• [1]

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

  • [2]

    S. W. Hawking, Nature 248(5443), 30 (1974)

  • [3]

    S. W. Hawking, Commun. Math. Phys. 43(3), 199 (1975)

  • [4]

    D. Christodoulou, Phys. Rev. Lett. 25(22), 1596 (1970)

  • [5]

    J. M. Bardeen, Nature 226(5240), 64 (1970)

  • [6]

    J. D. Bekenstein, Phys. Rev. D 7(8), 2333 (1973)

  • [7]

    G. W. Gibbons and M. J. Perry, Proc. R. Soc. Lond. A 358(1695), 467 (1978)

  • [8]

    G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15(10), 2738 (1977)

  • [9]

    J. Lu, Int. J. Theor. Phys. 38(7), 2029 (1999)

  • [10]

    B. Punsly, Phys. Rev. D 46(4), 1288 (1992)

  • [11]

    T. Damour and R. Ruffini, Phys. Rev. D 14(2), 332 (1976)

  • [12]

    D. G. Boulware, Phys. Rev. D 11(6), 1404 (1975)

  • [13]

    K. Schoutens, H. L. Verlinde, and E. P. Verlinde, Phys. Rev. D 48(6), 2670 (1993)

  • [14]

    E. Verlinde and H. Verlinde, Nucl. Phys. B 406(1-2), 43 (1993)

  • [15]

    A. Miković, Phys. Lett. B 304(1-2), 70 (1993)

  • [16]

    S. Iso, H. Umetsu, and F. Wilczek, Phys. Rev. Lett. 96(15), 151302 (2006)

  • [17]

    K. Murata and J. Soda, Phys. Rev. D 74(4), 044018 (2006)

  • [18]

    L. Susskind, L. Thorlacius, and J. Uglum, Phys. Rev. D 48(8), 3743 (1993)

  • [19]

    A. Almheiri, D. Marolf, J. Polchinski et al., J. High Energy Phys. 2013(2), 1 (2013)

  • [20]

    S. W. Hawking, M. J. Perry, and A. Strominger, J. High Energy Phys. 2017(5), 1 (2017)

  • [21]

    M. K. Parikh and F. Wilczek, Phys. Rev. Lett. 85(24), 5042 (2000)

  • [22]

    M. Parikh, Int. J. Mod. Phys. D 13(10), 2351 (2004)

  • [23]

    T. Padmanabhan, Phys. Rep. 406(2), 49 (2005)

  • [24]

    E. C. Vagenas, Phys. Lett. B 533(3-4), 302 (2002)

  • [25]

    Y. G. Miao, Z. Xue, and S. J. Zhang, Europhys. Lett. 96(1), 10008 (2011)

  • [26]

    M. R. Setare and E. C. Vagenas, Phys. Lett. B 584(1-2), 127 (2004)

  • [27]

    A. J. M. Medved and E. C. Vagenas, Mod. Phys. Lett. A 20(32), 2449 (2005)

  • [28]

    W. Liu, Phys. Lett. B 634(5-6), 541 (2006)

  • [29]

    C. Ding, M. Wang, and J. Jing, Phys. Lett. B 676(1-3), 99 (2009)

  • [30]

    S. Stotyn, K. Schleich, and D. M. Witt, Class. Quantum Grav. 26(6), 065010 (2009)

  • [31]

    W. Israel and Z. Yun, Phys. Rev. D 82(12), 124036 (2010)

  • [32]

    B. Zhang, Q. Y. Cai, L. You et al., Phys. Lett. B 675(1), 98 (2009)

  • [33]

    H. Kim, Phys. Lett. B 703(1), 94 (2011)

  • [34]

    S. Q. Wu and Q. Q. Jiang, J. High Energy Phys. 2006(03), 079 (2006)

  • [35]

    Q. Q. Jiang, S. Q. Wu, and X. Cai, Phys. Rev. D 73(6), 064003 (2006)

  • [36]

    D. Wu, Q. Jiang, S. Wu, L. Zhang et al., Sci. Sin. Phys. Mech. Astron. 43(2), 115 (2013)

  • [37]

    Q. Q. Jiang and X. Cai, J. High Energy Phys. 2009(11), 110 (2009)

  • [38]

    B. Bagchi and S. Sen, Int. J. Mod. Phys. A 37(02), 2150252 (2022)

  • [39]

    J. Zhang and Z. Zhao, J. High Energy Phys. 2005(10), 055 (2005)

  • [40]

    Y. Hu, J. Zhang, and Z. Zhao, Mod. Phys. Lett. A 21(28), 2143 (2006)

  • [41]

    M. Hajebrahimi and K. Nozari, Prog. Theor. Exp. Phys. 2020(4), 043E03 (2020)

  • [42]

    S. Saghafi and K. Nozari, Gen. Relativ. Gravit. 55(1), 20 (2023)

  • [43]

    C. Hu, X. X. Zeng, Y. W. Han et al., Int. J. Theor. Phys. 62(9), 204 (2023)

  • [44]

    X. X. Zeng and S. Z. Yang, Chin. Phys. B 18(2), 462 (2009)

  • [45]

    A. J. M. Medved, Phys. Rev. D 66(12), 124009 (2002)

  • [46]

    S. Hemming and E. Keski-Vakkuri, Phys. Rev. D 64(4), 044006 (2001)

  • [47]

    M. K. Parikh, Phys. Lett. B 546(3-4), 189 (2002)

  • [48]

    S. Kar, Phys. Rev. D 74(12), 126002 (2006)

  • [49]

    J. Zhang and Z. Zhao, Nucl. Phys. B 725(1-2), 173 (2005)

  • [50]

    D. Y. Chen, Q. Q. Jiang, and X. T. Zu, Phys. Lett. B 665(2-3), 106 (2008)

  • [51]

    G. E. Volovik, Int. J. Mod. Phys. D 18(08), 1227 (2009)

  • [52]

    H. S. Singh, C. Singha, and S. K. Upadhyaya, Eur. Phys. J. Plus 140(3), 1 (2025)

  • [53]

    B. Tan, Eur. Phys. J. C 85(4), 375 (2025)

  • [54]

    N. Altamirano, D. Kubizňák, and R. B. Mann, Phys. Rev. D 88(10), 101502 (2013)

  • [55]

    D. Kastor, S. Ray, and J. Traschen, Class. Quantum Grav. 26(19), 195011 (2009)

  • [56]

    X. M. Liu, H. B. Shao, and X. X. Zeng, Adv. High Energy Phys. 2017, 640210 (2017)

  • [57]

    N. Altamirano, D. Kubizňák, R. B. Mann et al., Class. Quantum Grav. 31(4), 042001 (2014)

  • [58]

    S. Arguedas Cuendis, J. Baier, K. Barth et al., Phys. Dark Univ. 26, 100367 (2019)

  • [59]

    M. M. Caldarelli, G. Cognola, and D. Klemm, Class. Quantum Grav. 17(2), 399 (2000)

  • [60]

    S. H. Hendi, S. Panahiyan, B. Eslam Panah et al., Annalen Phys. 528(11-12), 819 (2016)

  • [61]

    D. Kubizňák and R. B. Mann, J. High Energy Phys. 2012(7), 1 (2012)

  • [62]

    R. G. Cai, L. M. Cao, L. Li et al., J. High Energy Phys. 2013(9), 1 (2013)

  • [63]

    B. P. Dolan, A. Kostouki, D. Kubizňák et al., Class. Quantum Grav. 31(24), 242001 (2014)

  • [64]

    R. A. Hennigar, R. B. Mann, and E. Tjoa, Phys. Rev. Lett. 118(2), 021301 (2017)

  • [65]

    S. W. Wei, Y. X. Liu, and R. B. Mann, Phys. Rev. Lett. 123(7), 071103 (2019)

  • [66]

    X. X. Zeng, Y. W. Han, and D. Y. Chen, Chin. Phys. C 43(10), 105104 (2019)

  • [67]

    K. J. He, G. P. Li, and X. Y. Hu, Eur. Phys. J. C 80(3), 209 (2020)

  • [68]

    B. P. Dolan, Class. Quantum Grav. 28(23), 235017 (2011)

  • [69]

    P. Painlevé, Comptes Rendus Acad. Sci. 173, 677 (1921)