• [1]

    A. S. Eddington, Space, time and gravitation. an outline of the general relativity theory (1920)

  • [2]

    F. W. Dyson, A. S. Eddington, and C. Davidson, Phil. Trans. R. Soc. 220, 291 (1920)

  • [3]

    V. Perlick, Living Rev. Relativ. 7, 9 (2004)

  • [4]

    K. S. Virbhadra, Phys. Rev. D 79, 083004 (2009), arXiv: 0810.2109[gr-qc]

  • [5]

    K. S. Virbhadra and C. R. Keeton, Phys. Rev. D 77, 124014 (2008), arXiv: 0710.2333[gr-qc]

  • [6]

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

  • [7]

    K. S. Virbhadra, D. Narasimha, and S. M. Chitre, Astronomy and Astrophysics 337, 1 (1998), arXiv: 9801174[astro-ph]

  • [8]

    C. M. Will, Class. Quant. Grav. 32, 124001 (2015), arXiv: 1409.7812[physics.hist-ph]

  • [9]

    R. A. Vanderveld, M. J. Mortonson, W. Hu et al., Phys. Rev. D 85, 103518 (2012), arXiv: 1203.3195[astro-ph.CO]

  • [10]

    H. J. He and Z. Zhang, J. Cosmol. A. P. 2017, 036 (2017), arXiv: 1701.03418

  • [11]

    S. Jung and C. S. Shin, Phys. Rev. Lett 122, 041103 (2019), arXiv: 1712.01396[astro-ph.CO]

  • [12]

    P. Schneider, J. Ehlers, and E. E. Falco, Gravitational Lenses (1992)

  • [13]

    R. D. Blandford and R. Narayan, Annual Rev. Astron. Astrophys. 30, 311 (1992)

  • [14]

    A. O. Petters, H. Levine, and J. Wambsganss, Singularity theory and gravitational lensing (2001)

  • [15]

    C. R. Keeton and A. O. Petters, Phys. Rev. D 72, 104006 (2005), arXiv: gr-qc/0511019[gr-qc]

  • [16]

    M. C. Werner and A. O. Petters, Phys. Rev. D 76, 064024 (2007), arXiv: 0706.0132[gr-qc]

  • [17]

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

  • [18]

    M. Sereno and F. de Luca, Phys. Rev. D 74, 123009 (2006), arXiv: 0609435[astro-ph]

  • [19]

    V. Bozza, S. Capozziello, G. Iovane et al., Gen. Rel. Grav. 33, 1535 (2001), arXiv: 0102068[gr-qc]

  • [20]

    V. Bozza, Phys. Rev. D 66, 103001 (2002), arXiv: 0208075[gr-qc]

  • [21]

    N. U. Molla, H. Chaudhary, G. Mustafa et al., Eur. Phys. J. C 84, 574 (2024), arXiv: 2310.14234[gr-qc]

  • [22]

    S. E. Vázquez and E. P. Esteban, Nuovo Cim. B 119, 489 (2004), arXiv: gr-qc/0308023

  • [23]

    S. U. Islam, J. Kumar, and S. G. Ghosh, J. Cosmol. A. P. 2021, 013 (2021), arXiv: 2104.00696[gr-qc]

  • [24]

    J. Kumar, S. U. Islam, and S. G. Ghosh, Eur. Phys. J. C 82, 443 (2022), arXiv: 2109.04450[gr-qc]

  • [25]

    N. U. Molla, S. G. Ghosh, and U. Debnath, Phys. Dark Universe 44, 101495 (2024)

  • [26]

    S. S. Zhao and Y. Xie, Phys. Lett. B 774, 357 (2017)

  • [27]

    M. Sereno, Mon. Not. Roy. Astron. Soc. 344, 942 (2003), arXiv: astro-ph/0307243[astro-ph]

  • [28]

    A. S. Majumdar and N. Mukherjee, Mod. Phys. Lett. A 20, 2487 (2005), arXiv: astro-ph/0403405[astro-ph]

  • [29]

    A. Övgün, K. Jusufi, and İ. Sakallı, Annals of Physics 399, 193 (2018), arXiv: 1805.09431[gr-qc]

  • [30]

    R. C. Pantig and E. T. Rodulfo, Chin. J. Phys. 66, 691 (2020), arXiv: 2003.00764[gr-qc]

  • [31]

    G. Mustafa, F. Atamurotov, I. Hussain et al., Chin. Phys. C 46, 125107 (2022), arXiv: 2207.07608[gr-qc]

  • [32]

    H. Liu, J. Liang, and J. Jia, Class. Quant. Grav. 39, 195013 (2022), arXiv: 2204.04519[gr-qc]

  • [33]

    A. Övgün, Phys. Rev. D 99, 104075 (2019), arXiv: 1902.04411[gr-qc]

  • [34]

    W. Javed, M. B. Khadim, J. Abbas et al., Eur. Phys. J. Plus 135, 314 (2020), arXiv: 2004.00408[gr-qc]

  • [35]

    C. Y. Wang, Y. F. Shen, and Y. Xie, J. Cosmol. A. P. 2019, 022 (2019), arXiv: 1902.03789[gr-qc]

  • [36]

    G. N. Gyulchev and S. S. Yazadjiev, Phys. Rev. D 81, 023005 (2010), arXiv: 0909.3014[gr-qc]

  • [37]

    K. Jusufi, A. Övgün, J. Saavedra et al., Phys. Rev. D 97, 124024 (2018), arXiv: 1804.00643[gr-qc]

  • [38]

    N. Parbin, D. J. Gogoi, and U. D. Goswami, Phys. Dark Universe 41, 101265 (2023), arXiv: 2305.09157[gr-qc]

  • [39]

    V. Perlick, Ray Optics, Fermat's Principle, and Applications to General Relativity, Vol. 61 (2000)

  • [40]

    O. Y. Tsupko and G. S. Bisnovatyi-Kogan, Gravitation and Cosmology 15, 184 (2009)

  • [41]

    G. S. Bisnovatyi-Kogan and O. Y. Tsupko, Mon. Not. R. Astron. Soc. 404, 1790 (2010)

  • [42]

    A. Rogers, Mon. Not. Roy. Astron. Soc. 451, 17 (2015), arXiv: 1505.06790[gr-qc]

  • [43]

    X. Er and A. Rogers, Mon. Not. R. Astron. Soc. 475, 867 (2018), arXiv: 1712.06900[astro-ph.GA]

  • [44]

    F. Atamurotov, S. Shaymatov, P. Sheoran et al., J. Cosmol. A. P. 2021, 045 (2021), arXiv: 2105.02214[gr-qc]

  • [45]

    G. Z. Babar, F. Atamurotov, and A. Z. Babar, Phys. Dark Universe 32, 100798 (2021)

  • [46]

    G. Z. Babar, F. Atamurotov, S. Ul Islam et al., Rev. D 103, 084057 (2021), arXiv: 2104.00714[gr-qc]

  • [47]

    A. Hakimov and F. Atamurotov, Astrophys. Space. Sci. 361, 112 (2016)

  • [48]

    F. Atamurotov, A. Abdujabbarov, and J. Rayimbaev, Eur. Phys. J. C 81, 118 (2021)

  • [49]

    C. Benavides-Gallego, A. Abdujabbarov, and Bambi, Eur. Phys. J. C. 78, 694 (2018)

  • [50]

    O. Y. Tsupko, Phys. Rev D 103, 104019 (2021), arXiv: 2102.00553[gr-qc]

  • [51]

    W. Javed, I. Hussain, and A. Övgün, Eur. Phys. J. Plus 137, 148 (2022), arXiv: 2201.09879[gr-qc]

  • [52]

    G. Crisnejo and E. Gallo, Phys. Rev. D 97, 124016 (2018), arXiv: 1804.05473[gr-qc]

  • [53]

    H. Chakrabarty, A. B. Abdikamalov, A. A. Abdujabbarov et al., Phys. Rev. D. 98, 024022 (2018)

  • [54]

    H. Hoshimov, O. Yunusov, F. Atamurotov et al., Phys. Dark Universe 43, 101392 (2024), arXiv: 2312.10678[gr-qc]

  • [55]

    F. Atamurotov, O. Yunusov, A. Abdujabbarov et al., New Astronomy 105, 102098 (2024)

  • [56]

    F. Atamurotov, H. Alibekov, A. Abdujabbarov et al., Symmetry 15, 848 (2023)

  • [57]

    F. Atamurotov, I. Hussain, G. Mustafa et al., Chin. Phys. C 47, 025102 (2023)

  • [58]

    F. Atamurotov and S. G. Ghosh, Eur. Phys. J. Plus 137, 662 (2022)

  • [59]

    F. Atamurotov, F. Sarikulov, V. Khamidov et al., Eur. Phys. J. Plus 137, 567 (2022)

  • [60]

    V. Perlick and O. Y. Tsupko, Phys. Rev. D. 95, 104003 (2017), arXiv: 1702.08768[gr-qc]

  • [61]

    J. Badía and E. F. Eiroa, Phys. Rev. D 104, 084055 (2021), arXiv: 2106.07601[gr-qc]

  • [62]

    F. Atamurotov, B. Ahmedov, and A. Abdujabbarov, Phys. Rev. D 92, 084005 (2015), arXiv: 1507.08131[gr-qc]

  • [63]

    Q. Li, Y. Zhu, and T. Wang, Eur. Phys. J. C 82, 2 (2022), arXiv: 2102.00957[gr-qc]

  • [64]

    A. Chowdhuri and A. Bhattacharyya, Phys. Rev. D 104, 064039 (2021), arXiv: 2012.12914[gr-qc]

  • [65]

    Y. Pahlavon, F. Atamurotov, K. Jusufi et al., Phys. Dark Universe 45, 101543 (2024), arXiv: 2406.09431[gr-qc]

  • [66]

    F. Atamurotov, M. Jamil, and K. Jusufi, Chin. Phys. C 47, 035106 (2023), arXiv: 2212.12949[gr-qc]

  • [67]

    F. Atamurotov, I. Hussain, G. Mustafa et al., Eur. Phys. J. C 82, 831 (2022), arXiv: 2209.01652[gr-qc]

  • [68]

