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.
The successive approximations to the exact geodesic (b) have been shown in Figure 1. Lines (c) and (d) represent the first-order approximation, i.e., , and the second-order approximation, i.e., , respectively.
In the next section, we are going to obtain the components of from the first-order geodesic deviation, (5), for a circular orbit of charged particles. Then by substituting them into (11), we can solve the second-order geodesic deviation equations, . Finally, by substituting and into (8), we will find the relativistic trajectory of charged particles in Reissner-Nordstrom space-time.
3. The First-Order Geodesic Deviation
3.1. Circular Orbits in Reissner-Nordstrom Metric
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 and charge (in the natural coordinate with and )where Also, the vector potential and the electromagnetic field of Maxwell’s equations are [6]By assuming that , we are going to obtain the equation of motion for test particles which have mass and charge . Now, we consider a circular orbit with a constant radius . On the other hand, we know that the angular momentum of particles which are bounded to the spherically symmetric condition is limited to the equatorial plane. For this purpose and for simplicity, we limit the space to the plane of in which the angular momentum is in the direction. By using of the Euler-Lagrange equation, one can lead to the following constants of motion:where is the angular momentum per unit mass, is the angular velocity, and is the energy per unit mass.
Finally, from (12), (15), and (16) one obtains two constraints, namely, the conservation of the absolute four-velocity and the radial acceleration. Now, due to the fact that the radius of the circular orbit is constant (), two mentioned constraints vanish at all times and this creates two relations between , , and as follows:As we expect by eliminating charge, all obtained equations reduce to the similar equations in the Schwarzschild metric. In summary, we obtain the following four-velocity vector for a circular orbit with radius in an equatorial plane:In the next subsection, we obtain the orbital motion by using the higher-order geodesic deviation method and compare the results with the perturbation method.
3.2. First-Order Geodesic Deviation around the Circular Orbits
Now let us calculate the first-order geodesic deviation for the components , , , and , by using of (5) in a matrix formwhere the matrix elements are given by As can be seen, the geodesic deviation equation of component represents a harmonic oscillator equation with the angular frequency of . So we consider as follows:which is similar to the Schwarzschild case. So in this case we can neglect this solution (), because the new plane of orbit is a new one inclined, or just a change of coordinate system [4]. Now, by eliminating the derivatives of and in the differential equation of , we obtain the following oscillating equation:with the characteristic frequencyBy considering , we choose the following solution for :Also, from the and geodesic deviation equations, the solutions for and are given bywhere the amplitudes depend on In this way, the trajectory and the law of motion are obtained bywhere the argument phase of the cosine function is taken by for perihelion and for aphelion. Now, (30) can be written asBy direct solution of the Euler-Lagrange equations, the trajectory of motion for particles is obtained in terms of centrifugal inertia [25]Obviously, (33) and (34) show that we have the same results. It means that if we bring up the eccentricity to and the semimajor axis to , the same results are extracted, but there is also a difference that the circular frequency, , is lower than the circular frequency of the unperturbed circular motion, . So, if the circular frequency decreases, the period increases. Then we obtain an expression for the periastron shift per one revolution asIt can be seen from the above equation that the charge parameter, , decreases the perihelion advance. In the perturbation method (Einstein’s method), the orbital motion for charged particles moving in the equatorial plane of the Reissner-Nordstrom source is given by [20]and comparing (35) with (36) shows that the presented method can be used in the vicinity of very massive and compact objects which is having a nonnegligible ratio of .
When the source is neutral and for the small values of , (35) reduces to the standard formula for Perihelion advance of planets [23]. If we also compare (35) to (2), it is clear that, in the first-order deviation, we hold only the terms up to . In order to obtain for the higher values of the eccentricity, we must go beyond the first-order deviation equations. Therefore in the next section, we solve the second-order geodesic deviation equations in Reissner-Nordstrom space-time.
4. The Second-Order Geodesic Deviation
In this section, by using the first-order geodesic deviation equation and inserting (25), (26), and (27) into (10) and also doing a set of hard calculations, a linear equations system for the second-order geodesic deviation vector is obtained
where the constants , , , , , and contain quantities depending on , , , , , and
Here we have not used any approximation in , but in what follows we neglect terms of higher order of the small parameters , , and . Solving the matrix equation (10) for is similar to the approach used in the previous section (for the first-order geodesic deviation vector ) which contains the terms with characteristic frequency . Here we are only interested in a particular solution because of the oscillating general solution with the angular frequency already taken into account for . The particular solution of the above equation which is containing the oscillating terms with the angular frequency , the linear terms in the proper time , and constants. To obtain the trajectory according to (9), we need to calculate . Also for , the perihelion is extracted by and the aphelion is derived by , where .
In Appendix B, we have put the particular solution of the above equation, , the second-order geodesic deviation , and the semimajor axis and eccentricity , respectively.
Finally, successive approximation brings us to trajectory by substituting to
In the Schwarzschild limit, we have an elliptical orbit with [2]Also, for the Schwarzschild case the shape of the orbit is described up to second-order of () aswhich is in agreement with equation (62) of reference [2].
5. Third-Order Geodesic Deviation and Poincaré-Lindstedt’s Method
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 (9). Using the first and second-order solutions and third-order equation (A.2) for , we havewhere are defined in (20) and the coefficients , , , , , and , () are functions of , , , and As one can see the right-hand side of (48) has a frequency that is the same as the eigenvalues of the differential matrix in the left-hand side (resonant terms). This makes a new problem, i.e., infinite solution for which is called the secular term (growing without bound). For avoiding these unbounded deviations we use the Poincaré’s method. In this method by replacing by infinite series in power of the infinitesimal parameter asthe correction frequencies can be chosen such that Poincaré’s resonances vanish, by considering a differential equation for asNow, by developing both of the sides in terms of a series of the parameter , for avoiding the secular terms, we find some algebraic relations on . In the Schwarzschild limit, we have [4]where . The resonant terms will also appear at the fifth-order approximation; by terms , etc., this problem can be solved in a similar way.
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 [26–28]. The study of them will be the subject of the future investigations.
6. Conclusion and Discussion
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 [2].
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 [2, 4]. In the present paper by using of the higher-order geodesic deviation method for charged particles [7], we applied this approximation method to charged particles in the Reissner-Nordstrom space-time.
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 . In the next step, we solved the first- and second-order deviation equations which reduced to a system of the second-order linear differential equations with constant coefficients. The solutions are harmonic oscillators with characteristic frequency. From (37), the first- and second-order corrections are oscillating terms with angular frequency and , respectively.
Finally, we have obtained the new trajectory by adding the higher-order geodesic deviations (nonlinear effects) to the circular one (44). The advantage of this approach is to get the relativistic trajectories of planets without using Newtonian and post-Newtonian approximations for arbitrary values of quantity .
Appendix
A.
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 third-order geodesic deviation equation is related to the third-order geodesic deviation vector where . We derive the third-order geodesic deviation equation asand by substituting in term of into above equation, we obtain (72) for case [2].
B.
The second-order geodesic deviation vector isAs explained in Section 2 the second-order geodesic deviation, , is given byand, also, the semimajor axis and eccentricity arein which for massive central objects we have neglected all terms of order .
Data Availability
The data findings of this study are available within the article and could be open access.
Conflicts of Interest
The authors declare that they have no conflicts of interest.