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Motion of charged and magnetized particles around regular black holes immersed in an external magnetic field in modified gravity

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Abstract

We study properties of spacetime around regular black holes (BHs) in modified gravity, so-called regular MOG BH. The motion of the test electrically neutral and charged, and particles with magnetic dipole moment, near the regular MOG BH, considering the BH is immersed in an external, asymptotically uniform magnetic field, have also been investigated. In a study of charged particles’ dynamics, we have shown that the radius of the particle’s innermost stable circular orbits (ISCOs) increases with respect to increasing the parameter of MOG, while as cyclotron frequency grows the ISCO radius decreases. Effects of MOG field on synchrotron radiation by charged particles have also been studied and obtained that the existence of the MOG parameter essentially decreases the total luminosity of the radiation. Moreover, through studies of the dynamics of magnetic dipoles, we have shown that the increase of the MOG and the magnetic coupling parameters lead to an increase of the inner radius and the width of the accretion disk consisting of test magnetized particles. Finally, it is obtained that the range for values of the magnetic coupling parameter for neutron stars orbiting supermassive black holes causes their orbits to be stable.

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References

  1. C. Bambi, Testing black hole candidates with electromagnetic radiation. Rev. Mod. Phys. 89(2), 025001 (2017). https://doi.org/10.1103/RevModPhys.89.025001

    Article  ADS  MathSciNet  Google Scholar 

  2. K. Akiyama et al., First M87 event horizon telescope results. I. The shadow of the supermassive black hole. Astrophys. J. 875(1), L1 (2019). https://doi.org/10.3847/2041-8213/ab0ec7

    Article  ADS  Google Scholar 

  3. K. Akiyama et al., First M87 event horizon telescope results. VI. The shadow and mass of the central black hole. Astrophys. J. 875(1), L6 (2019). https://doi.org/10.3847/2041-8213/ab1141

    Article  ADS  Google Scholar 

  4. R. Vogt, The u.s. ligo Project, in Sixth Marcel Grossman Meeting on General Relativity (Proceedings, Kyoto, Japan, 1991). ed. by H. Sato, T. Nakamura (World Scientific, Singapore, 1992), pp.244–266

    Google Scholar 

  5. K. Schwarzschild,Über das Gravitationsfeld eines Massenpunktes nach der Einsteinchen Theorie. Sitzungsber Dtsch Akad Wiss Berlin, Kl Math. Phys. Tech. pp 189–196 (1916)

  6. G. Nordström, On the energy of the gravitation field in Einstein’s theory. Koninklijke Nederlandse Akademie van Wetenschappen Proceedings Series B Physical Sciences 20, 1238–1245 (1918)

    ADS  Google Scholar 

  7. H. Reissner, Über die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie. Ann. Phys. 355(9), 106–120 (1916). https://doi.org/10.1002/andp.19163550905

    Article  Google Scholar 

  8. J.M. Bardeen, Kerr metric black holes. Nature 226, 64 (1970). https://doi.org/10.1038/226064a0

    Article  ADS  Google Scholar 

  9. K.A. Bronnikov, Comment on “regular black hole in general relativity coupled to nonlinear electrodynamics’’. Phys. Rev. Lett. 85, 4641 (2000). https://doi.org/10.1103/PhysRevLett.85.4641

    Article  ADS  Google Scholar 

  10. I. Dymnikova, E. Galaktionov, Regular rotating electrically charged black holes and solitons in non-linear electrodynamics minimally coupled to gravity. Class. Quantum Gravity 32(16), 165015 (2015). https://doi.org/10.1088/0264-9381/32/16/165015. arXiv:1510.01353 [gr-qc]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. I. Dymnikova, M. Korpusik, Thermodynamics of regular cosmological black holes with the de sitter interior. Entropy 13, 1967–1991 (2011). https://doi.org/10.3390/e13121967

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. S.A. Hayward, Energy and entropy conservation for dynamical black holes. Phys. Rev. D 70(104), 027 (2004). arXiv:gr-qc/0408008

    MathSciNet  Google Scholar 

  13. J. Schee, Z. Stuchlík, Gravitational lensing and ghost images in the regular Bardeen no-horizon spacetimes. JCAP 6, 048 (2015). https://doi.org/10.1088/1475-7516/2015/06/048. arXiv:1501.00835 [astro-ph.HE]

