^{1}

^{2}

^{1}

^{1}

^{2}

^{3}.

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.

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 [

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 [

The orbital motions of neutral test particles via the higher-order geodesic deviation equations for Schwarzschild and Kerr metrics are studied in [

The structure of the paper is as follows. In Section

As is mentioned above, the higher-order geodesic deviation equations for charged particles have been derived in [

Deviation of two nearby geodesics in a gravitational field. Lines (a) and (b) represent the central geodesic

The successive approximations to the exact geodesic (b) have been shown in Figure

In the next section, we are going to obtain the components of

The Reissner-Nordstrom metric is a static exact solution of the Einstein-Maxwell equations which describes the space-time around a spherically nonrotating charged source with mass

Finally, from (

Now let us calculate the first-order geodesic deviation for the components

When the source is neutral and for the small values of

In this section, by using the first-order geodesic deviation equation and inserting (

where the constants

Here we have not used any approximation in

In Appendix

Finally, successive approximation brings us to trajectory by substituting

In the Schwarzschild limit, we have an elliptical orbit with [

In the previous section, we have calculated the trajectory of charged particles up to second-order. To find a more accurate trajectory, we need to obtain the higher-order terms of expansion (

Finally, we note that the electric charge of any celestial body is practically close to zero anyway. Therefore, it is worth investigating the geodesic deviation and higher-order geodesic deviations in a more realistic background such as the Schwarzschild metric in a strong magnetic dipole field or magnetized black holes [

Many of significant successes in general relativity are obtained by approximation methods. One of the most important approximation scheme in general relativity is the post-Newtonian approximation, an expansion with a small parameter which is the ratio of the velocity of matter to the speed of light. A novel approximation method was also proposed by Kerner et al. which is based on the world-line deviations [

The calculation of the perihelion advance by means of the higher-order geodesic deviation method for neutral particles in different gravitational fields such as Schwarzschild and Kerr metric was first studied in several papers [

We first started with an orbital motion which is close to a circular orbit with constant angular velocity which is considered as zeroth-approximation (unperturbed circular orbital motion) with the orbital frequency

Finally, we have obtained the new trajectory by adding the higher-order geodesic deviations (nonlinear effects) to the circular one (

For solving the third-order geodesic deviation equation, we should invoke to Poincare’s method. For this purpose, it is better to write the third-order geodesic deviation as

The second-order geodesic deviation vector

The data findings of this study are available within the article and could be open access.

The authors declare that they have no conflicts of interest.