On the light deflection and perihelion precession by BBMB black holes in HPM approximation


  • Victor Konstantinovich Shchigolev Department of Theoretical Physics, Ulyanovsk State University






In this paper, the homotopy perturbation method (HPM) is applied for calculating the weak deflection angle and the perihelion precession angle of planetary orbits in the gravitational field of Bocharova-Bronnikov-Melnikov-Bekenstein (BBMB) black hole. Follow the procedure of HPM, we obtain the approximate solutions for the null and time-like geodesics in the gravitational field of BBMB black hole. On the basis of these solutions and the general formulae for the angle of deflection and the perihelion precession angle, derived by the author earlier via HPM, the corresponding angles for the BBMB black hole are obtained and compared with the similar angles in Schwarzschild spacetime. Notably that if the deflection angle obtained in this article using the HPM confirms the results of other researchers, then the perihelion precession angle obtained here for the first time can be compared with those known for a classical non-rotating static black hole.


S. Weinberg, Gravitation and Cosmology: Principles and Applications of The General Theory of Relativity (John Wiley. Press, New York, 1972).

C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, W. H. Freeman and Co., San Francisco, Calif, USA, 1973.

R. M. Wald, General Relativity, University of Chicago Press, Chicago, Ill, USA, 1984.

G. V. Kraniotis, S. B. ”Whitehouse, Compact calculation of the Perihelion Precession of Mercury in General Relativity, the Cosmological Constant and

Jacobi’s Inversion problem”, Class. Quant. Grav., Vol. 20, 4817-4835, 2003. DOI:10.1088/0264-9381/20/22/007

Ya-Peng Hu, Hongsheng Zhang, Jun-Peng Hou, and Liang-Zun Tang, ”Perihelion Precession and Deflection of Light in the General Spherically

Symmetric Spacetime”, Advances in High Energy Physics, Vol. 2014, Article ID 604321, 7 pages. http://dx.doi.org/10.1155/2014/604321

W. Javed, M. B. Khadim, A. Ovg ¨ un, ”Weak gravitational lensing by Bocharova-Bronnikov-Melnikov-Bekenstein black holes using Gauss-Bonnet ¨

theorem”, Eur. Phys. J. Plus, Vol. 135 , 595, 2020. https://doi.org/10.1140/epjp/s13360-020-00619-x

M. V. Bebronne, P.G.Tinyakov, Erratum: ”Black hole solutions in massive gravity”, J. High Energ. Phys., Vol. 2011, 018, 2011.


R. S. Kuniyal, R. Uniyal, A. Biswas, et al., ”Null geodesics and red-blue shifts of photons emitted from geodesic particles around a noncommutative

black hole space-time”, Int. J. Modern Phys. A, Vol. 33, 1850098, 2018. https://doi.org/10.1142/S0217751X18500987

M. K. Mak, C. S. Leung, and T. Harko, ”Computation of the General Relativistic Perihelion Precession and of Light Deflection via the Laplace-Adomian

Decomposition Method”, Advances in High Energy Physics, Vol. 2018, Article ID 7093592, 15 pages. https://doi.org/10.1155/2018/7093592

G. W. Gibbons and M. C. Werner, ”Classical and Quantum Gravity Applications of the Gauss-Bonnet theorem to gravitational lensing”, Class. Quantum

Grav., Vol. 25, 235009, 2008. https://doi.org/10.1088/0264-9381/25/23/235009

I. Sakalli, A. Ovg ¨ un, ”Hawking radiation and deflection of light from Rindler modified Schwarzschild black hole”, EPL, Vol. 118, 60006, 2017. ¨


G. Crisnejo and E. Gallo, ”Weak lensing in a plasma medium and gravitational deflection of massive particles using the Gauss-Bonnet theorem. A

unified treatment”, Phys. Rev. D 97, 124016 (2018). https://doi.org/10.1103/PhysRevD.97.124016

J. Y. Kim, ”Deflection of light by a Coulomb charge in Born-Infeld electrodynamics”, Eur. Phys. J. C, Vol. 81, 508, 2021.


S. W. Kim and Y. M. Cho, In: Evolution of the Universe and its Observational Quest (Universal Academy Press, Tokyo, 1994), p. 353.

J. G. Cramer, R. L. Forward, M. S. Morris, et al., ”Natural wormholes as gravitational lenses”, Phys. Rev. D, Vol. 51, 3117, 1995.


K. Jusufi, N. Sarkar, F. Rahaman, et al., ”Deflection of light by black holes and massless wormholes in massive gravity”, Eur. Phys. J. C, Vol. 78, 349,


J. Poutanen, ”Accurate analytic formula for light bending in Schwarzschild metric”, A&A Vol. 640, A24, 2020. https://doi.org/10.1051/0004-


K. A. Bronnikov, K. A. Baleevskikh, ”On Gravitational Lensing by Symmetric and Asymmetric Wormholes”, Gravit. Cosmol., Vol. 25, 44-49, 2019.


C. Magnan, ”Complete calculations of the perihelion precession of Mercury and the deflection of light by the Sun in General Relativity”,


H. Arakida, ”Note on the Perihelion/Periastron Advance Due to Cosmological Constant”, Int. J. Theor. Phys., Vol. 52, 1408, 2013.


