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Higher-order geodesic deviations and orbital precession in a Kerr–Newman space–time

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Abstract

A novel approximation method in studying the perihelion precession and planetary orbits in general relativity is to use geodesic deviation equations of first and high-orders, proposed by Kerner et al. Using higher-order geodesic deviation approach, we generalize the calculation of orbital precession and the elliptical trajectory of neutral test particles to Kerr–Newman space–times. One of the advantage of this method is that, for small eccentricities, one obtains trajectories of planets without using Newtonian and post-Newtonian approximations for arbitrary values of quantity \({G M}/{R c^2}\).

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Acknowledgements

We would like to thank Richard Kerner for helpful discussions and Mohsen Khodadi for a careful reading of the manuscript and useful suggestions. We also thank the anonymous referee for valuable comments.

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Authors and Affiliations

  1. Department of Physics, Shahid Beheshti University, G. C., Evin, 19839, Tehran, Iran

    Mohaddese Heydari-Fard & Hamid Reza Sepangi

  2. Department of Physics, The University of Qom, 3716146611, Qom, Iran

    Malihe Heydari-Fard

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  1. Mohaddese Heydari-Fard
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  2. Malihe Heydari-Fard
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  3. Hamid Reza Sepangi
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Correspondence to Hamid Reza Sepangi.

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Appendix

Appendix

The coefficients of \(\delta ^2x^{\mu }\) in the second-order geodesic deviation equations are

$$\begin{aligned} \delta ^{2}t _{0}= & {} 0,\quad \delta ^{2}t _{2\theta }=\frac{(n_{0}^{\theta })^{2}}{\sqrt{f_{3}}}\frac{a^{2}}{R}\sqrt{\frac{M}{R}-\frac{Q^{2}}{R^{2}}} \left[ 1+\frac{Q^{2}}{2R^{2}}\left( \frac{M}{R}-\frac{Q^{2}}{R^{2}}\right) ^{-1}\right] , \end{aligned}$$
(57)
$$\begin{aligned} \delta ^{2}t _{2r}= & {} \frac{(n_{0}^{r})^{2}\sqrt{\frac{M}{R}-\frac{Q^{2}}{R^{2}}}}{Rf_{2}^2f_{6}^{\frac{3}{2}}} \Bigg \lbrace \left[ \left( 2-\frac{15M}{R}+\frac{14M^{2}}{R^{2}} +\frac{10Q^{2}}{R^{2}}-\frac{17MQ^{2}}{R^{3}}+\frac{4Q^{4}}{R^{4}}\right) \right. \nonumber \\&+\,\left. \frac{1}{2}\left( \frac{3Q^{2}}{R^{2}}-\frac{2Q^{4}}{R^{4}} -\frac{Q^{6}}{R^{6}}\right) \left( \frac{M}{R}-\frac{Q^{2}}{R^{2}}\right) ^{-1}\nonumber \right. \\&\left. -\frac{1}{2}\left( \frac{Q^{4}}{R^{4}}-\frac{2Q^{6}}{R^{6}}+\frac{Q^{8}}{R^{8}}\right) \left( \frac{M}{R}-\frac{Q^{2}}{R^{2}}\right) ^{-2}\right] \nonumber \\&+\,\frac{a}{R}\sqrt{\frac{M}{R}-\frac{Q^{2}}{R^{2}}}\left[ \left( 11+\frac{34M}{R}-\frac{64M^{2}}{R^{2}} +\frac{15Q^{2}}{R^{2}}+\frac{38MQ^{2}}{R^{3}}-\frac{26Q^{4}}{R^{4}}\right) \right. \nonumber \\&\left. -\frac{1}{2}\left( \frac{31Q^{2}}{R^{2}}-\frac{64Q^{4}}{R^{4}}+\frac{33Q^{6}}{R^{6}}\right) \left( \frac{M}{R}-\frac{Q^{2}}{R^{2}}\right) ^{-1}\nonumber \right. \\&\left. -\frac{1}{2}\left( \frac{5Q^{4}}{R^{4}}-\frac{10Q^{6}}{R^{6}}+\frac{5Q^{8}}{R^{8}}\right) \left( \frac{M}{R}-\frac{Q^{2}}{R^{2}}\right) ^{-2}\right] \nonumber \\&-\,\frac{a^{2}}{R^{2}}\left[ \left( 3+\frac{84M}{R}-\frac{126M^{2}}{R^{2}}-\frac{53Q^{2}}{R^{2}}+\frac{137MQ^{2}}{R^{3}}-\frac{36Q^{4}}{R^{4}}\right) \right. \nonumber \\&\left. -\frac{1}{2}\left( \frac{15Q^{2}}{R^{2}}-\frac{10Q^{4}}{R^{4}}-\frac{Q^{6}}{R^{6}}\right) \left( \frac{M}{R}-\frac{Q^{2}}{R^{2}}\right) ^{-1}\nonumber \right. \\&\left. +\frac{1}{2} \left( \frac{3Q^{4}}{R^{4}}-\frac{4Q^{6}}{R^{6}}+\frac{Q^{8}}{R^{8}}\right) \left( \frac{M}{R}-\frac{Q^{2}}{R^{2}}\right) ^{-2}\right] \nonumber \\&+\,\frac{a^{3}}{R^{3}}\sqrt{\frac{M}{R}-\frac{Q^{2}}{R^{2}}}\left[ \left( 56-\frac{66M}{R}+\frac{35Q^{2}}{R^{2}}\right) -\left( \frac{16Q^{2}}{R^{2}}-\frac{15Q^{4}}{R^{4}}\right) \left( \frac{M}{R}-\frac{Q^{2}}{R^{2}}\right) ^{-1}\right. \nonumber \\&-\,4\left. \left( \frac{Q^{4}}{R^{4}}-\frac{Q^{6}}{R^{6}}\right) \left( \frac{M}{R}-\frac{Q^{2}}{R^{2}}\right) ^{-2}\right] -\frac{a^{4}}{R^{4}}\left[ \left( 12+\frac{21M}{R}-\frac{19Q^{2}}{R^{2}}\right) \right. \nonumber \\&-\,\left. \frac{1}{2}\left( \frac{21Q^{2}}{R^{2}}-\frac{8Q^{4}}{R^{4}}\right) \left( \frac{M}{R}-\frac{Q^{2}}{R^{2}}\right) ^{-1} +\frac{1}{2}\left( \frac{3Q^{4}}{R^{4}}-\frac{2Q^{6}}{R^{6}}\right) \left( \frac{M}{R}-\frac{Q^{2}}{R^{2}}\right) ^{-2}\right] \nonumber \\&+\,\frac{a^5}{R^5}\sqrt{\frac{M}{R}-\frac{Q^2}{R^2}}\left[ 29-\frac{17Q^2}{2R^2}\left( \frac{M}{R}-\frac{Q^2}{R^2}\right) ^{-1} -\frac{3Q^4}{2R^4}\left( \frac{M}{R}-\frac{Q^2}{R^2}\right) ^{-2}\right] \nonumber \\&-\,\frac{a^6}{R^6}\left[ 27-\frac{9Q^2}{2R^2}\left( \frac{M}{R}-\frac{Q^2}{R^2}\right) ^{-1} +\frac{Q^4}{2R^4}\left( \frac{M}{R}-\frac{Q^2}{R^2}\right) ^{-2}\right] \Bigg \rbrace \end{aligned}$$
(58)
$$\begin{aligned} \delta ^{2}r_{0}= & {} 0,\quad \delta ^{2}r_{2r}=\frac{-(n_{0}^{r})^{2}}{Rf_{6}} \Bigg \lbrace \left[ \left( 1-\frac{7M}{R}+\frac{5Q^{2}}{R^{2}}\right) +\left( \frac{Q^{2}}{R^{2}}-\frac{Q^{4}}{R^{4}}\right) \left( \frac{M}{R}-\frac{Q^{2}}{R^{2}}\right) ^{-1}\right] \nonumber \\&+\,\frac{10a}{R}\sqrt{\frac{M}{R}-\frac{Q^{2}}{R^{2}}}\left[ 1-\frac{Q^{2}}{5R^{2}}\left( \frac{M}{R}-\frac{Q^{2}}{R^{2}}\right) ^{-1}\right] \nonumber \\&-\,\frac{4a^{2}}{R^{2}}\left[ 1-\frac{Q^{2}}{2R^{2}}\left( \frac{M}{R}-\frac{Q^{2}}{R^{2}}\right) ^{-1}\right] \Bigg \rbrace , \end{aligned}$$
(59)
$$\begin{aligned} \delta ^{2}\theta _{-}= & {} -\delta ^{2}\theta _{+}=\frac{2(n_{0}^{r})(n_{0}^{\theta })}{R}\frac{\sqrt{f_3}}{\sqrt{f_6}} \end{aligned}$$
(60)
$$\begin{aligned} \delta ^{2}\varphi _{0}= & {} 0,\quad \delta ^{2}\varphi _{2\theta }=\frac{(n_{0}^{\theta })^{2}}{\sqrt{f_{3}}} \left[ 1-\frac{2a}{R}\sqrt{\frac{M}{R}-\frac{Q^{2}}{R^{2}}} \left( 1+\frac{Q^{2}}{2R^{2}} \left( \frac{M}{R}-\frac{Q^{2}}{R^{2}}\right) ^{-1}\right) \right] , \end{aligned}$$
(61)
$$\begin{aligned} \delta ^{2}\varphi _{2r}= & {} \frac{-2(n_{0}^{r})^{2}}{R^{2}f^{2}_{2}f_{6}^{\frac{3}{2}}}\Bigg \lbrace \left[ \left( 5-\frac{32M}{R}\right) \left( 1-\frac{2M}{R}\right) ^{2}-\frac{9Q^{2}}{R^{2}} +\frac{140MQ^{2}}{R^{3}}-\frac{204M^{2}Q^{2}}{R^{4}}\right. \nonumber \\&-\,\frac{63Q^{4}}{R^{4}}+\frac{128MQ^{4}}{R^{5}} \nonumber \\&-\,\left. \frac{19Q^{6}}{R^{6}} -5\left( \frac{Q^{2}}{R^{2}}-\frac{3Q^{4}}{R^{4}}+\frac{3Q^{6}}{R^{6}}-\frac{Q^{8}}{R^{8}}\right) \left( \frac{M}{R}-\frac{Q^{2}}{R^{2}}\right) ^{-1}\right] \nonumber \\&-\,\frac{2a}{R}\sqrt{\frac{M}{R}-\frac{Q^{2}}{R^{2}}}\left[ \left( 26-\frac{119M}{R}+\frac{126M^{2}}{R^{2}} +\frac{66Q^{2}}{R^{2}}-\frac{137MQ^{2}}{R^{3}}+\frac{36Q^{4}}{R^{4}}\right) \right. \nonumber \\&+\,\frac{1}{2}\left( \frac{3Q^{2}}{R^{2}}-\frac{2Q^{4}}{R^{4}}-\frac{Q^{6}}{R^{6}}\right) \left( \frac{M}{R}-\frac{Q^{2}}{R^{2}}\right) ^{-1}\nonumber \\&\left. -\frac{1}{2}\left( \frac{Q^{4}}{R^{4}}-\frac{2Q^{6}}{R^{6}}+\frac{Q^{8}}{R^{8}}\right) \left( \frac{M}{R}-\frac{Q^{2}}{R^{2}}\right) ^{-2}\right] \nonumber \\&+\,\frac{2a^{2}}{R^{2}}\left[ \left( 8-\frac{61M}{R}+\frac{66M^{2}}{R^{2}}-\frac{101MQ^{2}}{R^{3}}+\frac{20Q^{4}}{R^{4}}+\frac{68Q^{2}}{R^{2}}\right) \nonumber \right. \\&\left. -4\left( \frac{Q^{2}}{R^{2}}-\frac{2Q^{4}}{R^{4}}+\frac{Q^{6}}{R^{6}}\right) \left( \frac{M}{R}-\frac{Q^{2}}{R^{2}}\right) ^{-1}\right] \nonumber \\&+\,\frac{2a^{3}}{R^{3}}\sqrt{\frac{M}{R}-\frac{Q^{2}}{R^{2}}}\left[ \left( 5+\frac{21M}{R}-\frac{19Q^{2}}{R^{2}}\right) -\left( \frac{6Q^{2}}{R^{2}}-\frac{4Q^{4}}{R^{4}}\right) \left( \frac{M}{R}-\frac{Q^{2}}{R^{2}}\right) ^{-1}\right. \nonumber \\&\left. +\left( \frac{Q^{4}}{R^{4}}-\frac{Q^{6}}{R^{6}}\right) \left( \frac{M}{R}-\frac{Q^{2}}{R^{2}}\right) ^{-2}\right] +\frac{a^{4}}{R^{4}}\left[ \left( 5-\frac{58M}{R}+\frac{75Q^{2}}{R^{2}}\right) \right. \nonumber \\&-\,\left. \left( \frac{Q^2}{R^2}-\frac{Q^4}{R^4}\right) \left( \frac{M}{R}-\frac{Q^2}{R^2}\right) ^{-1}\right] +\frac{a^{5}}{R^{5}}\sqrt{\frac{M}{R}-\frac{Q^{2}}{R^{2}}}\left[ 14-\frac{9Q^{2}}{R^{2}}\left( \frac{M}{R}-\frac{Q^{2}}{R^{2}}\right) ^{-1}\right. \nonumber \\&+\,\left. \frac{Q^{4}}{R^{4}}\left( \frac{M}{R}-\frac{Q^{2}}{R^{2}}\right) ^{-2}\right] \Bigg \rbrace . \end{aligned}$$
(62)

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Heydari-Fard, M., Heydari-Fard, M. & Sepangi, H.R. Higher-order geodesic deviations and orbital precession in a Kerr–Newman space–time. Gen Relativ Gravit 51, 77 (2019). https://doi.org/10.1007/s10714-019-2557-7

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