Study On The Motion Of Test Particles In Kerr Field And The Accuracy Of The Post-Newtonian Approximation

Einstein's field equations contain four Bianchi identities.For specific problems,we need additional four conditions to solve Einstein's field equations.Harmonic-coordinate condition is the most important condition for general relativity.The academician of the Soviet Academy of Sciences Fock and the Chinese academician Zhou even believe that under these conditions,the solution of the Einstein's field equation has special physical significance.This paper can be divided into two parts,which are summarized as follows.In the first part,we study the motion of the test particles in the gravitational field of Kerr black hole under the harmonic coordinate,and compare it with that under the Boyer-Lindquist coordinate,which has commonly been employed in literatures.We discuss the difference of orbits under different coordinates and the effects on the fitting of the parameters of the gravitational source.Numerical simulation results show that for the observed S0-2 / S2 and S0-102 stars near the black hole at the Galactic center,the orbital differences between these two coordinates are relatively small.But for the stars being closer to the black hole,the orbit differences will be large,and we will arrive at the different parameters ofthe black hole when we use them to fit the observed stars' orbit.The post-Newtonian?PN?approximation is an important method for dealing with the weak field problems in general relativity.Usually its accuracy is estimated by the PN-order analysis.The second part of this thesis is about the quantitative study on the accuracy of the PN approximation.We consider the motion of the test particles in Kerr space-time,and compare the 1.5PN analytical and numerical solutions of orbits as well as the orbital period and the perihelion precession to those of the numerical solution of the exact geodesic equation.The results show that the real error of the PN approximation is usually much larger than the theoretical estimations.For large eccentricity orbits,the real error may be hundreds of times the latter.For example,when the orbital perihelion point is 50 times the Schwarzschild radius and the eccentricity is 0.8,the1.5PN error from the theoretically estimation is about 10^-3,but the 1.5PNanalytical period is only about one third of the real one.At the same time,the error of the 1.5PN analytical orbital solution is also much larger than the theoretical estimation for the cases with the eccentricity being very close to 0.