A. Tursunov, A. Eckart, et al., arXiv preprint (2018), arXiv:1812.03574[astro-ph.GA].

  • [22]

    M. D. Johnson, V. L. Fish, S. S. Doeleman, et al., arXiv preprint (2015), arXiv:1512.01220[astro-ph.HE].

  • [23]

    R. M. Wald, Phys. Rev. D 10, 1680 (1974)

  • [24]

    Event Horizon Telescope Collaboration, Astrophys. J. Lett. 964, L25 (2024)

  • [25]

    B. Narzilloev, D. Malafarina, A. Abdujabbarov, and C. Bambi, Eur. Phys. J. C 80, 784 (2020), arXiv: 2003.11828[gr-qc]

  • [26]

    B. Narzilloev, D. Malafarina, A. Abdujabbarov, B. Ahmedov, and C. Bambi, Phys. Rev. D 104, 064016 (2021), arXiv: 2105.09174[gr-qc]

  • [27]

    B. Narzilloev, I. Hussain, A. Abdujabbarov, and B. Ahmedov, Eur. Phys. J. Plus 137, 645 (2022), arXiv: 2205.11760[gr-qc]

  • [28]

    B. Narzilloev and B. Ahmedov, New Astron. 98, 101922 (2023)

  • [29]

    B. Narzilloev and B. Ahmedov, Int. J. Mod. Phys. A 38, 2350026 (2023)

  • [30]

    H. Alibekov, B. Narzilloev, A. Abdujabbarov, and B. A, Symmetry 15 (2023), 10.3390/sym15071414.

  • [31]

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

  • [32]

    A. Davlataliev, B. Narzilloev, I. Hussain, A. Abdujabbarov, and B. Ahmedov, Phys. Dark Univ. 46, 101569 (2024)

  • [33]

    A. N. Aliev and N. Özdemir, Mon. Not. R. Astron. Soc. 336, 241 (2002)

  • [34]

    B. Saidov, B. Narzilloev, A. Abdujabbarov, M. Khudoyberdieva, and B. Ahmedov, Universe 10, 454 (2024)

  • [35]

    J. Rayimbaev, A. Demyanova, U. Camci, A. Abdujabbarov, and B. Ahmedov, International Journal of Modern Physics D 30, 2150019 (2021), arXiv: 2010.04969[gr-qc]

  • [36]

    M. Kološ, M. Shahzadi, and A. Tursunov, Eur. Phys. J. C 83, 323 (2023)

  • [37]

    S. Kenzhebayeva, S. Toktarbay, A. Tursunov, and M. Kološ, Phys. Rev. D 109, 063005 (2024)

  • [38]

    C. Li, P. Zhang, et al., Eur. Phys. J. C 82, 494 (2022)

  • [39]

    V. V. Kiselev, Classical and Quantum Gravity 20, 1187 (2003), arXiv: gr-qc/0210040[gr-qc]

  • [40]

    S. Shaymatov, D. Malafarina, and B. Ahmedov, Physics of the Dark Universe 34, 100891 (2021)

  • [41]

    M. Boltatoshova, B. Narzilloev, and B. Ahmedov, Communications in Theoretical Physics 77, 085403 (2025)

  • [42]

    A. Abdujabbarov, B. Ahmedov, and A. Hakimov, arXiv preprint (2011), arXiv:1101.4741[gr-qc].

  • [43]

    M. Kološ, Z. Stuchlík, and A. Tursunov, Classical and Quantum Gravity 32, 165009 (2015), arXiv: 1506.06799[gr-qc]

  • [44]

    M. Takahashi and H. Koyama, The Astrophysical Journal 693, 472 (2009), arXiv: 0807.0277[astro-ph]

  • [45]

    O. Kopáček, V. Karas, J. Kovář, and Z. Stuchlík, Astrophys. J. 722, 1240 (2010), arXiv: 1008.4650[astro-ph.HE]

  • [46]

    O. Kopáček and V. Karas, Astrophys. J. 787, 117 (2014), arXiv: 1404.5495[astro-ph.HE]

  • [47]

    D. Li and X. Wu, Eur. Phys. J. Plus 134, 96 (2019), arXiv: 1803.02119[gr-qc]

  • [48]

    M. Yi and X. Wu, Phys. Scr. 95, 085008 (2020)

  • [49]

    M. Azreg-Aïnou, The European Physical Journal C 76, 414 (2016), arXiv: 1603.07894[gr-qc]

  • [50]

    X. Sun, X. Wu, Y. Wang, C. Deng, B. Liu, and E. Liang, Universe 7, 410 (2021)

  • [51]

    A. Davlataliev, B. Narzilloev, I. Hussain, A. Abdujabbarov, and B. Ahmedov, The European Physical Journal C 84, 694 (2024)

  • [52]

    S. U. Khan and Z.-M. Chen, The European Physical Journal C 83, 704 (2023)

  • [53]

    V. P. Frolov and A. A. Shoom, Phys. Rev. D 82, 084034 (2010), arXiv: 1008.2985[gr-qc]

  • [54]

    I. D. Novikov and K. S. Thorne, in Les Houches Summer School of Theoretical Physics: Black Holes (1973) pp. 343–550.

  • [55]

    D. N. Page and K. S. Thorne, Astrophys. J. 191, 499 (1974)

  • [56]

    K. S. Thorne, Astrophysical Journal, Vol. 191, pp. 507-520 (1974) 191, 507 (1974).