弱场Schwarzschild测地线的解析解
An analytic solution for weak-field Schwarzschild geodesics
摘要：
众所周知，经典引力二体问题可以通过正则化转化为球谐振子问题。我们发现，正则化变换的一个修改有一个类似的结果，在广义相对论的领导秩序。在所得到的谐振子中，首阶相对论扰动形式上是一个负离心力。净离心力在三个史瓦西半径处改变符号，这有趣地模仿了完整史瓦西问题的最内层稳定圆轨道（ISCO）。将谐振子解变换回空间坐标，对于类时弱场史瓦西测地线和零弱场史瓦西测地线，都可以得到$t,r,ϕ$的解，它可以用一个可以解释为广义偏心距的变量的初等函数表示。相对论性旋进和光偏转的教科书表达式很容易恢复。我们建议如何将这种解决方案与其他扰动相结合，适合于相对论吸积或银河系中心恒星动力学等应用的数值方法。
英文摘要：
It is well known that the classical gravitational two body problem can be transformed into a spherical harmonic oscillator by regularization. We find that a modification of the regularization transformation has a similar result to leading order in general relativity. In the resulting harmonic oscillator, the leading-order relativistic perturbation is formally a negative centrifugal force. The net centrifugal force changes sign at three Schwarzschild radii, which interestingly mimics the innermost stable circular orbit (ISCO) of the full Schwarzschild problem. Transforming the harmonic-oscillator solution back to spatial coordinates yields, for both timelike and null weak-field Schwarzschild geodesics, a solution for $t,r,ϕ$ in terms of elementary functions of a variable that can be interpreted as a generalized eccentric anomaly. The textbook expressions for relativistic precession and light deflection are easily recovered. We suggest how this solution could be combined with additional perturbations into numerical methods suitable for applications such as relativistic accretion or dynamics of the Galactic-centre stars.
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