L. S. Ridao, R. Avalos, M. D. De Cicco and M. Bellini, ”Perihelion advances for the orbits of Mercury, Earth and Pluto from Extended Theory of

General Relativity (ETGR)”, https://doi.org/10.48550/arXiv.1402.4511

A. A. Vankov, ”General Relativity Problem of Mercury’s Perihelion Advance Revisited”, http://arxiv.org/abs/1008.1811

G. V. Kraniotis, S. B. Whitehouse, ”Compact calculation of the Perihelion Precession of Mercury in General Relativity, the Cosmological Constant and

Jacobi’s Inversion problem”, Class. Quantum Grav., Vol. 20, 4817-4835, 2003. https://doi.org/10.1088/0264-9381/20/22/007

T. Ono, T. Suzuki, N. Fushimi, K. Yamada, H. Asada, ”Marginally stable circular orbit of a test body in spherically symmetric and static spacetimes”,


S.S. Ovcherenko, Z.K. Silagadze, ”Comment on perihelion advance due to cosmological constant”, Ukr. J. Phys., Vol. 61, 342-344, 2016.


M. L. Ruggiero, ”Perturbations of Keplerian orbits in. stationary spherically symmetric spacetimes”, Int. J. Mod. Phys. D, Vol. 23, 1450049, 2014.


S. Chakraborty, S. SenGupta, ”Solar system constraints on alternative gravity theories”, Phys. Rev. D, Vol. 89, 026003, 2014.


R. R. Cuzinatto, P. J. Pompeia, M. de Montigny, et al., ”Classic tests of General Relativity described by brane-based spherically symmetric solutions”,

Eur. Phys. J. C, Vol. 74, 3017, 2014. https://doi.org/10.1140/epjc/s10052-014-3017-x

A. S. Fokas, C. G. Vayenas, D. Grigoriou, ”Analytical computation of the Mercury perihelion precession via the relativistic gravitational law and

comparison with general relativity”, http://arxiv.org/abs/1509.03326

Haotian Liu, Jinning Liang, Junji Jia, ”Deflection and Gravitational lensing of null and timelike signals in the Kiselev black hole spacetime in the weak

field limit”, https://doi.org/10.48550/arXiv.2204.04519

N. M. Bocharova, K. A. Bronnikov, and V. N. Melnikov, ”An exact solution of the system of Einstein equations and mass-free scalar field”, Vestn. Mosk.

Univ. Ser. III Fiz. Astron., Vol. 6, 706, 1970.

J. D. Bekenstein, ”Exact solutions of Einstein-conformal scalar equations”, Annals of Physics, Vol. 82(2), 535-547 (1974).

J.-H. He, ”Homotopy perturbation technique”, Comput. Meth. Appl. Mech. Eng., Vol. 178, 257, 1999. https://doi.org/10.1016/S0045-7825(99)00018-3

J.-H. He, ”A coupling method of a homotopy technique and a perturbation technique for non-linear problems”, Int. J. Nonlinear Mech., Vol. 35, 37,


L. Cveticanin, ”Homotopy-perturbation method for pure nonlinear differential equation”, Chaos Soliton Fract., Vol. 30, 1221, 2006.


M. Zare, O. Jalili and M. Delshadmanesh, ”Two binary stars gravitational waves: homotopy perturbation method”, Indian J. Phys., Vol. 86, 855, 2012.


V. Shchigolev, ”Homotopy Perturbation Method for Solving a Spatially Flat FRW Cosmological Model”, Univ. J. Applied Math., Vol. 2, 99, 2014. DOI:


V. Shchigolev, ”Analytical Computation of the Perihelion Precession in General Relativity via the Homotopy Perturbation Method”, Univ. J. Comput.

Math., Vol. 3, 45, 2015. DOI: 10.13189/ujcmj.2015.030401

F. Rahaman, S. Ray, A. Aziz, S. R. Chowdhury, D. Deb, ”Exact Radiation Model For Perfect Fluid Under Maximum Entropy Principle”,


V. K. Shchigolev, ”Calculating luminosity distance versus redshift in FLRW cosmology via homotopy perturbation method”, Gravit.Cosmol., Vol. 23,

-148, 2017. https://doi.org/10.1134/S0202289317020098

V. K. Shchigolev, D. N. Bezbatko, ”On HPM approximation for the perihelion precession angle in general relativity”, Int. J. Adv. Astron., Vol. 5(1), 38,


V. K. Shchigolev, D. N. Bezbatko, ”Studying Gravitational Deflection of Light by Kiselev Black Hole via Homotopy Perturbation Method”, Gen. Rel.

Grav., Vol. 51, 34, 2019. ttps://doi.org/10.1007/s10714-019-2521-6

V. K. Shchigolev, ”Exact solutions to the null-geodesics in Ellis-Bronnikov wormhole spacetime via (G’/G)-expansion method”, Modern Physics Letters

A, Vol. 37, No. 20, 2250124, 2022. https://doi.org/10.1142/S0217732322501243

V. K. Shchigolev, ”Exact analytical solutions to the geodesic equations in general relativity via (G’/G) - expansion method”, Gen Relativ Gravit, Vol. 54,

2022. https://doi.org/10.1007/s10714-022-02964-x

View Full Article: