An analytic solution for weak-field Schwarzschild geodesics

Understand

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.