• [1]

    A. Einstein, Ann. Phys. (Berlin) 354, 769 (1916)

  • [2]

    A. Einstein and N. Rosen, Phys. Rev. 48(1), 73 (1935)

  • [3]

    A. Einstein, L. Infeld, and B. Hoffmann, Ann. Math. 39(1), 65 (1938)

  • [4]

    W. Greiner, Quantum mechanics: an introduction, (Springer Science, 2011)

  • [5]

    J. Weber, Phys. Rev. 117(1), 306 (1960)

  • [6]

    P. Ball, Nat. Mater 21(2), 132 (2022)

  • [7]

    T. D. Ladd, F. Jelezko, R. Laflamme et al., Nature 464(7285), 45 (2010)

  • [8]

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

  • [9]

    S. A. Hayward, Phys. Rev. D 49(12), 6467 (1994)

  • [10]

    T. Jacobson, G. Kang, and R. C. Myers, Phys. Rev. D 49(12), 6587 (1994)

  • [11]

    A. Ashtekar, T. Pawlowski, and P. Singh, Phys. Rev. Lett. 96(14), 141301 (2006)

  • [12]

    S. Chandrasekhar, Rev. Mod. Phys. 15(1), 1 (1943)

  • [13]

    R. Bousso and S. W. Hawking, Phys. Rev. D 54(10), 6312 (1996)

  • [14]

    A. Paggi, G. Fabbiano, F. Civano et al., Astrophys. J. 823(2), 112 (2016)

  • [15]

    G. T. Horowitz and A. R. Steif, Phys. Rev. D 42(6), 1950 (1990)

  • [16]

    D. Piriz, M. Roy, and J. Wudka, Phys. Rev. D 54(2), 1587 (1996)

  • [17]

    Y. N. Obukhov, A. J. Silenko, and O. V. Teryaev, Phys. Rev. D 84(2), 024025 (2011)

  • [18]

    O. Y. Tsupko and G. S. Bisnovatyi-Kogan, Gravit. Cosmol. 20(3), 220 (2014)

  • [19]

    A. Chatelain, and M. C. Volpe, Phys. Lett. B 801, 135150 (2020)

  • [20]

    T. D. Le, Chin. J. Phys. 73, 147 (2021)

  • [21]

    R. P. Kerr, Phys. Rev. Lett. 11(5), 237 (1963)

  • [22]

    K. Schwarzschild, Gen. Relativ. Gravit 35(5), 951 (2003)

  • [23]

    K. Akiyama et al. (Event Horizon Telescope Collaboration), Astrophys. J. Lett. 875(1), L1 (2019)

  • [24]

    A. Abdujabbarov and B. Ahmedov, Phys. Rev. D 81, 044022 (2010)

  • [25]

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

  • [26]

    A. Ditta, X. Tiecheng, F. Atamurotov et al., Commun. Theor. Phys. 75, 125404 (2023)

  • [27]

    A. Ditta, X. Tiecheng, and M. Yasir, Int. J. Mod. Phys. A 38, 2350041 (2023)

  • [28]

    A. Ditt, X. Tiecheng, S. Mumtaz et al., Phys. Dark Universe 41, 101248 (2023)

  • [29]

    A. Ashraf, A. Ditta, D. Sofuoğlu et al., Phys. Scr. 99, 065011 (2024)

  • [30]

    B. Turimov, J. Rayimbaev, A. Abdujabbarov et al., Phys. Rev. D 102, 064052 (2020)

  • [31]

    M. Boboqambarova, B. Turimov, and A. Abdujabbarov, Mod. Phys. Lett. A 38, 2350071 (2023)

  • [32]

    K. Isomura, R. Suzuki, S. Tomizawa, Phys. Rev. D 107, 084003 (2023)

  • [33]

    J. L. Synge, Principles of Classical Mechanics and Field Theory/Prinzipien der Klassischen Mechanik und Feldtheorie, (New York: Springer, 1960)

  • [34]

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

  • [35]