    Article  ADS  Google Scholar 

  14. Z. Stuchlík, J. Schee, Circular geodesic of Bardeen and Ayon-Beato-Garcia regular black-hole and no-horizon spacetimes. Int. J. Mod. Phys. D 24, 1550020 (2015). https://doi.org/10.1142/S0218271815500200. arXiv:1501.00015 [astro-ph.HE]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. B. Toshmatov, B. Ahmedov, A. Abdujabbarov et al., Rotating regular black hole solution. Phys. Rev. D 89(10), 104017 (2014). https://doi.org/10.1103/PhysRevD.89.104017. arXiv:1404.6443 [gr-qc]

    Article  ADS  Google Scholar 

  16. B. Toshmatov, A. Abdujabbarov, Z. Stuchlík et al., Quasinormal modes of test fields around regular black holes. Phys. Rev. D 91(8), 083008 (2015). https://doi.org/10.1103/PhysRevD.91.083008. arXiv:1503.05737 [gr-qc]

    Article  ADS  Google Scholar 

  17. B. Toshmatov, C. Bambi, B. Ahmedov et al., Scalar perturbations of nonsingular nonrotating black holes in conformal gravity. Phys. Rev. D 96(6), 064028 (2017). https://doi.org/10.1103/PhysRevD.96.064028. arXiv:1705.03654 [gr-qc]

    Article  ADS  Google Scholar 

  18. B. Toshmatov, Z. Stuchlík, B. Ahmedov, Rotating black hole solutions with quintessential energy. Eur. Phys. J. Plus 132, 98 (2017). https://doi.org/10.1140/epjp/i2017-11373-4

    Article  MATH  Google Scholar 

  19. B. Toshmatov, Z. Stuchlík, B. Ahmedov, Comment on “construction of regular black holes in general relativity’’. Phys. Rev. D 98(2), 028501 (2018). https://doi.org/10.1103/PhysRevD.98.028501. arXiv:1807.09502 [gr-qc]

    Article  ADS  MathSciNet  Google Scholar 

  20. B. Toshmatov, Z. Stuchlík, B. Ahmedov, Electromagnetic perturbations of black holes in general relativity coupled to nonlinear electrodynamics: polar perturbations. Phys. Rev. D 98(8), 085021 (2018). https://doi.org/10.1103/PhysRevD.98.085021. arXiv:1810.06383 [gr-qc]

    Article  ADS  MathSciNet  Google Scholar 

  21. B. Toshmatov, Z. Stuchlík, J. Schee et al., Electromagnetic perturbations of black holes in general relativity coupled to nonlinear electrodynamics. Phys. Rev. D 97(8), 084058 (2018). https://doi.org/10.1103/PhysRevD.97.084058. arXiv:1805.00240 [gr-qc]

    Article  ADS  MathSciNet  Google Scholar 

  22. J. Rayimbaev, P. Tadjimuratov, A. Abdujabbarov et al., Dynamics of test particles and twin peaks QPOs around regular black holes in modified gravity. Galaxies 9(4), 75 (2021). https://doi.org/10.3390/galaxies9040075. arXiv:2010.12863 [gr-qc]

    Article  ADS  Google Scholar 

  23. J.W. Moffat, M.H. Zhoolideh Haghighi, Modified gravity (MOG) and the Abell 1689 cluster acceleration data. Eur. Phys. J. Plus 132(10), 417 (2017). https://doi.org/10.1140/epjp/i2017-11684-4

    Article  Google Scholar 

  24. A. Jawad, F. Ali, M. Jamil et al., Dynamics of particles around a regular black hole with nonlinear electrodynamics. Commun. Theor. Phys. 66(5), 509 (2016). https://doi.org/10.1088/0253-6102/66/5/509. arXiv:1610.07411 [gr-qc]

    Article  ADS  MathSciNet  Google Scholar 

  25. S. Hussain, M. Jamil, Timelike geodesics of a modified gravity black hole immersed in an axially symmetric magnetic field. Phys. Rev. D 92(4), 043008 (2015). https://doi.org/10.1103/PhysRevD.92.043008. arXiv:1508.02123 [gr-qc]

    Article  ADS  MathSciNet  Google Scholar 

  26. G.Z. Babar, M. Jamil, Y.K. Lim, Dynamics of a charged particle around a weakly magnetized naked singularity. Int. J. Mod. Phys. D 25(2), 1650024 (2016). https://doi.org/10.1142/S0218271816500243. arXiv:1504.00072 [gr-qc]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. M. Bañados, J. Silk, S.M. West, Kerr black holes as particle accelerators to arbitrarily high energy. Phys. Rev. Lett. 103(11), 111102 (2009). https://doi.org/10.1103/PhysRevLett.103.111102

    Article  ADS  Google Scholar 

  28. B. Majeed, M. Jamil, Dynamics and center of mass energy of colliding particles around black hole in f(R) gravity. Int. J. Mod. Phys. D 26(5), 1741017 (2017). https://doi.org/10.1142/S0218271817410176. arXiv:1705.04167 [gr-qc]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. A. Zakria, M. Jamil, Center of mass energy of the collision for two general geodesic particles around a Kerr–Newman–Taub-NUT black hole. J. High Energy Phys. 2015, 147 (2015). https://doi.org/10.1007/JHEP05(2015)147. arXiv:1501.06306 [gr-qc]

    Article  MathSciNet  MATH  Google Scholar 

  30. I. Brevik, M. Jamil, Black holes in the turbulence phase of viscous rip cosmology. Int. J. Geom. Methods Mod. Phys. 16(2), 1950030 (2019). https://doi.org/10.1142/S0219887819500300. arXiv:1901.00002 [gr-qc]

    Article  MathSciNet  MATH  Google Scholar 

  31. S. Chen, M. Wang, J. Jing, Chaotic motion of particles in the accelerating and rotating black holes spacetime. J. High Energy Phys. 9, 82 (2016). https://doi.org/10.1007/JHEP09(2016)082. arXiv:1604.02785 [gr-qc]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. K. Hashimoto, N. Tanahashi, Universality in chaos of particle motion near black hole horizon. Phys. Rev. D 95(2), 024007 (2017). https://doi.org/10.1103/PhysRevD.95.024007. arXiv:1610.06070 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  33. S. Dalui, B.R. Majhi, P. Mishra, Presence of horizon makes particle motion chaotic. Phys. Lett. B 788, 486–493 (2019). https://doi.org/10.1016/j.physletb.2018.11.050. arXiv:1803.06527 [gr-qc]

    Article  ADS  Google Scholar 

  34. W. Han, Chaos and dynamics of spinning particles in Kerr spacetime. Gen. Relativ. Gravit. 40(9), 1831–1847 (2008). https://doi.org/10.1007/s10714-007-0598-9. arXiv:1006.2229 [gr-qc]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. A.P.S. de Moura, P.S. Letelier, Chaos and fractals in geodesic motions around a nonrotating black hole with halos. Phys. Rev. E 61(6), 6506–6516 (2000). https://doi.org/10.1103/PhysRevE.61.6506. arXiv:chao-dyn/9910035 [nlin.CD]

    Article  ADS  Google Scholar 

  36. R.M. Wald, Black hole in a uniform magnetic field. Phys. Rev. D 10, 1680–1685 (1974). https://doi.org/10.1103/PhysRevD.10.1680

    Article  ADS  Google Scholar 

  37. A.N. Aliev, D.V. Galtsov, V.I. Petukhov, Negative absorption near a magnetized black hole–Black hole masers. Astrophys. Space Sci. 124, 137–157 (1986). https://doi.org/10.1007/BF00649756

    Article  ADS  Google Scholar 

  38. A.N. Aliev, D.V. Gal’tsov, Reviews of topical problems: “magnetized’’ black holes. Soviet Phys. Uspekhi 32, 75–92 (1989). https://doi.org/10.1070/PU1989v032n01ABEH002677

    Article  ADS  Google Scholar 

  39. A.N. Aliev, N. Özdemir, Motion of charged particles around a rotating black hole in a magnetic field. Mon. Not. R. Astron. Soc. 336, 241–248 (2002). https://doi.org/10.1046/j.1365-8711.2002.05727.x. arXiv:gr-qc/0208025

    Article  ADS  Google Scholar 

  40. V.P. Frolov, P. Krtouš, Charged particle in higher dimensional weakly charged rotating black hole spacetime. Phys. Rev. D 83(2), 024016 (2011). https://doi.org/10.1103/PhysRevD.83.024016. arXiv:1010.2266 [hep-th]

    Article  ADS  Google Scholar 

  41. V.P. Frolov, Weakly magnetized black holes as particle accelerators. Phys. Rev. D 85(2), 024020 (2012). https://doi.org/10.1103/PhysRevD.85.024020. arXiv:1110.6274 [gr-qc]

