广义相对论二体问题比内方程的精确解探讨  

Discussion onexact solution of Binet equation for two-body problem of general relativity

作　　者：程有度[1] CHENG You-du(Kunming Institute of Physics,Kunming,Yunnan 650223,China)

机构地区：[1]昆明物理研究所红外探测器中心,云南昆明650223

出　　处：《大学物理》2020年第11期31-35,共5页College Physics

摘　　要：根据广义相对论二体问题比内微分方程的函数积分形式的精确解,得到3类精确轨迹曲线,其中两类可以积出解析解.爱因斯坦就是通过对这个函数积分形式的精确解直接进行近似简化计算,得到了与迭代近似解方法高度一致的水星进动值的高精度计算结果,这相互印证了不同解法的计算精度.通过计算广义相对论比内方程的极点,得到行星轨道远日点、近日点的精确计算公式,以及圆形轨道的精确半径,与牛顿力学结果相差约5 km,差值与行星轨道半径关系不大,主要由太阳质量决定.上述差值也是近似解极径的最大绝对误差.运用高精度数学计算软件计算数值积分,可以得到广义相对论二体问题的精确数值轨迹.According to the exact solution of Binet differential equation for general relativity two-body problem in function integral form,three kinds of precise trajectory curves are obtained,two of which can be integrated to get analytic solutions.Einstein directly approximated and simplified this exact solution which is in function integral form,and obtained the high-precision calculation results of mercury precession highly consistent with the iterative approximate solution,which mutually verified the calculation accuracy of different solutions.By calculating the poles of Binet equation of general relativity,the exact formulas for calculating the aphelion and perihelion of the orbit of the planet and the exact radius of the circular orbit are obtained.There is a difference of about 5 km with the results of Newtonian mechanics,which is mainly determined by the mass of the sun,and has little to do with planetary orbit radius.The above difference is also the maximum absolute error of the approximate solution for polar radius.The precise numerical trajectory of the general relativistic two-body problem can be obtained by using the high-precision mathematical calculating software to calculate the numerical integral.

关 键 词：广义相对论 二体问题 比内方程 水星进动 精确解 

分 类 号：O412[理学—理论物理]