Analytical Solution Of The Test Particle's Motion In The Post-newtonian Approximation

The bodies' motion is a classical problem in astronomy and physics,and it is also the basis to calculate the gravitational-waves radiation.In this work,we mainly investigate the analytical solutions of the test particle's motion in the post-Newtonian approximation.First,the first-order post-Newtonian(1PN)solutions for the quasi-Keplerian motion under the generally parameterized force are derived in the Wagoner-Will-Epstein-Haugan representation and the Brumberg-Damour-Deruelle representation respectively by two different methods.The achieved results can be applied to not only the motion of a test particle in various gravity theories(including general relativity),but also to the relative motion of a binary system in a broad spectrum of gravitation theories.Then,we further derive the 2PN solution for the quasi-Keplerian motion in the Schwarzschild spacetime under the Wagoner-Will-Epstein-Haugan representation.Second,based on the Reissner-Nordstr?m(RN)spacetime metric in harmonic coordinates,we derive the analytical post-Newtonian solutions for the quasi-Keplerian motion of the test particle in the RN spacetime,including the lowest order effect of black hole charge.The solutions are formulated in the Wagoner-Will-Epstein-Haugan representation and the Brumberg-Damour-Deruelle representation,respectively.In addition,the relations between the Keplerian parameters in the Wagoner-Will-Epstein-Haugan representation and post-Keplerian parameters in the Brumberg-Damour-Deruelle representation are given,as well as their relations to the orbital energy and angular momentum are also provided.Third,we study the 2PN solution for the equatorial motion of a test particle in the Kerr spacetime and the analytical solution of the circular orbit frequency.We obtained the effect of spin-induced quadrupole term of Kerr black hole on the equatorial motion of test particle,which can reach the order of 2PN.Fourth,at present,the analytical PN solutions for the motion of the test particle involves the 3PN effect of the gravitational source mass and the 1.5PN effect of the spin.In the thesis,we also derive the second-and-half post-Newtonian solutions for the generally inclined motion in Kerr spacetime when the spin-induced quadrupole of the black hole is negligible.The analytical solutions include the 2.5PN effect of next-to-leading spin-orbit coupling.Starting from the metric of Kerr black hole in harmonic coordinates,according to Taylor expansion,we keep the metric to the order of 2.5PN with the spin-induced-quadrupole term being dropped.Based on the 2.5PN Kerr metric,we calculate the Lagrangian of the test particle's motion,and further derive the conserved quantities of the orbital energy and angular momentum through the Euler-Lagrange equation.Then,with these two conserved quantity,the radial equation and angular equation of the orbital motion are derived by the iterative method and function-fitting method.Two slightly different but equivalent 2PN formulations for the quasi-Keplerian equations are presented,which can be used not only to fit the motions of the extremely-mass-ratio-inspiral systems,but also to calculate the radiated gravitational wave.It has important application in constructing the theoretical template of gravitational wave.Finally,we derive the post-Minkowskian(PM)solution of the small-deflection motion of the test particle in Kerr-Newman spacetime.Based on the 2PM metric of Kerr Newman spacetime in harmonic coordinates,we derive the trajectory and velocity of the smalldeflection motion of the test particle in the gravitational field of the classical rotating charged black hole via an iterative method.When the initial position and velocity of the test particle are given,the achieved solution can directly give the particle's position and velocity at any time.It is worth emphasizing that the test particles can be either photons or particles with mass.As a by-product,we also present the analytical expression of 2PM velocity of photon at any position in Kerr-Newman spacetime.The research in this thesis will help people to understand the laws of the bodies' motion.At the same time,the relevant results can not only be used to fit the motion of the extreme mass ratio system,but also be used to calculate the gravitational wave radiation,so it has important scientific significance and application value.