    Article  ADS  Google Scholar 

  42. C.A. Benavides-Gallego, A. Abdujabbarov, D. Malafarina et al., Charged particle motion and electromagnetic field in \(\gamma \) spacetime. Phys. Rev. D 99(044), 012 (2019). https://doi.org/10.1103/PhysRevD.99.044012. arXiv:1812.04846 [gr-qc]

    Article  MathSciNet  Google Scholar 

  43. S. Shaymatov, B. Ahmedov, Z. Stuchlík et al., Effect of an external magnetic field on particle acceleration by a rotating black hole surrounded with quintessential energy. Int. J. Mod. Phys. D 27, 1850088 (2018). https://doi.org/10.1142/S0218271818500888

    Article  ADS  MathSciNet  Google Scholar 

  44. Z. Stuchlík, J. Schee, A. Abdujabbarov, Ultra-high-energy collisions of particles in the field of near-extreme Kehagias–Sfetsos naked singularities and their appearance to distant observers. Phys. Rev. D 89(10), 104048 (2014). https://doi.org/10.1103/PhysRevD.89.104048

    Article  ADS  Google Scholar 

  45. A. Abdujabbarov, B. Ahmedov, Test particle motion around a black hole in a braneworld. Phys. Rev. D 81(4), 044022 (2010). https://doi.org/10.1103/PhysRevD.81.044022. arXiv:0905.2730 [gr-qc]

    Article  ADS  Google Scholar 

  46. A. Abdujabbarov, B. Ahmedov, A. Hakimov, Particle motion around black hole in Hořava–Lifshitz gravity. Phys. Rev. D 83(4), 044053 (2011). https://doi.org/10.1103/PhysRevD.83.044053. arXiv:1101.4741 [gr-qc]

    Article  ADS  Google Scholar 

  47. A.A. Abdujabbarov, B.J. Ahmedov, V.G. Kagramanova, Particle motion and electromagnetic fields of rotating compact gravitating objects with gravitomagnetic charge. Gen. Relativ. Gravit. 40, 2515–2532 (2008). https://doi.org/10.1007/s10714-008-0635-3. arXiv:0802.4349 [gr-qc]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. A.A. Abdujabbarov, B.J. Ahmedov, S.R. Shaymatov et al., Penrose process in Kerr-Taub-NUT spacetime. Astrophys. Space Sci. 334, 237–241 (2011). https://doi.org/10.1007/s10509-011-0740-8. arXiv:1105.1910 [astro-ph.SR]

    Article  ADS  MATH  Google Scholar 

  49. V. Karas, J. Kovar, O. Kopacek, et al, Regular and Chaotic Motion in General Relativity. Case of Magnetized Black Hole and a Massive Magnetic Dipole, in American Astronomical Society Meeting Abstracts #220, p 430.07 (2012)

  50. Z. Stuchlík, M. Kološ, Acceleration of the charged particles due to chaotic scattering in the combined black hole gravitational field and asymptotically uniform magnetic field. Eur. Phys. J. C 76, 32 (2016). https://doi.org/10.1140/epjc/s10052-015-3862-2. arXiv:1511.02936 [gr-qc]

    Article  ADS  Google Scholar 

  51. J. Kovář, O. Kopáček, V. Karas et al., Off-equatorial orbits in strong gravitational fields near compact objects II: halo motion around magnetic compact stars and magnetized black holes. Class. Quantum Gravity 27(13), 135006 (2010). https://doi.org/10.1088/0264-9381/27/13/135006. arXiv:1005.3270 [astro-ph.HE]

    Article  ADS  MATH  Google Scholar 

  52. J. Kovář, P. Slaný, C. Cremaschini et al., Electrically charged matter in rigid rotation around magnetized black hole. Phys. Rev. D 90(4), 044029 (2014). https://doi.org/10.1103/PhysRevD.90.044029. arXiv:1409.0418 [gr-qc]

    Article  ADS  Google Scholar 

  53. M. Kološ, A. Tursunov, Z. Stuchlík, Possible signature of magnetic fields related to quasi-periodic oscillation observed in microquasars. Eur. Phys. J. C 77, 860 (2017). arXiv:1707.02224 [astro-ph.HE]

    Article  ADS  Google Scholar 

  54. J. Rayimbaev, P. Tadjimuratov, Can modified gravity silence radio-loud pulsars? Phys. Rev. D 102(2), 024019 (2020). https://doi.org/10.1103/PhysRevD.102.024019

    Article  ADS  MathSciNet  Google Scholar 

  55. J. Rayimbaev, B. Turimov, B. Ahmedov, Braneworld effects in plasma magnetosphere of a slowly rotating magnetized neutron star. Int. J. Mod. Phys. D 28(10), 1950128–209 (2019). https://doi.org/10.1142/S0218271819501281

    Article  ADS  MathSciNet  MATH  Google Scholar 

  56. J. Rayimbaev, B. Turimov, S. Palvanov, Plasma magnetosphere of slowly rotating magnetized neutron star in branewold. Int. J. Mod. Phys. Conf. Ser. 19, 60–209 (2019). https://doi.org/10.1142/S201019451960019X

    Article  Google Scholar 

  57. J. Rayimbaev, B. Turimov, F. Marcos et al., Particle acceleration and electromagnetic field of deformed neutron stars. Mod. Phys. Lett. A 35(9), 2050056 (2020). https://doi.org/10.1142/S021773232050056X

    Article  ADS  MathSciNet  Google Scholar 

  58. F. de Felice, F. Sorge, Magnetized orbits around a Schwarzschild black hole. Class. Quantum Gravity 20, 469–481 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  59. F. de Felice, F. Sorge, S. Zilio, Magnetized orbits around a Kerr black hole. Class. Quantum Gravity 21, 961–973 (2004). https://doi.org/10.1088/0264-9381/21/4/016

    Article  ADS  MathSciNet  MATH  Google Scholar 

  60. J.R. Rayimbaev, Magnetized particle motion around non-Schwarzschild black hole immersed in an external uniform magnetic field. Astrophys. Space Sci. 361, 288 (2016). https://doi.org/10.1007/s10509-016-2879-9

    Article  ADS  MathSciNet  Google Scholar 

  61. N. Juraeva, J. Rayimbaev, A. Abdujabbarov et al., Distinguishing magnetically and electrically charged Reissner–Nordström black holes by magnetized particle motion. Eur. Phys. J. C 81(1), 70 (2021). https://doi.org/10.1140/epjc/s10052-021-08876-5

    Article  ADS  Google Scholar 

  62. J. Rayimbaev, A. Abdujabbarov, M. Jamil et al., Dynamics of magnetized particles around Einstein–Æther black hole with uniform magnetic field. Nucl. Phys. B 966, 115364 (2021). https://doi.org/10.1016/j.nuclphysb.2021.115364. arXiv:2009.04898 [gr-qc]

    Article  MATH  Google Scholar 

  63. B. Toshmatov, A. Abdujabbarov, B. Ahmedov et al., Motion and high energy collision of magnetized particles around a Hořava–Lifshitz black hole. Astrophys. Space Sci. 360, 19 (2015). https://doi.org/10.1007/s10509-015-2533-y

    Article  ADS  Google Scholar 

  64. A. Abdujabbarov, B. Ahmedov, O. Rahimov et al., Magnetized particle motion and acceleration around a Schwarzschild black hole in a magnetic field. Phys. Scr. 89(8), 084008 (2014). https://doi.org/10.1088/0031-8949/89/8/084008

    Article  ADS  Google Scholar 

  65. O.G. Rahimov, A.A. Abdujabbarov, B.J. Ahmedov, Magnetized particle capture cross section for braneworld black hole. Astrophys. Space Sci. 335(2), 499–504 (2011). https://doi.org/10.1007/s10509-011-0755-1. arXiv:1105.4543 [astro-ph.SR]

    Article  ADS  MATH  Google Scholar 

  66. O.G. Rahimov, Magnetized particle motion around black hole in braneworld. Mod. Phys. Lett. A 26, 399–408 (2011). https://doi.org/10.1142/S0217732311034931. arXiv:1012.1481 [gr-qc]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  67. K. Haydarov, A. Abdujabbarov, J. Rayimbaev, et al, Magnetized particle motion around black holes in conformal gravity: can magnetic interaction mimic spin of black holes? Universe 6(3). https://doi.org/10.3390/universe6030044, https://www.mdpi.com/2218-1997/6/3/44 (2020)

  68. K. Haydarov, J. Rayimbaev, A. Abdujabbarov et al., Magnetized particle motion around magnetized Schwarzschild-MOG black hole. Eur. Phys. J. C 80(5), 399 (2020). https://doi.org/10.1140/epjc/s10052-020-7992-9. arXiv:2004.14868 [gr-qc]

    Article  ADS  Google Scholar 

  69. A. Abdujabbarov, J. Rayimbaev, B. Turimov et al., Dynamics of magnetized particles around 4-D Einstein Gauss–Bonnet black hole. Phys. Dark Univ. 30, 100715 (2020). https://doi.org/10.1016/j.dark.2020.100715

    Article  Google Scholar 

  70. B. Narzilloev, J. Rayimbaev, S. Shaymatov et al., Can the dynamics of test particles around charged stringy black holes mimic the spin of Kerr black holes? Phys. Rev. D 102(4), 044013 (2020). https://doi.org/10.1103/PhysRevD.102.044013. arXiv:2007.12462 [gr-qc]

    Article  ADS  MathSciNet  Google Scholar 

  71. J. Rayimbaev, M. Figueroa, Z. Stuchlík et al., Test particle orbits around regular black holes in general relativity combined with nonlinear electrodynamics. Phys. Rev. D 101(10), 104045 (2020). https://doi.org/10.1103/PhysRevD.101.104045

    Article  ADS  MathSciNet  Google Scholar 

  72. B. Turimov, J. Rayimbaev, A. Abdujabbarov et al., Test particle motion around a black hole in Einstein–Maxwell–Scalar theory. Phys. Rev. D 102(064), 052 (2020). https://doi.org/10.1103/PhysRevD.102.064052

    Article  MathSciNet  Google Scholar 

  73. V.S. Morozova, L. Rezzolla, B.J. Ahmedov, Nonsingular electrodynamics of a rotating black hole moving in an asymptotically uniform magnetic test field. Phys. Rev. D 89(10), 104030 (2014). https://doi.org/10.1103/PhysRevD.89.104030. arXiv:1310.3575 [gr-qc]

    Article  ADS  Google Scholar 

  74. J. Vrba, A. Abdujabbarov, A. Tursunov et al., Particle motion around generic black holes coupled to non-linear electrodynamics. Eur. Phys. J. C 79(9), 778 (2019). https://doi.org/10.1140/epjc/s10052-019-7286-2. arXiv:1909.12026 [gr-qc]

    Article  ADS  Google Scholar 

  75. J. Vrba, A. Abdujabbarov, M. Kološ et al., Charged and magnetized particles motion in the field of generic singular black holes governed by general relativity coupled to nonlinear electrodynamics. Phys. Rev. D 101(12), 124039 (2020). https://doi.org/10.1103/PhysRevD.101.124039

    Article  ADS  MathSciNet  Google Scholar 

  76. J.W. Moffat, V.T. Toth, The bending of light and lensing in modified gravity. Mon. Not. R. Astron. Soc. 397(4), 1885–1892 (2009). https://doi.org/10.1111/j.1365-2966.2009.14876.x. arXiv:0805.4774 [astro-ph]

    Article  ADS  Google Scholar 

  77. J.W. Moffat, S. Rahvar, The MOG weak field approximation and observational test of galaxy rotation curves. Mon. Not. R. Astron. Soc. 436(2), 1439–1451 (2013). https://doi.org/10.1093/mnras/stt1670. arXiv:1306.6383 [astro-ph.GA]

    Article  ADS  Google Scholar 

  78. J.W. Moffat, Modified gravity black holes and their observable shadows. Eur. Phys. J. C 75, 130 (2015). https://doi.org/10.1140/epjc/s10052-015-3352-6. arXiv:1502.01677 [gr-qc]

    Article  ADS  Google Scholar 

  79. J.W. Moffat, Black holes in modified gravity (MOG). Eur. Phys. J. C 75, 175 (2015). https://doi.org/10.1140/epjc/s10052-015-3405-x. arXiv:1412.5424 [gr-qc]

    Article  ADS  Google Scholar 

  80. J.W. Moffat, V.T. Toth, Rotational velocity curves in the Milky Way as a test of modified gravity. Phys. Rev. D 91(4), 043004 (2015). https://doi.org/10.1103/PhysRevD.91.043004. arXiv:1411.6701 [astro-ph.GA]

