Advances in High Energy Physics

Volume 2019, Article ID 1879568, 10 pages

https://doi.org/10.1155/2019/1879568

## Perihelion Advance and Trajectory of Charged Test Particles in Reissner-Nordstrom Field via the Higher-Order Geodesic Deviations

^{1}Department of Physics, The University of Qom, P.O. Box 37185-359, Qom, Iran^{2}Department of Physics, Shahid Beheshti University, G. C., Evin, Tehran 19839, Iran

Correspondence should be addressed to Malihe Heydari-Fard; ri.ca.moq@drafiradyeh

Received 20 February 2019; Revised 4 May 2019; Accepted 20 May 2019; Published 3 June 2019

Guest Editor: Saibal Ray

Copyright © 2019 Malihe Heydari-Fard et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

By using the higher-order geodesic deviation equations for charged particles, we apply the method described by Kerner et.al. to calculate the perihelion advance and trajectory of charged test particles in the Reissner-Nordstrom space-time. The effect of charge on the perihelion advance is studied and we compared the results with those obtained earlier via the perturbation method. The advantage of this approximation method is to provide a way to calculate the perihelion advance and orbit of planets in the vicinity of massive and compact objects without considering Newtonian and post-Newtonian approximations.

#### 1. Introduction

The problem of planets motion in general relativity is the subject of many studies in which the planet has been considered as a test particle moving along its geodesic [1]. Einstein made the first calculations in this regard for the planet Mercury in the Schwarzschild space-time which resulted in the equation for the perihelion advancewhere is the gravitational constant, is the mass of the central body, is the length of semi-major axis for planet’s orbit, and is eccentricity. Derivation of perihelion advance by using this method leads to a quasielliptic integral whose calculation is very difficult, which is then evaluated after expanding the integrand in a power series of the small parameter . For the low-eccentricity trajectories of planets, one can obtain the following approximate formula for the perihelion advance:even for the case of Mercury up to second-order of eccentricity, the perihelion advance differs only by error from its actual value [2]. It should be noted again that Einstein’s method is only valid for the small values of .

In what follows, we show that one can obtain the same results (without taking the complex integrals) only by considering the successive approximations around a circular orbit in the equatorial plane as the initial geodesic with constant angular velocity, which leads to an iterative process of the solving the geodesic deviation equations of first, second, and higher-orders [3–5]. Here, instead of the parameter the eccentricity, , plays the role of the small parameter which is controlling the maximal deviation from the initial circular orbit. In this method, we have no constraint on anymore. So, one can determine the value of perihelion advance for large mass objects and write it in the higher-order of .

The orbital motions of neutral test particles via the higher-order geodesic deviation equations for Schwarzschild and Kerr metrics are studied in [2] and [4], respectively. Also, for massive charged particles in Reissner-Nordstrom metric, geodesic deviations have been extracted up to first order [6]. In this paper, by using the higher-order geodesic deviations for charged particles [7], we are going to obtain the orbital motion and trajectory of charged particles. We also expect that our calculations reduce to similar one in Schwarzschild metric [2] by elimination of charge. In fact, we generalize the novel method used in [2] for neutral particles in the Schwarzschild metric to the charged particles in the Reissner-Nordstrom metric. Recently, an analytical computation of the perihelion advance in general relativity via the Homotopy perturbation method has been proposed in [8]. Also, one can study the perihelion advance of planets in general relativity and modified theories of gravity by using different methods in [8–21].

The structure of the paper is as follows. In Section 2, by using the approximation method introduced in [7], we derive the higher-order geodesic deviation for charged particles. By using the first-order geodesic deviation equations, the orbital motion of charged particles is found in Section 3. In Section 4, we obtain the second-order geodesic deviations and derive the semi-major axis, eccentricity, and trajectory using the Taylor expansion around a central geodesic. The obtained results are discussed in Section 5.

#### 2. The Higher-Order Geodesic Deviation Method

As is mentioned above, the higher-order geodesic deviation equations for charged particles have been derived in [7] for the first time. In this section, we are going to derive the geometrical set-up used in our work. The geodesic deviation equation for charged particles is [6]where is the covariant derivative along the curve and is the separation vector between two particular neighboring geodesics (see Figure 1). Here, is the tangent vector to the geodesic, is the curvature tensor of space-time, and are charge and mass of particles (particles have the same charge-to-mass ratio, ), and is the electromagnetic force acting on the charged particles. For neutral particles, the above equation reduces to the following geodesic deviation [22, 23]:which is the well-known equation (Jacobi equation) in general relativity. We introduce the four-velocity as the time-like tangent vector to the world-line and as the deviation four-vector as well. Practically it is often convenient to work with the nontrivial covariant form. It can be obtained by replacement of the trivial expressions for the covariant derivatives, the Riemann curvature tensor, and use of the equation of motion in the left-hand side of (3) [6]The geodesic deviation can be used to compose geodesics near a given reference geodesic , by an iterative method as follows. Considering this, one can write Taylor expansion of around the central geodesic and obtain the first-order and higher-order geodesic deviations for charged particlesand our aim is to obtain an expression in terms of the deviation vector. As shown in the above equation, the second term, , is the definition of deviation vector and shows the first-order geodesic deviation. But in the third term, is not vector anymore. Therefore, we define the vector as follows:to change into the expression showing the second-order geodesic deviation. By substituting (7) into (6), one can obtain the expression in terms of the order of vector deviationIn the above expression, one can make some changes for simplification. We consider as -th order of geodesic deviation and by assuming as a small quantity, ; we rewrite (8) as follows:where is the first-order geodesic deviation and is the second-order geodesic deviation. In order to obtain the second-order geodesic deviation equation, one can apply the definition of the covariant derivative on (7)(for more details see [7] and appendix therein)Similar to the first-order geodesic deviation (5), we can write (10) in the nonmanifest covariant formAs it clears, the left-hand side of the second-order geodesic deviation equation (11) is same to the left-hand side of (5). As in the case of the second-order geodesic deviation, the higher-order geodesic deviation equations have the same left-hand side and different right-hand side. A nonmanifest covariant form of the third-order geodesic deviation equation is given in Appendix A.