We study the equations of motion of the massive and massless particles in the Schwarzschild geometry of general relativity by using the Laplace-Adomian Decomposition Method, which proved to be extremely successful in obtaining series solutions to a wide range of strongly nonlinear differential and integral equations. After introducing a general formalism for the derivation of the equations of motion in arbitrary spherically symmetric static geometries and of the general mathematical formalism of the Laplace-Adomian Decomposition Method, we obtain the series solution of the geodesics equation in the Schwarzschild geometry. The truncated series solution, containing only five terms, can reproduce the exact numerical solution with a high precision. In the first order of approximation we reobtain the standard expression for the perihelion precession. We study in detail the bending angle of light by compact objects in several orders of approximation. The extension of this approach to more general geometries than the Schwarzschild one is also briefly discussed.

1. Introduction

General relativity is a very successful theory of the gravitational field, whose predictions are in excellent agreement with a large number of astronomical observations and experiments performed at the scale of the Solar System. In particular, three fundamental tests of general relativity, the perihelion precession of planet Mercury [1, 2], the bending of light by the Sun [3, 4], and the radar echo delay experiment [5, 6] have all fully confirmed, within the range of observational/experimental errors, the predictions of Einstein’s theory of gravity. But the importance of these effects goes far beyond the limits of the Solar System. A fast full general relativistic method to simultaneously constrain the mass of massive black holes, their spin, and the spin direction by considering both the motion of a star and the propagation of photons from the star to a distant observer was developed in [7]. The spin-induced effects on the projected trajectory and redshift curve of a star depend on both the value and the direction of the spin. The maximum effects over a full orbit can differ by a factor up to more than one order of magnitude for cases with significantly different spin directions. In [8] it was shown that the spin of the massive black hole at the Galactic Center can be constrained with 1 error or even greater than 0.02 by monitoring the orbital motion of a star with semi major axis AU and eccentricity over a period shorter than a decade through future facilities. An improvement in astrometric precision would be more effective at improving the quality of constraining the spin than an improvement in velocity precision. Short-period stars orbiting around the supermassive black hole in our Galactic Center can successfully be used to probe the gravitational theory in a strong regime [9]. By using 19 years of observations of the two best measured short-period stars orbiting our Galactic Center constraints on a hypothetical fifth force that arises in some extended theories of gravity or in some models of dark matter and dark energy were obtained in [9]. No deviations from general relativity were found, and the fifth force strength was restricted to an upper 95% confidence limit of at a length scale of astronomical units. Moreover, 95% confidence upper limit on a linear drift of the argument of periastron of the short-period star S0-2 was obtained, a result that opens the possibility of testing gravitational theories using orbital dynamic in the strong gravitational regime of a supermassive black hole. The S-star cluster in the Galactic Center allows the study of physics close to a supermassive black hole, including distinctive dynamical tests of general relativity [10], where a new and practical method for the investigation of the relativistic orbits of stars in the gravitational field near Sgr was developed, by using a first-order post-Newtonian approximation to calculate the stellar orbits with a broad range of periapse distance . For in depth discussions of the experimental and Solar System tests of general relativity, see [11] and [12], respectively.

Due to its importance in many applications, the study of the motion of massive or massless particles in different geometries obtained as solutions of Einstein’s gravitational field equations and of their extensions is a fundamental field of general relativity. The first exact solution of the vacuum field equations was the Schwarzschild solution [13], which can be used efficiently to explain all astronomical observations at the scale of the Solar System. The exact equation of motion in Schwarzschild geometry is highly nonlinear, and therefore to obtain the observable physical parameters approximate methods must be used. The first-order approximation of the equation of motion already gives the correct approximation of the perihelion precession of Mercury and of the deflection of light by the Sun [11]. However, due to the importance of the problem many mathematical techniques for the study of the astrometric properties of the planetary motions and of the light have been developed. A standard approach is based on the solution of the Hamilton-Jacobi equation [13]where are the components of the metric tensor and is the mass of the particle. By representing as , where and are the constants of the energy and angular momentum, one can obtain the full solution of the equation of motion in Schwarzschild geometry in an integral form as [13]

