An analytic solution for weak-field Schwarzschild geodesics

Abstract

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 $t,r,\phi$ 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.