    J. M. Bardeen, Black Holes, (1973), p. 215 - 239

  • [36]

    H. Falcke, F. Melia, and E. Agol, Astrophys. J. 528, L13 (2000)

  • [37]

    A. Abdujabbarov, F. Atamurotov, Y. Kucukakca et al., Astrophys. Space Sci. 344, 429 (2013)

  • [38]

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

  • [39]

    F. Atamurotov, A. Abdujabbarov, and B. Ahmedov, Astrophys. Space Sci. 348, 179 (2013)

  • [40]

    P. V. P. Cunha, N. A. Eiró, C. A.R. Herdeiro et al, J. Cosmol. Astropart. Phys. 2020, 035 (2020)

  • [41]

    P. Kocherlakota et al. (Event Horizon Telescope Collaboration), Phys. Rev. D 103, 104047 (2021)

  • [42]

    P. Z. He, Q. Q. Fan, H. R. Zhang et al, Eur. Phys. J. C 80, 1195 (2020)

  • [43]

    A. de Vries, Class. Quant. Grav. 17, 123 (2000)

  • [44]

    A. A. Abdujabbarov, L. Rezzolla, and B. J. Ahmedov, Mon. Not. R. Astron. Soc. 454, 2423 (2015)

  • [45]

    A. Grenzebach, V. Perlick, and C. Lämmerzahl, Phys. Rev. D 89, 124004 (2014)

  • [46]

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

  • [47]

    F. Atamurotov, S. G. Ghosh, and B. Ahmedov, Eur. Phys. J. C 76, 273 (2016)

  • [48]

    G. Bozza, F. De Luca, and G. Scarpetta, Phys. Rev. D 74, 063001 (2006)

  • [49]

    E. F. Eiroa and D. F. Torres, Phys. Rev. D 69, 063004 (2004)

  • [50]

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

  • [51]

    K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D 65, 103004 (2002)

  • [52]

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

  • [53]

    F. de Felice and F. Sorge, Class. Quant. Grav. 20, 469 (2003)

  • [54]

    F. de Felice, F. Sorge, and S. Zilio, Class. Quant. Grav. 21, 961 (2004)

  • [55]

    J. R. Rayimbaev, Astrophys. Space Sci. 361, 288 (2016)

  • [56]

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

  • [57]

    A. N. Aliev, D. V. Galtsov, and V. I. Petukhov, Astrophys. Space Sci. 124, 137 (1986)

  • [58]

    A. N. Aliev and D. V. Gal'tsov, Sov. Phys. Usp. 32, 75 (1989)

  • [59]

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

  • [60]

    C. A. Benavides-Gallego, A. Abdujabbarov, D. Malafarina et al, Phys. Rev. D 99, 044012 (2019)

  • [61]

    Z. Stuchlík, J. Schee, and A. Abdujabbarov, Phys. Rev. D 89, 104048 (2014)

  • [62]

    Z. Stuchlík and M. Kološ, Eur. Phys. J. C 76, 32 (2016)

  • [63]

    J. Rayimbaev and P. Tadjimuratov, Phys. Rev. D 102, 024019 (2020)

  • [64]

    J. Rayimbaev, B. Turimov, F. Marcos et al, Mod. Phys. Lett. A 35, 2050056 (2019)

  • [65]

    J. Rayimbaev, B. Turimov, and B. Ahmedov, Int. J. Mod. Phys. D 28, 1950128 (2019)

  • [66]

    J. R. Rayimbaev, B. J. Ahmedov, N. B. Juraeva et al, Astrophys. Space Sci. 356, 301 (2015)

  • [67]

    S. Chen, M. Wang, and J. Jing, JHEP 2016(9), 82 (2016)

  • [68]

    K. Hashimoto and N. Tanahashi, Phys. Rev. D 95, 024007 (2017)

  • [69]

    S. Dalui, B. R. Majhi, and P. Mishra, Phys. Lett. B 788, 486 (2019)

  • [70]

    W. Han, Gen. Relativ. Gravit. 40, 1831 (2008)

  • [71]

    A. P. S. de Moura, and P. S. Letelier, Phys. Rev. E 61, 6506 (2000)

  • [72]

    V. S. Morozova, L. Rezzolla, and B. J. Ahmedov, Phys. Rev. D 89, 104030 (2014)

  • [73]

    B. Narzilloev, J. Rayimbaev, A. Abdujabbarov et al, Eur. Phys. J. C 80, 1074 (2020)

  • [74]

    Z. Stuchlík, M. Kološ, J. Kovář et al, Universe 6, 26 (2020)

  • [75]

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

  • [76]

    J. D. Bekenstein, Phys. Rev. D 9, 3292 (1974)

  • [77]

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

  • [78]

    N. Altamirano, D. Kubiznak, R. B. Mann et al., Galaxies 2, 89 (2014)

  • [79]

    L. Susskind, J. Math. Phys. 36, 6377 (1995)

  • [80]

    R. Bousso, Rev. Mod. Phys. 74, 825 (2002)

  • [81]

    D. Bak and S. J. Rey, Class. Quant. Grav. 17, L1 (2000)

  • [82]

    S. K. Rama, Phys. Lett. B 457, 268 (1999)

  • [83]

    A. Ashtekar, Lectures on Non-perturbative Canonical Gravity, (Singapore: World Scientific , 1991)

  • [84]

    T. R. Govindarajan, R. K. Kaul, and V. Suneeta, Class. Quant. Grav. 18, 2877 (2001)

  • [85]

    S. Carlip, Class. Quant. Grav. 17, 4175 (2000)

  • [86]

    R. B. Mann and S. N. Solodukhin, Nucl. Phys. B 523, 293 (1998)

  • [87]

    A. J. M. Medved and G. Kunstatter, Phys. Rev. D 60, 104029 (1999)

  • [88]

    A. J. M. Medved and G. Kunstatter, Phys. Rev. D 63, 104005 (2001)

  • [89]

    S. N. Solodukhin, Phys. Rev. D 57, 2410 (1998)

  • [90]

    A. Sen, JHEP 04, 156 (2013)

  • [91]

    A. Sen, Entropy 13, 1305 (2011)

  • [92]

    D. A. Lowe and S. Roy, Phys. Rev. D 82, 063508 (2010)

  • [93]

    J. Jing and M. L. Yan, Phys. Rev. D 63, 084028 (2001)

  • [94]

    D. Birmingham and S. Sen, Phys. Rev. D 63, 47501 (2001)

  • [95]

    S. Fernando, Gen. Relativ. Gravit. 37, 461 (2005)

  • [96]

    T. Harmark, J. Natario, and R. Schiappa, Adv. Theor. Math. Phys. 14, 727 (2010)

  • [97]

    P. Boonserm, C. H. Chen, T. Ngampitipan et al., Phys. Rev. D 104, 084054 (2021)

  • [98]

    K. Jusufi, M. Amir, M. S. Ali, et al., Phys. Rev. D 102, 064020 (2020)

  • [99]

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

  • [100]

    P. Boonserm, T. Ngampitipan, and P. Wongjun, Eur. Phys. J. C 79, 330 (2019)

  • [101]

    I. Sakalli, Phys. Rev. D 94, 084040 (2016)

  • [102]

    A. Al-Badawi, I. Sakalli, and S. Kanzi, Ann. Phys. 412, 168026 (2020)

  • [103]

    A. Al-Badawi, S. Kanzi, and I. Sakalli, Eur. Phys. J. Plus 135, 219 (2020)

  • [104]

    H. Gursel and I. Sakalli, Eur. Phys. J. C 80, 234 (2020)

  • [105]

    S. Kanzi and İ. Sakallı, Eur. Phys. J. C 81, 501 (2021)

  • [106]

    F. J. Zerill, Phys. Rev. D 9, 860 (1974)

  • [107]

    S. S. Gubser, Commun. Math. Phys. 203, 325 (1999)

  • [108]

    J. M. Maldacena and A. Strominger, Phys. Rev. D 55, 861 (1997)

  • [109]

    I. R. Klebanov and S. D. Mathur, Nucl. Phys. B 500, 115 (1997)

  • [110]

    Y. S. Myung and H. W. Lee, Class. Quantum Grav. 20, 3533 (2003)

  • [111]

    P. R. Anderson, A. Fabbri, and R. Balbinot, Phys. Rev. D 91, 064061 (2015)

  • [112]

    S. S. Gubser and I. R. Klebanov, Phys. Rev. Lett. 77, 4491 (1996)

  • [113]

    M. Cvetic and F. Larsen, Phys. Rev. D 56, 4994 (1997)

  • [114]

    W.T. Kim and J. J. Oh, Phys. Lett. B 461, 189 (1999)

  • [115]

    L. H. Ford, Phys. Rev. D 12, 2963 (1975)

  • [116]

    A. D. Sakharov, Sov. J. Exp. Theor. Phys. 22, 241 (1966)

  • [117]

    E.B. Gliner, Sov. J. Exp. Theor. Phys. 22, 378 (1966)

  • [118]

    J. Bardeen, in Proceedings of the International Conference GR5, (Russia, 1968)

  • [119]

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

  • [120]

    K. A. Bronnikov, Phys. Rev. Lett. 85, 4641 (2000)

  • [121]

    K. A. Bronnikov, Phys. Rev. D 63, 044005 (2001)

  • [122]

    S. Ansoldi, arXiv: 0802.0330

  • [123]

    J. P. S. Lemos and V. T. Zanchin, Phys. Rev. D 83, 124005 (2011)

  • [124]

    J. Schee and Z. Stuchlik, J. Cosmol. Astropart. Phys. 06, 48 (2015)

  • [125]

    M. S. Ali and S. G. Ghosh, Phys. Rev. D 98, 084025 (2018)

  • [126]

    A. Kumar, D.V. Singh, and S.G. Ghosh, Eur. Phys. J. C 79, 275 (2019)

  • [127]

    I. Dymnikova, Gen. Relativ. Gravit. 24, 235 (1992)

  • [128]

    I. Dymnikova, Class. Quant. Grav. 21, 4417 (2004)

  • [129]

    L. Xiang, Y. Ling, and Y. G. Shen, Int. J. Mod. Phys. D 22, 1342016 (2013)

  • [130]

    L. Balart and E. C. Vagenas, Phys. Lett. B 730, 14 (2014)

  • [131]

    L. Balart and E.C. Vagenas, Phys. Rev. D 90, 124045 (2014)

  • [132]

    J. C. S. Neves and A. Saa, Phys. Lett. B 734, 44 (2014)

  • [133]

    S. Fernando, Int. J. Mod. Phys. D 26, 1750071 (2017)

  • [134]

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

  • [135]

    S. G. Ghosh, A. Kumar, and D. V. Singh, Phys. Dark Univ. 30, 100660 (2020)

  • [136]

    A. Kumar, D. V. Singh, and S. G. Ghosh, Ann. Phys. 419, 168214 (2020)

  • [137]

    A. Kumar, D. Baboolal, and S. G. Ghosh, Universe 8, 244 (2022)

  • [138]

    V. P. Frolov and A. Zelnikov, Phys. Rev. D 95, 124028 (2017)

  • [139]

    V. P. Frolov, Phys. Rev. D 94, 104056 (2016)

  • [140]

    K. Lin, J. Li, and S. Yang, Int. J. Theor. Phys. 52, 3771 (2013)

  • [141]

    T. Chiba and M. Kimura, Prog. Theor. Exp. Phys. 2017, 043E01 (2017)

  • [142]

    S. H. Mehdipour and M. H. Ahmadi, Nucl. Phys. B 926, 49 (2018)

  • [143]

    S. S. Zhao and Y. Xie, Eur. Phys. J. C 77, 272 (2017)

  • [144]

    A. Kumar, S. G. Ghosh, and A. Beesham, Eur. Phys. J. Plus 139, 439 (2024)

  • [145]

    A. Kumar, R. K. Walia, and S. G. Ghosh, Universe 8, 232 (2022)

  • [146]

    A. Kumar, S. G. Ghosh, and A. Wang, Phys. Dark Univ. 46, 101608 (2024)

  • [147]

    A. Ashraf, A. Bouzenada, S. K. Maurya et al, Phys. Dark Univ. 47, 101787 (2025)

  • [148]

    G. Mustafa, P. Channuie, F. Javed et al, Phys. Dark Univ. 47, 101765 (2025)

  • [149]

    G. Mustafa, S. K. Maurya, P. Channuie et al, Phys. Dark Univ. 47, 101753 (2025)

  • [150]

    A. Ashraf, A. S. Alqahtani, F. Javed et al, Phys. Dark Univ. 47, 101725 (2025)

  • [151]

    A. Ashraf, A. Ditta, A. Bouzenada et al, Phys. Dark Univ. 48, 101836 (2025)

  • [152]

    A. Ashraf, A. Ditta, A. Bouzenada et al., Phys. Dark Univ. 47, 101823 (2025)

  • [153]

    A. Ditta, A. Bouzenada, G. Mustafa et al, Phys. Dark Univ. 47, 101818 (2025)

  • [154]

    A. Ditta, A. Bouzenada, G. Mustafa et al, Phys. Dark Univ. 46, 101573 (2024)

  • [155]

    A. Ashraf, A. Bouzenada, S. K. Maurya et al, Phys. Dark Univ. 48, 101874 (2025)

  • [156]

    A. Ashraf, A. Ditta, A. Bouzenada et al, Eur. Phys. J. C 85(4), 383 (2025)

  • [157]

    A. Ashraf, T. Naseer, H. Chaudhary et al, Nucl. Phys. B 1014, 116873 (2025)

  • [158]

    A. Kumar, A. Sood, S. G. Ghosh et al., Particles 7, 1017 (2024)

  • [159]

    F. Atamurotov, M. Alloqulov, A. Abdujabbarov et al., Eur. Phys. J. Plus 137, 634 (2022)

  • [160]

    P. V. P. Cunha and C. A. R. Herdeiro, Gen. Relativ. Gravit. 50, 42851 (2018)

  • [161]

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

  • [162]

    K. Hioki, and K. I. Maeda, Phys. Rev. D 80, 024042 (2009)

  • [163]

    T. S. Biro and V. G. Czinner, Phys. Lett. B 726, 861 (2013)

  • [164]

    B. Pourhassan and M. Faizal, Europhys. Lett. 111, 40006 (2015)

  • [165]

    G. W. Gibbons, S. W. Hawking, and J. M. Perry, Nucl. Phys. B 138, 141 (1978)

  • [166]

    J. B. Hartle and S. W. Hawking, Phys. Rev. D 13, 2188 (1976)

  • [167]

    R. F. Sobreiro and V. J. V. Otoya, Class. Quant. Grav. 24, 4937 (2007)

  • [168]

    L. Bonora and A. A. Bytsenko, Nucl. Phys. B 852, 508 (2011)

  • [169]

    Z. Akhtar, R. Babar, and R. Ali, Ann. Phys. 448, 169190 (2023)

  • [170]

    G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2752 (1977)

  • [171]

    V. Iyer and R. M. Wald, Phys. Rev. D 52, 4430 (1995)

  • [172]

    S. Das, P. Majumdar, and R. K. Bhaduri, Class. Quant. Grav. 19, 2355 (2002)

  • [173]

    J. Sadeghi, B. Pourhassan, and F. Rahimi, Canad. J. Phys. 92, 1638 (2014)

  • [174]

    E. Berti, V. Cardoso, and M. Casals, Phys. Rev. D 73, 024013 (2006)

  • [175]

    C. Flammer, Spheroidal Wave Functions, (Stanford: Stanford University Press, 1957)