The parameters of the orbits can be obtained by solving the integrals by using some approximate methods. The geodesic equations obtained from the Schwarzschild gravitational metric in the presence of a cosmological constant were solved exactly and expressed in a closed form in [14] aswhere , is an arbitrary integration constant, is the Weierstrass function that gives the inversion of the elliptic integral , by the Weierstrass function, . In this approach the exact expression of the perihelion precession is given by , where and is a root of the cubic equation in the integral. The perihelion precession and deflection of light have been investigated in the four-dimensional general spherically symmetric spacetime in [15], where the master equation has also been obtained. As an application of this master equation, the Reissner-Nordstorm solution and Clifton-Barrow solution in gravity have been investigated. The homotopy perturbation method, which was introduced in [16], was applied for calculating the perihelion precession angle of planetary orbits in general relativity in [17, 18]. The basic ideas behind the homotopy perturbation method are as follows [16]. We start from the nonlinear differential equation , where is a general differential operator and is a known analytic function, with the boundary conditions , where is a boundary operator and is the boundary of the domain . We assume that the operator can be divided into two parts and , and we reformulate our initial equation as . Then the homotopy is constructed in the following way: , where and is an imbedding parameter, and is an initial approximation of the equation. Since , it can be considered as a small parameter, and one can assume that the solution of the equation can be expressed as a power series in as . When , then this series becomes the approximate solution of the equation, that is, . This series is generally convergent.

The study and the applications of Adomian’s Decomposition Method (ADM) [1922], which allows investigating the solutions of many kinds of ordinary, partial, stochastic differential, and integral equations that describe numerous physical and/or mathematical problems, have attracted a lot of attention in recent years. Historically, the ADM was first proposed and applied in the 1980s [2325]. An essential advantage of the ADM is that with its use one can obtain analytical approximations to the solutions of a rather wide class of nonlinear (and stochastic) differential and integral equations, without the need of linearization, perturbation, closure approximations, or discretization methods. Usually the application of these methods could lead to the necessity of intensive numerical computation. Moreover, to make solvable and to obtain closed-form analytical solutions of a nonlinear problem imply the necessity of introducing some simplifying and restrictive assumptions.

It is important to mention that ADM can generate the solution of a given equation in the form of a power series. The terms of the series are obtained by recursive relations using the Adomian polynomials. Another important advantage of the ADM is that usually the series solution of the differential equation converges fast, and therefore the use of this method saves a lot of computational time. Moreover, in the ADM there is no need to linearize or discretize the differential equation. Reviews of ADM and its applications in applied mathematics and physics can be found in [19, 20], respectively. Many studies have been devoted to the modification and improvement of the ADM in an attempt to increase its accuracy and/or to extend the applicability of the initial method [21, 22, 2640]. An important improvement of the ADM is represented by the Laplace-Adomian Decomposition Method [41], in which the Adomian Decomposition Method is applied not to initial equation, but to the Laplace transformed one.

Even that the ADM has been extensively used in the study of many problems in different fields of physics and engineering, it has been applied very little in astronomy, astrophysics, or general relativity, the only exception from this “rule” known to authors being the papers [42, 43]. It is the purpose of the present paper to investigate the equations describing the perihelion precession and light bending in general relativity for static gravitational fields by using the Adomian Decomposition Method, representing a very powerful mathematical method for the investigation of the solutions of nonlinear differential equations. For the geometry outside a compact, stellar type object (the Sun) we adopt some specific static and spherically symmetric vacuum solutions of general relativity. In particular the power series solution of the equation describing the motion of massive and massless particles in Schwarzschild geometry is investigated in detail. As a first step in our analysis we derive the equations of motion for particles in arbitrary spherically symmetric spacetimes, and we develop a general formalism for obtaining the equations of motion that can be used for any given metric. As the next step in our study we adopt the Schwarzschild form of the metric, and we apply the Laplace-Adomian Decomposition Method to obtain its approximate analytical power series solution for both massive and massless particles. We compare our solutions with the exact numerical solutions of the equations of motion, and it turns out that by truncating our series to five terms only we obtain a very good description of the solution of the equation of motion. Moreover, in the first approximation we can reobtain easily the standard expressions of the perihelion precession and the bending angle of light.

The present paper is organized as follows. We derive the equations of motion of massive and massless particles in arbitrary static spherically symmetric spacetimes in Section 2. The application of the Laplace-Adomian Decomposition Method to the case of second-order nonlinear differential equations is presented in Section 3. The power series solution of the equation of motion of massive particles by using the Laplace-Adomian Decomposition Method is obtained in Section 4, where the comparison with the exact numerical solution is also performed. The motion of photons in Schwarzschild geometry is investigated in Section 5. Finally, in Section 6, we discuss and conclude our results.

2. Particle Motion in Arbitrary Spherically Symmetric Static Spacetimes

In the following we will restrict our analysis to the case of static and spherically symmetric metrics, given bywhere we have denoted and and are the standard coordinates on the three-sphere. The time variable takes real values only, while the radial coordinate ranges over a finite open interval , so that . Moreover, we also require that the functions and are strictly positive and that on the interval they are (at least piecewise) differentiable. This form of the metric is relevant for the study of the dynamics of particles (both massive and massless) in the Solar System.

Important observational evidence for the correctness of the theory of general relativity is provided, at the level of the Solar System, by three fundamental tests, which also allow the testing of its extensions and generalization, as well as of alternative theories of gravitation. These three essential tests are the perihelion precession of Mercury, the deflection of photons by the Sun, and the radar echo delay observations. These three effects have been successfully used to test the Schwarzschild solution of general relativity, as well as other predictions of the theory. However, it is also important to study these physical phenomena in arbitrary static spherically symmetric spacetimes for any given metric. In the present Section, we develop a formalism that can be used for obtaining the equations of motion and compute the perihelion precession and light bending angle in any static spherically symmetric metric. This formalism was first introduced to study the Solar System tests for some modified gravity vacuum solutions in [4447].

2.1. The Equation of Motion of Massive Test Particles

The geodesic equations of motion of a massive test particle in the gravitational field of the metric given by (5) can be derived with the use of the variational principlewhere by a dot we have denoted . It can be easily checked that the orbit is planar, and therefore without any loss of generality we can take . Hence is the only the angular coordinate in this problem. Since and do not appear explicitly in (6), their conjugate momenta give two constants of motion, denoted by and , so thatThe constant gives the energy of the particle, while the constant is related to its angular momentum.

From the line element (5) we obtain the following equation of motion for :

Substituting and from (7) gives the following relation:

We introduce now a new variable , defined as , as well as the transformation . Then (9) takes the form

We represent formally as , thus obtaining

We take the derivative of the above equation with respect to , which giveswhere

Equation (12) gives the equation of motion of a particle in an arbitrary spherically symmetric geometry.

2.1.1. The Precession of the Perihelion

The root of the equation gives a circular orbit with . Any deviation from it can be obtained from the substitution of into (12), which gives the equationIn the first order of approximation and hence

Therefore, to first order in , the trajectory of the massive particle can be obtained aswhere and are two arbitrary constants of integration. The angles for which is minimum are the angles of the perihelia of the orbit. Therefore they are determined from the condition that or is maximum. Hence, from one perihelion to the next the orbital angle varies by a quantity , given by

The parameter introduced in the previous equation is called the perihelion advance. From a physical point of view it represents the rate of advance of the perihelion after one rotation. As the test particle advances through radians in its orbit, its perihelion precesses by radians. From (17), can be expressed asor for small , as

For a complete rotation of the planet we obtain . Hence the advance of the perihelion isTo be able to obtain effective estimations of the perihelion precession we must know the expression of the angular momentum of the particle as a function of the geometric parameters of the orbit. We will obtain now the expression of in the Newtonian limit [4447].

Let us assume that the planet moves on a Keplerian ellipse, with semiaxes and , respectively, where and by we have denoted the eccentricity of the orbit. The ellipse has a surface area . The oriented surface area of the ellipse is , and consequently the areolar velocity of the planet is given by , where is the period of the planetary motion. On the other hand can be obtained from Kepler’s third law as [48]. In the Newtonian limit of small velocities , and the conservation equation of the angular momentum reduces to . Hence we obtain first , and hence

2.1.2. The Equation of Motion of Massive Particles in Schwarzschild Geometry

As a first astronomical application of the formalism introduced in the previous section we obtain the precession of the perihelion of a planet in the Schwarzschild geometry, withThen we immediately obtain . On the other hand since for this geometry , we easily findandrespectively. Therefore the equation of motion of a massive test particle in Schwarzschild geometry is given by

The radius of the circular orbit is found as the solution of the algebraic equationwith the only physically acceptable solution given by

Thereforewhich is the standard general relativistic result [13].

2.2. Equation of Motions of Photons and the Deflection of Light

In a gravitational field a photon follows a null geodesic, given by . In this case the affine parameter along the trajectory of the photon can be taken as an arbitrary quantity . In the following we denote again by a dot the derivatives with respect to it. Similar to the case of the motion of massive particles we have two constants of motion, the energy and the angular momentum , which can be obtained from (7).

The equation of motion of the photon is given byBy using the constants of motion the above equation can be transformed into

We change now the independent variable to . With the use of the conservation equations we eliminate the derivative with respect to the affine parameter, thus obtainingWe take the derivative of (31) with respect to , and thus we find the basic equation of the photon in an arbitrary static spherically symmetric geometry, as given bywhere we have denoted

In the particular case of the Schwarzschild geometry we have and , giving and , respectively. Hence the equation of motion of photons in the Schwarzschild metric is obtained as

2.2.1. The Deflection Angle of Light

In the lowest order of approximation we can neglect the term on the right-hand side of (32). Then the solution is given by a straight line,whereby we have denoted the distance of the closest approach to the central massive gravitating object. In the next order of approximation (35) is substituted into the right-hand side of (32). Hence the equation of the trajectory is given by a second-order linear inhomogeneous differential equation,which has a general solution given by . The photons travel towards the star from infinity at the asymptotic angle and are deflected to infinity at the asymptotic angle . The angle can be computed by solving the algebraic equation . For the total deflection angle of the photon beam we find .

(a) The Deflection of Light in Schwarzschild Geometry. We consider now the case of the Schwarzschild geometry. In the lowest order of approximation from (35) and (36) we obtain for the photon trajectory the second-order linear differential equationhaving the general solution given by

By substituting , into (38) we easily findwhere we have used the simple trigonometric relations , , and the approximations and , respectively. The total deflection angle of light in the Schwarzschild geometry is thus , a well-known result in general relativity [13].

3. The Laplace-Adomian Method for Nonlinear Second-Order Ordinary Differential Equations

3.1. The General Formalism

Let us consider a nonlinear differential equation of the formwhere and are constants and is an arbitrary nonlinear function of dependent variable . Equation (40) must be integrated with the initial conditions and , respectively.

In the Laplace-Adomian method we first apply the Laplace transformation operator to (40), thus obtaining

With the use of the properties of the Laplace transformation we easily find

With the use of the initial conditions for our problem we obtain

As a next step in our analysis we assume that the solution can be represented in the form of an infinite series,where the terms are computed recursively. As for the nonlinear operator , it is decomposed aswhere ’s are the so-called Adomian polynomials, defined generally as [20]

The first five Adomian polynomials can be obtained in the following form:

Substituting (44) and (45) into (43) we obtain

Matching both sides of (52) yields the following iterative algorithm for the power series solution of (40),

By applying the inverse Laplace transformation to (53), we obtain the value of . Substituting into (47) to find the first Adomian polynomial , then we substitute into (54), and we evaluate the Laplace transform of the quantities on the right-hand side of it. The application of the inverse Laplace transformation yields then the value of . The other terms , can be computed recursively in a similar step by step approach.

3.2. The Particular Case

Let us consider a nonlinear differential equation of the formwhere , , and , , are constants. Equation (58) must be integrated with the initial conditions and , respectively. By applying the Laplace-Adomian method we first take the Laplace transform to (58), thus obtaining

Now we easily findthus obtaining

From (61) we obtain in the form

We assume a power series solution for as , and we write the nonlinear terms aswhere are the Adomian polynomial corresponding to . Then we obtain

We rewrite (64) in the form

Equation (65) can be written as the recursive relations

For the function a few Adomian polynomials are

For we obtain the first-order approximation to the solution asFor we find givesFinally, for we obtainHence we have obtained the truncated power series solution of (58) as given by

4. The Solution of the Equation of the Motion of the Massive Test Particles in Schwarzschild Geometry by the Laplace-Adomian Decomposition Method

In the following we will use a system of units with . Then (25), describing the motion of a massive test particle in the Schwarzschild geometry, takes the formIn order to simplify the mathematical formalism we rescale the function so thatThus (78) becomeswhere we have denoted . Equation (80) must be solved with the initial conditions and , respectively.

4.1. Power Series Solution of the Equation of Motion

Assume that the solution of (80) can be obtained in power series form,

Now taking Laplace transform to (80) yields

Hence we obtain

We write down a few Adomian polynomials for ,Substituting (81) and into (84) gives the relationor equivalently,

Next we rewrite (90) in the recursive forms

With the help of the explicit expressions of the Adomian polynomials, we obtain

The power series solution of the equation of motion of the massive test particles in Schwarzschild geometry is thus given by

4.2. Comparison with the Exact Numerical Solution

In order to estimate the results obtained by the Laplace-Adomian Decomposition Method, in Figure 1 we present the comparison between the exact numerical solution of (80) and the analytical, power series representation, given by (105), for , , and , respectively.

The absolute difference , defined asbetween the numerical solution and the truncated Laplace-Adomian power series solution, is represented in Figure 2.

4.3. Application: The Motion of Planet Mercury

In order to integrate the equation of motion of the massive test particles in Schwarzschild geometry we need to know the initial conditions of the motion. For the initial value of and we adopt the valueswhere is the eccentricity of the orbit. The angular momentum can be expressed in terms of the geometric parameters of the orbit by using (21). Thus in physical units we obtain

Therefore for this choice of parameters and initial conditions the successive terms in the power series solution of the equation of motion can be obtained by the Laplace-Adomian Decomposition Method as follows: