Variational iteration method for studying perihelion precession and deflection of light in General Relativity


  • V.K. Shchigolev Department of Theoretical Physics, Ulyanovsk State University





Approximate solution, Deflection of light, Perihelion precession, Spherically symmetric spacetime, Variational iteration method.


A new approach in studying the planetary orbits and deflection of light in General Relativity (GR) by means of the Variational iteration method (VIM) is proposed in this paper. For this purpose, a brief review of the nonlinear geodesic equations in the spherical symmetry spacetime and the main ideas of VIM are given. The appropriate correct functionals are constructed for the geodesics in the spacetime of Schwarzschild, Reissner-Nordström and Kiselev black holes. In these cases, the Lagrange multiplier is obtained from the stationary conditions for the correct functionals. Then, VIM leads to the simple problem of computation of the integrals in order to obtain the approximate solutions of the geodesic equations. On the basis of these approximate solutions, the perihelion shift and the light deflection have been obtained for the metrics mentioned above.


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

[2] G. V. Kraniotis, S. B. Whitehouse, â€Compact calculation of the Perihelion Precession of Mercury in General Relativity, the Cosmological Constant and Jacobi’s Inversion problemâ€,

[3] Classical and Quantum Gravity, 20 (2003), 4817-4835.


[5] J.-H. He, â€Homotopy perturbation techniqueâ€, Computer Methods in Applied Mechanics and Engineering, 178 (1999), 257-262.

[6] J. H. He, â€Variational iteration method - a kind of non-linear analytical technique: some examplesâ€, International Journal of Non-Linear Mechanics, 34(4) (1999), 699-708.

[7] J. H. He, â€Variational iteration method for autonomous ordinary differential systemsâ€, Applied Mathematics and Computation, 114(2-3) (2000), 115-123.

[8] J. H. He, â€Variational iteration method-Some recent results and new interpretationsâ€, Journal of Computational and Applied Mathematics, 207(1) (2007), 3-17.

[9] V. Shchigolev, â€Homotopy Perturbation Method for Solving a Spatially Flat FRW Cosmological Modelâ€, Universal Journal of Applied Mathematics, 2(2) (2014), 99-103.


[11] V. Shchigolev, â€Analytical Computation of the Perihelion Precession in General Relativity via the Homotopy Perturbation Methodâ€, Universal Journal of Computational Mathematics, 3(4) (2015), 45-49.

[12] V. K. Shchigolev, â€Calculating Luminosity Distance versus Redshift in FLRW Cosmology via Homotopy Perturbation Methodâ€,

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

[14] Abdul Aziz, Saibal Ray, Farook Rahaman, â€A generalized model for compact starsâ€, European Physical Journal C, 76 (2016), 248.


[16] Ya-Peng Hu, Hongsheng Zhang, Jun-Peng Hou, and Liang-Zun Tang. â€Perihelion Precession and Deflection of Light in the General Spherically Symmetric Spacetimeâ€, Advances in High

[17] Energy Physics, Volume 2014, Article ID 604321, 7 pages.


[19] Hideyoshi Arakida, â€Note on the Perihelion/Periastron Advance Due to Cosmological Constantâ€, International Journal of Theoretical Physics, 52 (2013), 1408-1414.

[20] Christian Magnan, â€Complete calculations of the perihelion precession of Mercury and the deflection of light by the Sun in General Relativityâ€,

[21] A.S. Fokas, C.G. Vayenas, D. Grigoriou, â€Analytical computation of the Mercury perihelion precession via the relativistic gravitational law and comparison with general relativityâ€,


  1. G. Riess , et al. , â€Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constantâ€, Astronomical Journal, Vol. 116 (1998), 1009.

[23] S. Perlmutter, et al., â€Measurements of Omega and Lambda from 42 High-Redshift Supernovaeâ€, Astrophysical Journal, Vol. 517 (1999), 565.

[24] D. N. Spergel, et al., â€First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parametersâ€, Astrophysical Jounal Supplement Series, 148 (2003), 175- 194.

[25] M. Tegmark, M. A. Strauss, et al., â€Cosmological parameters from SDSS and WMAPâ€, Physical Review D, 69 (2004), 103501.

[26] S. W. Allen, R. W. Schmidt, H. Ebeling, et al. , â€Constraints on dark energy from Chandra observations of the largest relaxed galaxy clustersâ€, Monthly Notices of the Royal Astronomical Society, 353 (2004), 457-467.


[28] V. V. Kiselev, â€Quintessence and black holesâ€, Classical and Quantum Gravity, 20 (2003), 1187-1198.

[29] Azka Younas, Mubasher Jamil, Sebastian Bahamonde, Saqib Hussain, â€Strong Gravitational Lensing by Kiselev Black Holeâ€, Physical Review D, 92 (2015), 084042.

[30] Lei Jiao and Rong-Jia Yang, â€Accretion onto a Kiselev black holeâ€,

[31] Bushra Majeed, Mubasher Jamil, Parthapratim Pradhan, â€Thermodynamic Relations for Kiselev and Dilaton Black Holeâ€, Advances in High Energy Physics, 2015 (2015), 124910.


[33] Ibrar Hussain, Sajid Ali, â€Marginally Stable Circular Orbits in Schwarzschild Black Hole Surrounded by Quintessence Matterâ€,

[34] M. Tari and M. Dehghan, â€On the Convergence of He’s Variational Iteration Methodâ€, Journal of Computational and Applied Mathematics, 207(1) (2007), 121-128.

[35] J. I. Ramos, â€On the Variational Iteration Method and Other Iterative Techniques for Nonlinear Differential Equations,†Applied Mathematics and Computation, 199(1) (2008), 39-69.

[36] Ernest Scheiber, â€On the Convergence of the Variational Iteration Methodâ€,

View Full Article: