Journal of Holography Applications in Physics
Gravitational Lensing by Kalb-Ramond Black Holes Coupled by Nonlinear Electrodynamics
Document Type : Review article
Authors
Department of Physics, Institute of Applied Science and Humanities, GLA University, Mathura 281406, India
Abstract
In this paper, we study the black hole solution coupled with the Kalb–Ramond (KR) field and nonlinear electrodynamics (NLED). The obtained black hole solutions interpolate between a Kalb-Ramond black hole in the absence of NLED and the Schwarzschild black hole in the limit of vanishing KR and NLED field. We study gravitational lensing by a black hole solution. In addition, we estimate the mass parameter and constrain it using observational data.
Keywords
Main Subjects
Article PDF
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[7] The Event Horizon Telescope Collaboration et al, “First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring”, The Astrophysical Journal Letters 875(1), L5 (2019). DOI: 10.3847/2041-8213/ab0f43
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[11] The Event Horizon Telescope Collaboration et al, “First M87 Event Horizon Telescope Results. IX. Detection of Near-horizon Circular Polarization”, The Astrophysical Journal Letters 957(2), L20 (2023). DOI: 10.3847/2041-8213/acff70
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[14] V. Bozza and L. Mancini, “Observing Gravitational Lensing Effects by Sgr A* with Gravity”, Astrophys. J. 753(1), 56 (2012). DOI: 10.1088/0004-637X/753/1/56
[15] S. Pietroni and V. Bozza, “The impact of gravitational lensing in the reconstruction of stellar orbits around Sgr A*”, JCAP 2022(12), 018 (2022). DOI: 10.1088/1475- 7516/2022/12/018
[16] N. Tsukamoto, ”Deflection angle of a light ray reflected by a general marginally unstable photon sphere in a strong deflection limit”, Phys. Rev. D, 102, 104029 (2020). DOI: 10.1103/PhysRevD.102.104029
[17] N. Tsukamoto, “Gravitational lensing in the Simpson-Visser black-bounce spacetime in a strong deflection limit”, Phys. Rev. D 103, 024033 (2021). DOI: 10.1103/PhysRevD.103.024033
[18] N. Tsukamoto, “Gravitational lensing by a photon sphere in a Reissner-Nordström naked singularity spacetime in strong deflection limits”, Phys. Rev. D 104, 124016 (2021). DOI: 10.1103/PhysRevD.104.124016
[19] C. Furtado, J. R. Nascimento, A. Y. Petrov, P. J. Porfírio and A. R. Soares, ”Strong gravitational lensing in a spacetime with topological charge within the Eddingtoninspired Born-Infeld gravity”, Phys. Rev. D 103, 044047 (2021). DOI: 10.1103/PhysRevD.103.044047
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[21] D. Colladay and V. A. Kosteleckỳ, ”Lorentz violating extension of the standard model”, Phys. Rev. D 58, 116002 (1998). DOI: 10.1103/PhysRevD.58.116002
[22] V. A. Kosteleckỳ and S. Samuel, ”Phenomenological Gravitational Constraints on Strings and Higher Dimensional Theories”, Phys. Rev. Lett. 63, 224 (1989). DOI: doi.org/10.1103/PhysRevLett.63.224
[23] R. Gambini and J. Pullin, ”Nonstandard optics from quantum space-time”, Phys. Rev. D 59, 124021 (1999). DOI: 10.1103/PhysRevD.59.124021
[24] E. Battista, ”Quantum Schwarzschild geometry in effective field theory models of gravity”, Phys. Rev. D 109, 026004 (2024). DOI: 10.1103/PhysRevD.109.026004
[25] V. A. Kosteleckỳ and S. Samuel, ”Gravitational Phenomenology in Higher Dimensional Theories and Strings”, Phys. Rev. D 40, 1886 (1989). DOI: 10.1103/PhysRevD.40.1886
[26] M. Kalb and P. Ramond, ”Classical direct interstring action”, Phys. Rev. D 9, 2273 (1974). DOI: 10.1103/PhysRevD.9.2273
[27] A. D. Sakharov, ”The Initial Stage of an Expanding Universe and the Appearance of a Nonuniform Distribution of Matter”, JETP 22(1), 241 (1966).
