【作者】 丁亚明；

【导师】 邓剑波；

【作者基本信息】 兰州大学 ， 理论物理， 2009， 硕士

【摘要】 众所周知,爱因斯坦于1915年创立的广义相对论是研究引力和时空结构的理论。本文从爱因斯坦场方程出发,运用作用角变量方法研究行星进动,并对广义Mφller变换进行了讨论.我们首先介绍了测地线方程和黎曼曲率张量,这是我们求解爱因斯坦场方程的数学基础.其次,我们介绍了经典力学中的正则变换、哈密顿-雅可比理论及作用角变量方法,为后面研究行星进动问题提供了方法。在第三章中,我们介绍了广义相对论的基本原理和基本任务,并介绍了广义相对论的一个新的研究方向——数值广义相对论。我们具体讨论了运用作用角变量方法研究行星分别在近圆和椭圆轨道下的近日点进动值,并给出其进动值的明确表述.在回顾了爱因斯坦场方程的Schwarzschild真空解之后,我们介绍了质点在Schwarzschild场中的运动方程.从这组完备的质点动力学微分方程出发,我们探讨了质点运动相应的作用变量的一般形式。并且,分别讨论了在近圆轨道和椭圆轨道两种情况下,质点运动的作用变量的共轭变量——角变量的性质,从而研究出,经过一个周期后,质点(行星)的轨道进动值。我们还研究了广义Mφller变换——由惯性参考系到任意变速参考系的一种变换。重点讨论了广义Mφller变换的微分形式.我们首先考虑一个沿某一方向变速运动的参考系中的引力场。从爱因斯坦场方程出发,求出该引力场的时空度规。通过分析计算,我们寻求到通过精确对钟的方法来确定某一参考系是否是惯性系,并研究出广义Mφller变换的微分形式。更多还原

【Abstract】 It is well known that, general relativity is the theory of space, time, and gravitation formulated by Einstein in 1915. In this dissertation, based on the Einstein’s field equation, the action-angle variables method was used to study the precession of planets, and the general M(?)ller transformation was discussed.Firstly, we gave an introduction of Riemann curvature tensor and the geodesic equation, which are very important for comprehending general relativity and solving Einstein’s field equation. Secondly, canonical transformations, Hamilton-Jacobi theory and action-angle variables were investigated, which made a foundation to study the precession of planets. Thirdly, the fundamental principles and the basic mission of general relativity were introduced, and a brief introduction to a rising direction for general relativity - Numerical Relativity was presented.We concretely gave a detailed study of the perihelion precession of planets, in nearly-circular orbital and elliptical orbital respectively, with action-angle variables method. And, the explicit expressions of orbital precession were presented. After reviewed the Schwarzschild vacuum solution of Einstein’s field equation, the motion equations of a particle in this field were investigated. Based on this set of self-contained equations, we presented the general integral form of action variables and discussed the properties of the angle variables in cases of nearly-circular orbital and elliptical orbital respectively. Then, the orbital precessions of the particle (a planet) in a cycle were come out by some analysis.On the other hand, we also gave a detailed discussion on the general M(?)ller transformation, which is a transformation from inertial reference frame to arbitrary time-varied reference system. And its differential form was em- phatically discussed. We considered a gravitational field in a reference system moving variably along a certain direction, the metric of which was derived later based on Einstein’s equation. After the analysis and calculation, it came out that an inertial reference system can be determined by precisely checking the distantly separated watches, and the differential form of general M(?)ller transformation was presented.更多还原