Computation of the General Relativistic Perihelion Precession and of Light Deflection via the Laplace-Adomian Decomposition Method

Man Kwong Mak,Tiberiu Harko,Chun Sing Leung

doi:10.1155/2018/7093592

Abstract

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.

Highlights

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
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 A∗ was developed, by using a first-order post-Newtonian approximation to calculate the stellar orbits with a broad range of periapse distance rp
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

Summary

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. 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. As the step in our study we adopt the Schwarzschild form of the metric, and we apply the LaplaceAdomian Decomposition Method to obtain its approximate analytical power series solution for both massive and massless particles.

Particle Motion in Arbitrary Spherically Symmetric Static Spacetimes

Equation of Motions of Photons and the Deflection of

Application

Solving the Equation of Motion for Light in Schwarzschild Geometry

Discussions and Concluding Remarks