[28] E. B. Gliner, ”Algebraic Properties of the Energy-momentum Tensor and Vacuum-like States of Matter”, JETP 22(2), 378 (1966).
[29] J. Bardeen, ”Non-singular general relativistic gravitational collapse” in Proceedings of GR5 (Tiflis, U.S.S.R., 1968).
[30] E. Ayon-Beato, A. Garcia, ”The Bardeen model as a nonlinear magnetic monopole”, Phys. Lett. B 493(1-2), 149 (2000). DOI: 10.1016/S0370-2693(00)01125-4
[31] E. Ayon-Beato and A. Garcia, ”Non-Singular Charged Black Hole Solution for Non-Linear Source”, General Relativity and Gravitation 31, 629 (1999). DOI: 10.1023/A:1026640911319
[32] E. Ayon-Beato and A. Garcia, ”Four-parametric regular black hole solution”, General Relativity and Gravitation 37, 635 (2005). DOI: 10.1007/s10714-005-0050-y
[33] E. Ayon-Beato and A. Garcia, ”Regular black hole in general relativity coupled to nonlinear electrodynamics”, Phys. Rev. Lett. 80, 5056 (1998). DOI: 10.1103/PhysRevLett.80.5056
[34] L. Xiang, Y. Ling and Y. G. Shen, ”SINGULARITIES AND THE FINALE OF BLACK HOLE EVAPORATION”, International Journal of Modern Physics D 22(12), 1342016 (2013). DOI: 10.1142/S0218271813420169
[35] H. Culetu, ”Nonsingular black hole with a nonlinear electric source”, International Journal of Modern Physics D 24(09), 1542001 (2015). DOI: 10.1142/S0218271815420018
[36] L. Balart and E. C. Vagenas, ”Regular black holes with a nonlinear electrodynamics source”, Phys. Rev. D 90, 124045 (2014). DOI: 10.1103/PhysRevD.90.124045
[37] M. S. Ma, R. Zhao and Y. Q. Ma, ”Thermodynamic stability of black holes surrounded by quintessence”, General Relativity and Gravitation 49(79), (2017). DOI: 10.1007/s10714-017-2245-4
[38] S. H. Hendi, S. Panahiyan and B. Eslam Panah, ”Charged black hole solutions in Gauss-Bonnet-massive gravity”, J. High Energy Phys. 2016, 129 (2016). DOI: 10.1007/JHEP01(2016)129
[39] S. Ansoldi, ”Spherical black holes with regular center: A Review of existing models including a recent realization with Gaussian sources”, [arXiv:0802.0330 [gr-qc]].
[40] Md. Sabir Ali and S. G. Ghosh, ”Exact d-dimensional Bardeen-de Sitter black holes and thermodynamics”, Phys. Rev. D 98, 084025 (2018). DOI: 10.1103/PhysRevD.98.084025
[41] A. Kumar, D. V. Singh and S. Upadhyay, ”Impact of Perfect Fluid Dark Matter on the Thermodynamics of AdS Ayón–Beato–García Black Holes”, JHAP 4(4), 85 (2024). [arXiv:2503.04805v1 [gr-qc]].
[42] A. Kumar, D. V. Singh and S. Upadhyay, ”Ayón–Beato–García black hole coupled with a cloud of strings: thermodynamics, shadows and quasinormal modes”,Int. J. Mod. Phys. A 39(31), 2450136 (2024). DOI: 10.1142/S0217751X24501367
[43] H. K. Sudhanshu, D. V. Singh, S. Upadhyay, Y. Myrzakulov and K. Myrzakulov, ”Thermodynamics of a newly constructed black hole coupled with nonlinear electrodynamics and cloud of strings”, Phys. Dark Univ. 46, 101648 (2024). DOI: 10.1016/j.dark.2024.101648
[44] B. Singh, D. Veer Singh and B. Kumar Singh, ”Thermodynamics, phase structure and quasinormal modes for AdS Heyward massive black hole”, Phys. Scripta 99(2), 025305 (2024). DOI: 10.1088/1402-4896/ad1da4
[45] P. Paul, S. Upadhyay, Y. Myrzakulov, D. V. Singh and K. Myrzakulov, ”More exact thermodynamics of nonlinear charged AdS black holes in 4D critical gravity”, Nucl. Phys. B 993, 116259 (2023). DOI: 10.1016/j.nuclphysb.2023.116259
[46] D. V. Singh, A. Shukla and S. Upadhyay, ”Quasinormal modes, shadow and thermodynamics of black holes coupled with nonlinear electrodynamics and cloud of strings”, Annals Phys. 447(1), 169157 (2022). DOI: 10.1016/j.aop.2022.169157
[47] D. V. Singh, S. G. Ghosh and S. D. Maharaj, ”Exact nonsingular black holes and thermodynamics”, Nucl. Phys. B 981, 11584 (2022). DOI: 10.1016/j.nuclphysb.2022.115854
[48] D. V. Singh and S. Siwach, Phys. Lett. B 808, 135658 (2020). DOI: 10.1016/j.physletb.2020.135658 [arXiv:2003.11754 [gr-qc]].
