On HPM approximation for the perihelion preces-sion angle in general relativity

  • Authors

    • Victor Shchigolev Department of Theoretical Physics, Ulyanovsk State University
    • Dmitrii Bezbatko Department of Theoretical Physics, Ulyanovsk State University
    2017-02-25
    https://doi.org/10.14419/ijaa.v5i1.7279
  • General Relativit, Homotopy Perturbation Method, Perihelion Precession, Kiselev Black Hole.
  • Abstract

    In this paper, the homotopy perturbation method (HPM) is applied for calculating the perihelion precession angle of planetary orbits in General Relativity. The HPM is quite efficient and is practically well suited for use in many astrophysical and cosmological problems. For our purpose, we applied HPM to the approximate solutions for the orbits in order to calculate the perihelion shift. On the basis of the main idea of HPM, we construct the appropriate homotopy that leads to the problem of solving the set of linear algebraic equations. As a result, we obtain a simple formula for the angle of precession avoiding any restrictions on the smallness of physical parameters. First of all, we consider the simple examples of the Schwarzschild metric and the Reissner - Nordström spacetime of a charged star for which the approximate geodesics solutions are known. Furthermore, the implementation of HPM has allowed us to readily obtain the precession angle for the orbits in the gravitational field of Kiselev black hole.

  • References

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

      [2] 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, Volume 2014, Article ID 604321, 7 pages. https://doi.org/10.1155/2014/604321.

      [3] Christian Magnan, "Complete calculations of the perihelion precession of Mercury and the deflection of light by the Sun in General Relativity", http://arxiv.org/abs/0712.3709

      [4] Hideyoshi Arakida, "Note on the Perihelion Periastron Advance Due to Cosmological Constant", International Journal of Theoretical Physics, 52 (2013), 1408. https://doi.org/10.1007/s10773-012-1458-2.

      [5] A. A. Vankov, "General Relativity Problem of Mercury's Perihelion Advance Revisite", http://arxiv.org/abs/1008.1811

      [6] 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", Classical and Quantum Gravity, 20 (2003), 4817-4835. https://doi.org/10.1088/0264-9381/20/22/007.

      [7] 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

      [8] M. Pejic, "Calculating perihelion precession using the multiple scales method", http://math.berkeley.edu/ mpejic/pdfdocuments/PerihelionPrecession.pdf

      [9] M. L. Ruggiero, "Perturbations of Keplerian orbits in stationary spherically symmetric spacetimes", International Journal of Modern Physsics D, 23(5) (2014), 1450049. https://doi.org/10.1142/S0218271814500497.

      [10] L. Iorio, E. N. Saridakis, "Solar system constraints on gravity", Monthly Notices of the Royal Astronomical Society, 427 (2012), 1555. https://doi.org/10.1111/j.1365-2966.2012.21995.x.

      [11] Sumanta Chakraborty, Soumitra SenGupta, "Solar system constraints on alternative gravity theories", Physical Review D, 89 (2014) 026003. https://doi.org/10.1103/PhysRevD.89.026003.

      [12] R. R. Cuzinatto, P. J. Pompeia, M. de Montigny, F. C. Khanna, J. M. Hoff da Silva, "Classic tests of General Relativity described by brane-based spherically symmetric solutions", European Physical Journal C 74 (2014), 3017. https://doi.org/10.1140/epjc/s10052-014-3017-x.

      [13] 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.

      [14] 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", Classical and Quantum Gravity, 20 (2003), 4817-4835. https://doi.org/10.1088/0264-9381/20/22/007.

      [15] V. K. Shchigolev, D. N. Bezbatko, "Studying Gravitational Deflection of Light by Kiselev Black Hole via Homotopy Perturbation Method", http://arxiv.org/abs/1612.07279

      [16] J.-H. He, "Homotopy perturbation technique", Computer Methods in Applied Mechanics and Engineering, 178 (1999), 257-262. https://doi.org/10.1016/S0045-7825(99)00018-3.

      [17] J.-H. He, "A coupling method of homotopy technique and perturbation technique for nonlinear problems", International Journal of Non-Linear Mechanics, 35 (1) (2000), 37-43. https://doi.org/10.1016/S0020-7462(98)00085-7.

      [18] J.-H. He, "Asymptotic Methods for Solitary Solutions and Compactions", Abstract and Applied Analysis, Volume 2012, Article ID 916793, 130 pages. https://doi.org/10.1155/2012/916793.

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

      [20] V. K. Shchigolev, "Analytical Computation of the Perihelion Precession in General Relativity via the Homotopy Perturbation Method", Universal Journal of Computational Mathematics, 3(4) (2015), 45-49.

      [21] F. Rahaman, S. Ray, A. Aziz, S. R. Chowdhury, D. Deb, "Exact Radiation Model For Perfect Fluid Under Maximum Entropy Principle", http://arxiv.org/abs/1504.05838

      [22] Abdul Aziz, Saibal Ray, Farook Rahaman, "A generalized model for compact stars", European Physical Journal C, 76 (2016), 248. https://doi.org/10.1140/epjc/s10052-016-4090-0.

      [23] V. V. Kiselev, "Quintessence and black holes", Classical and Quantum Gravity, 20 (2003), 1187-1198. https://doi.org/10.1088/0264-9381/20/6/310.

      [24] V. K. Shchigolev, "Variational iteration method for studying perihelion precession and deflection of light in General Relativity", International Journal of Physical Research, 4 (2) (2016), 52-57. https://doi.org/10.14419/ijpr.v4i2.6530.

      [25] J.-H. He, "A coupling method of homotopy technique and perturbation technique for nonlinear problems", International Journal of Non-Linear Mechanics, 35 (1) (2000), 37-43. https://doi.org/10.1016/S0020-7462(98)00085-7.

      [26] L. Cveticanin, "Homotopy-perturbation method for pure nonlinear differential equation", Chaos, Solitons Fractals, 30(5) (2006), 1221 - 1230. https://doi.org/10.1016/j.chaos.2005.08.180.

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  • How to Cite

    Shchigolev, V., & Bezbatko, D. (2017). On HPM approximation for the perihelion preces-sion angle in general relativity. International Journal of Advanced Astronomy, 5(1), 38-43. https://doi.org/10.14419/ijaa.v5i1.7279

    Received date: 2017-01-24

    Accepted date: 2017-02-18

    Published date: 2017-02-25