    Article  ADS  Google Scholar 

  81. M.F. Wondrak, P. Nicolini, Superradiance in modified gravity (MOG). JCAP 12, 021 (2018). https://doi.org/10.1088/1475-7516/2018/12/021. arXiv:1809.07509 [gr-qc]

    Article  ADS  MATH  Google Scholar 

  82. J.R. Mureika, J.W. Moffat, M. Faizal, Black hole thermodynamics in modified gravity (MOG). Phys. Lett. B 757, 528–536 (2016). https://doi.org/10.1016/j.physletb.2016.04.041. arXiv:1504.08226 [gr-qc]

    Article  ADS  MATH  Google Scholar 

  83. E. Ayon-Beato, New regular black hole solution from nonlinear electrodynamics. Phys. Lett. B 464, 25–29 (1999). https://doi.org/10.1016/S0370-2693(99)01038-2. arXiv:hep-th/9911174

    Article  ADS  MathSciNet  MATH  Google Scholar 

  84. E. Ayón-Beato, A. García, Regular black hole in general relativity coupled to nonlinear electrodynamics. Phys. Rev. Lett. 80, 5056–5059 (1998). https://doi.org/10.1103/PhysRevLett.80.5056. arXiv:gr-qc/9911046

    Article  ADS  Google Scholar 

  85. J.W. Moffat, LIGO GW150914 and GW151226 gravitational wave detection and generalized gravitation theory (MOG). Phys. Lett. B 763, 427–433 (2016). https://doi.org/10.1016/j.physletb.2016.10.082. arXiv:1603.05225 [gr-qc]

    Article  ADS  Google Scholar 

  86. S. Rahvar, J.W. Moffat, Propagation of electromagnetic waves in MOG: gravitational lensing. Mon. Not. R. Astron. 482(4), 4514–4518 (2019). https://doi.org/10.1093/mnras/sty3002. arXiv:1807.07424 [gr-qc]

    Article  ADS  Google Scholar 

  87. J.W. Moffat, V.T. Toth, Masses and shadows of the black holes sagittarius A* and M87* in modified gravity. Phys. Rev. D 101(2), 024014 (2020). https://doi.org/10.1103/PhysRevD.101.024014. arXiv:1904.04142 [gr-qc]

    Article  ADS  MathSciNet  Google Scholar 

  88. J.W. Moffat, S. Rahvar, The MOG weak field approximation and observational test of galaxy rotation curves. Mon. Not. R. Astron. 436(2), 1439–1451 (2013). https://doi.org/10.1093/mnras/stt1670. arXiv:1306.6383 [astro-ph.GA]

    Article  ADS  Google Scholar 

  89. L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields, Course of Theoretical Physics, vol. 2 (Elsevier Butterworth-Heinemann, Oxford, 2004)

    Google Scholar 

  90. Event Horizon Telescope. Collaboration, K. Akiyama, A. Alberdi et al., First M87 event horizon telescope results. I. The shadow of the supermassive black hole. Astrophys. J. Lett. 875(1), L1 (2019). https://doi.org/10.3847/2041-8213/ab0ec7. arXiv:1906.11238 [astro-ph.GA]

    Article  ADS  Google Scholar 

  91. L. Rezzolla, O. Zanotti, An improved exact Riemann solver for relativistic hydrodynamics. J. Fluid Mech. 449, 395 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  92. K. Mori, E.V. Gotthelf, S. Zhang et al., NuSTAR discovery of a 3.76 s transient magnetar near sagittarius A*. Astronom. J. Lett. 770(2), L23 (2013). https://doi.org/10.1088/2041-8205/770/2/L23. arXiv:1305.1945 [astro-ph.HE]

    Article  ADS  Google Scholar 

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Acknowledgements

This research is supported by Grants F-FA-2021-432, F-FA-2021-510, and MRB-2021-527 of the Uzbekistan Ministry for Innovative Development. JR thanks to the ERASMUS+ project 608715-EPP-1-2019-1-UZ-EPPKA2-JP (SPACECOM). DB thanks the Silesian University projects SGS/26/2022 and GAČR, grant number 23-07043S.

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Rayimbaev, J., Abdujabbarov, A., Bardiev, D. et al. Motion of charged and magnetized particles around regular black holes immersed in an external magnetic field in modified gravity. Eur. Phys. J. Plus 138, 358 (2023). https://doi.org/10.1140/epjp/s13360-023-03979-2

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