[49] V. Bozza, ”Gravitational lensing in the strong field limit”, Physical Review D 66, 103001 (2002). DOI: 10.1103/PhysRevD.66.103001
[2] F. W. Dyson, A. S. Eddington and C. Davidson, “A Determination of the Deflection of Light by the Sun’s Gravitational Field, from Observations Made at the Total Eclipse of May 29, 1919”, Philos Trans. A Math. Phys. Eng. Sci. 220(571-581), 291 (1920). DOI: 10.1098/rsta.1920.0009
[3] The Event Horizon Telescope Collaboration et al, “First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole”, The Astrophysical Journal Letters 875(1), L1 (2019). DOI: 10.3847/2041-8213/ab0ec7
[4] The Event Horizon Telescope Collaboration et al, “First M87 Event Horizon Telescope Results. II. Array and Instrumentation”, The Astrophysical Journal Letters 875(1), L2 (2019). DOI: 10.3847/2041-8213/ab0c96
[5] The Event Horizon Telescope Collaboration et al, “First M87 Event Horizon Telescope Results. III. Data Processing and Calibration”, The Astrophysical Journal Letters 875(1), L3 (2019). DOI: 10.3847/2041-8213/ab0c57
[6] The Event Horizon Telescope Collaboration et al, “First M87 Event Horizon Telescope Results. IV. Imaging the Central Supermassive Black Hole”, The Astrophysical Journal Letters 875(1), L4 (2019). DOI: 10.3847/2041-8213/ab0e85
[7] The Event Horizon Telescope Collaboration et al, “First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring”, The Astrophysical Journal Letters 875(1), L5 (2019). DOI: 10.3847/2041-8213/ab0f43
[8] The Event Horizon Telescope Collaboration et al, “First M87 Event Horizon Telescope Results. VI. The Shadow and Mass of the Central Black Hole”, The Astrophysical Journal Letters 875(1), L6 (2019). DOI: 10.3847/2041-8213/ab1141
[9] The Event Horizon Telescope Collaboration et al, “First M87 Event Horizon Telescope Results. VII. Polarization of the Ring”, The Astrophysical Journal Letters 910(1), L12 (2021). DOI: 10.3847/2041-8213/abe71d
[10] The Event Horizon Telescope Collaboration et al, “First M87 Event Horizon Telescope Results. VIII. Magnetic Field Structure near The Event Horizon”, The Astrophysical Journal Letters 910(1), L13 (2021). DOI: 10.3847/2041-8213/abe4de
[11] The Event Horizon Telescope Collaboration et al, “First M87 Event Horizon Telescope Results. IX. Detection of Near-horizon Circular Polarization”, The Astrophysical Journal Letters 957(2), L20 (2023). DOI: 10.3847/2041-8213/acff70
[12] K. S. Virbhadra and G. F. R. Ellis, ”Schwarzschild black hole lensing”, Phys. Rev. D 62, 084003 (2000). DOI: 10.1103/PhysRevD.62.084003
[13] C. M. Claudel, K. S. Virbhadra and G. F. R. Ellis, “The Geometry of photon surfaces”, J. Math. Phys. 42, 818 (2001). DOI: 10.1063/1.1308507