    F. Tello-Ortiz, M. Ali Raza, M. Zubair et al., Phys. Dark Univ. 44, 101460 (2024)

  • [69]

    M. Zubair, M. A. Raza, F. Sarikulov et al., JCAP 10, 058 (2023), arXiv: 2305.16888[gr-qc]

  • [70]

    K. Akiyama and et al. (Event Horizon Telescope Collaboration), Astrophys. J. 875, L1 (2019), arXiv: 1906.11238[astro-ph.GA]

  • [71]

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

  • [72]

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

  • [73]

    K. Akiyama and et al. (Event Horizon Telescope Collaboration), Astronomy and Astrophysics 681, A79 (2024)

  • [74]

    A. Sen, Phys. Rev. Lett. 69, 1006 (1992), arXiv: hep-th/9204046

  • [75]

    M. Guo, S. Song, and H. Yan, Phys. Rev. D 101, 024055 (2020), arXiv: 1911.04796[gr-qc]

  • [76]

    S. G. Ghosh, R. Kumar, and S. U. Islam, J. Cosmol. A. P. 2021, 056 (2021), arXiv: 2011.08023[gr-qc]

  • [77]

    S. V. M. C. B. Xavier, P. V. P. Cunha et al., Int. J. Mod. Phys. D 29, 2041005 (2020), arXiv: 2003.14349[gr-qc]

  • [78]

    M. Zhang and J. Jiang, Nucl. Phys. B 964, 115313 (2021), arXiv: 2004.03367[gr-qc]

  • [79]

    M. Zhang and J. Jiang, Phys. Rev. D 102, 124012 (2020), arXiv: 2004.11087[gr-qc]

  • [80]

    A. Belhaj, A. El Balali, W. El Hadri et al., Int. J. Geom. Meth. Mod. Phys. 17, 2050169 (2020), arXiv: 2007.16083[hep-th]

  • [81]

    A. Carleo, G. Lambiase, and L. Mastrototaro, Eur. Phys. J. C 82, 776 (2022), arXiv: 2206.12988[gr-qc]

  • [82]

    W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936), arXiv: physics/0605038

  • [83]

    M. Kalb and P. Ramond, Phys. Rev. D 9, 2273 (1974)

  • [84]

    L. A. Lessa, J. E. G. Silva, R. V. Maluf et al., Eur. Phys. J. C 80, 335 (2020), arXiv: 1911.10296[gr-qc]

  • [85]

    M. A. Raza, M. Zubair, and E. Maqsood, JCAP 05, 047 (2024), arXiv: 2401.04779[gr-qc]

  • [86]

    M. Zubair, M. A. Raza, and E. Maqsood, Phys. Dark Univ. 42, 101334 (2023), arXiv: 2310.12325[gr-qc]

  • [87]

    G. W. Gibbons and K.-i. Maeda, Nucl. Phys. B 298, 741 (1988)

  • [88]

    D. Garfinkle, G. T. Horowitz, and A. Strominger, Phys. Rev. D 43, 3140 (1991)

  • [89]

    G. T. Horowitz and A. Strominger, Nucl. Phys. B 360, 197 (1991)

  • [90]

    A. D. Shapere, S. Trivedi, and F. Wilczek, Mod. Phys. Lett. A 6, 2677 (1991)

  • [91]

    J. L. Synge, ed., Relativity: The General theory (1960)

  • [92]

    V. Perlick, O. Y. Tsupko, and G. S. Bisnovatyi-Kogan, Phys. Rev. D 92, 104031 (2015), arXiv: 1507.04217[gr-qc]

  • [93]

    F. Atamurotov, K. Jusufi, M. Jamil et al., Phys. Rev. D 104, 064053 (2021), arXiv: 2109.08150[gr-qc]

  • [94]

    M. A. Raza, J. Rayimbaev, F. Sarikulov et al., Phys. Dark Univ. 44, 101488 (2024), arXiv: 2311.15784[gr-qc]

  • [95]

    B. BEZDĚKOVÁ, Electromagnetic Waves in Dispersiveand Refractive Relativistic Systems, Master's thesis, Diplomová práce, vedoucí Bičák, Jiří. Praha: Univerzita Karlova, Matematicko-fyzikální fakulta, Ústav teoretické fyziky, (2019)

  • [96]

    B. Turimov, B. Ahmedov, A. Abdujabbarov et al., Int. J. Mod. Phys. D 28, 2040013 (2019), arXiv: 1802.03293[gr-qc]

  • [97]

    V. S. Morozova, B. J. Ahmedov, and A. A. Tursunov, Astrophys. Space Sci. 346, 513 (2013)

  • [98]

    A. A. A. Filho, (2024), arXiv: 2406.11582[gr-qc]

  • [99]

    M. Fathi, M. Olivares, and J. R. Villanueva, Eur. Phys. J. C 80, 51 (2020), arXiv: 1910.12811[gr-qc]

  • [100]

    K. Yang, Y. Z. Chen, Z. Q. Duan et al., Phys. Rev. D 108, 124004 (2023), arXiv: 2308.06613[gr-qc]