[14] V. Bozza and L. Mancini, “Observing Gravitational Lensing Effects by Sgr A* with Gravity”, Astrophys. J. 753(1), 56 (2012). DOI: 10.1088/0004-637X/753/1/56
[15] S. Pietroni and V. Bozza, “The impact of gravitational lensing in the reconstruction of stellar orbits around Sgr A*”, JCAP 2022(12), 018 (2022). DOI: 10.1088/1475- 7516/2022/12/018
[16] N. Tsukamoto, ”Deflection angle of a light ray reflected by a general marginally unstable photon sphere in a strong deflection limit”, Phys. Rev. D, 102, 104029 (2020). DOI: 10.1103/PhysRevD.102.104029
[17] N. Tsukamoto, “Gravitational lensing in the Simpson-Visser black-bounce spacetime in a strong deflection limit”, Phys. Rev. D 103, 024033 (2021). DOI: 10.1103/PhysRevD.103.024033
[18] N. Tsukamoto, “Gravitational lensing by a photon sphere in a Reissner-Nordström naked singularity spacetime in strong deflection limits”, Phys. Rev. D 104, 124016 (2021). DOI: 10.1103/PhysRevD.104.124016
[19] C. Furtado, J. R. Nascimento, A. Y. Petrov, P. J. Porfírio and A. R. Soares, ”Strong gravitational lensing in a spacetime with topological charge within the Eddingtoninspired Born-Infeld gravity”, Phys. Rev. D 103, 044047 (2021). DOI: 10.1103/PhysRevD.103.044047
[20] V. A. Kosteleckỳ and R. Potting, ”CPT and strings”, Nucl. Phys. B 359(2-3), 545 (1991). DOI: 10.1016/0550-3213(91)90071-5
[21] D. Colladay and V. A. Kosteleckỳ, ”Lorentz violating extension of the standard model”, Phys. Rev. D 58, 116002 (1998). DOI: 10.1103/PhysRevD.58.116002
[22] V. A. Kosteleckỳ and S. Samuel, ”Phenomenological Gravitational Constraints on Strings and Higher Dimensional Theories”, Phys. Rev. Lett. 63, 224 (1989). DOI: doi.org/10.1103/PhysRevLett.63.224
[23] R. Gambini and J. Pullin, ”Nonstandard optics from quantum space-time”, Phys. Rev. D 59, 124021 (1999). DOI: 10.1103/PhysRevD.59.124021
[24] E. Battista, ”Quantum Schwarzschild geometry in effective field theory models of gravity”, Phys. Rev. D 109, 026004 (2024). DOI: 10.1103/PhysRevD.109.026004
[25] V. A. Kosteleckỳ and S. Samuel, ”Gravitational Phenomenology in Higher Dimensional Theories and Strings”, Phys. Rev. D 40, 1886 (1989). DOI: 10.1103/PhysRevD.40.1886
[26] M. Kalb and P. Ramond, ”Classical direct interstring action”, Phys. Rev. D 9, 2273 (1974). DOI: 10.1103/PhysRevD.9.2273
[27] A. D. Sakharov, ”The Initial Stage of an Expanding Universe and the Appearance of a Nonuniform Distribution of Matter”, JETP 22(1), 241 (1966).
[28] E. B. Gliner, ”Algebraic Properties of the Energy-momentum Tensor and Vacuum-like States of Matter”, JETP 22(2), 378 (1966).
[29] J. Bardeen, ”Non-singular general relativistic gravitational collapse” in Proceedings of GR5 (Tiflis, U.S.S.R., 1968).
[30] E. Ayon-Beato, A. Garcia, ”The Bardeen model as a nonlinear magnetic monopole”, Phys. Lett. B 493(1-2), 149 (2000). DOI: 10.1016/S0370-2693(00)01125-4
[31] E. Ayon-Beato and A. Garcia, ”Non-Singular Charged Black Hole Solution for Non-Linear Source”, General Relativity and Gravitation 31, 629 (1999). DOI: 10.1023/A:1026640911319
[32] E. Ayon-Beato and A. Garcia, ”Four-parametric regular black hole solution”, General Relativity and Gravitation 37, 635 (2005). DOI: 10.1007/s10714-005-0050-y
[33] E. Ayon-Beato and A. Garcia, ”Regular black hole in general relativity coupled to nonlinear electrodynamics”, Phys. Rev. Lett. 80, 5056 (1998). DOI: 10.1103/PhysRevLett.80.5056
[34] L. Xiang, Y. Ling and Y. G. Shen, ”SINGULARITIES AND THE FINALE OF BLACK HOLE EVAPORATION”, International Journal of Modern Physics D 22(12), 1342016 (2013). DOI: 10.1142/S0218271813420169
[35] H. Culetu, ”Nonsingular black hole with a nonlinear electric source”, International Journal of Modern Physics D 24(09), 1542001 (2015). DOI: 10.1142/S0218271815420018
[36] L. Balart and E. C. Vagenas, ”Regular black holes with a nonlinear electrodynamics source”, Phys. Rev. D 90, 124045 (2014). DOI: 10.1103/PhysRevD.90.124045
[37] M. S. Ma, R. Zhao and Y. Q. Ma, ”Thermodynamic stability of black holes surrounded by quintessence”, General Relativity and Gravitation 49(79), (2017). DOI: 10.1007/s10714-017-2245-4
[38] S. H. Hendi, S. Panahiyan and B. Eslam Panah, ”Charged black hole solutions in Gauss-Bonnet-massive gravity”, J. High Energy Phys. 2016, 129 (2016). DOI: 10.1007/JHEP01(2016)129
[39] S. Ansoldi, ”Spherical black holes with regular center: A Review of existing models including a recent realization with Gaussian sources”, [arXiv:0802.0330 [gr-qc]].
[40] Md. Sabir Ali and S. G. Ghosh, ”Exact d-dimensional Bardeen-de Sitter black holes and thermodynamics”, Phys. Rev. D 98, 084025 (2018). DOI: 10.1103/PhysRevD.98.084025
[41] A. Kumar, D. V. Singh and S. Upadhyay, ”Impact of Perfect Fluid Dark Matter on the Thermodynamics of AdS Ayón–Beato–García Black Holes”, JHAP 4(4), 85 (2024). [arXiv:2503.04805v1 [gr-qc]].
[42] A. Kumar, D. V. Singh and S. Upadhyay, ”Ayón–Beato–García black hole coupled with a cloud of strings: thermodynamics, shadows and quasinormal modes”,Int. J. Mod. Phys. A 39(31), 2450136 (2024). DOI: 10.1142/S0217751X24501367
[43] H. K. Sudhanshu, D. V. Singh, S. Upadhyay, Y. Myrzakulov and K. Myrzakulov, ”Thermodynamics of a newly constructed black hole coupled with nonlinear electrodynamics and cloud of strings”, Phys. Dark Univ. 46, 101648 (2024). DOI: 10.1016/j.dark.2024.101648
[44] B. Singh, D. Veer Singh and B. Kumar Singh, ”Thermodynamics, phase structure and quasinormal modes for AdS Heyward massive black hole”, Phys. Scripta 99(2), 025305 (2024). DOI: 10.1088/1402-4896/ad1da4
[45] P. Paul, S. Upadhyay, Y. Myrzakulov, D. V. Singh and K. Myrzakulov, ”More exact thermodynamics of nonlinear charged AdS black holes in 4D critical gravity”, Nucl. Phys. B 993, 116259 (2023). DOI: 10.1016/j.nuclphysb.2023.116259
[46] D. V. Singh, A. Shukla and S. Upadhyay, ”Quasinormal modes, shadow and thermodynamics of black holes coupled with nonlinear electrodynamics and cloud of strings”, Annals Phys. 447(1), 169157 (2022). DOI: 10.1016/j.aop.2022.169157
[47] D. V. Singh, S. G. Ghosh and S. D. Maharaj, ”Exact nonsingular black holes and thermodynamics”, Nucl. Phys. B 981, 11584 (2022). DOI: 10.1016/j.nuclphysb.2022.115854
[48] D. V. Singh and S. Siwach, Phys. Lett. B 808, 135658 (2020). DOI: 10.1016/j.physletb.2020.135658 [arXiv:2003.11754 [gr-qc]].
[49] V. Bozza, ”Gravitational lensing in the strong field limit”, Physical Review D 66, 103001 (2002). DOI: 10.1103/PhysRevD.66.103001
)
Volume 6, Issue 2
January 2026Pages 104-118
- Receive Date: 16 November 2025
- Revise Date: 15 December 2025
- Accept Date: 22 December 2025