Exact Results for the Kepler Problem in General Relativity

Abstract

Exact results are derived, specifically the perihelion shift and the Kepler orbit, for a bound test particle in the Schwarzschild metric with cosmological constant $\Lambda=0$. A series expansion, of $\Delta\phi = 2(2(1-2M/p(3-e))^{-1/2} K((4eM/p)/(1-2M/p(3-e)))-\pi)$, the exact perihelion shift, admits the standard approximation $\Delta\phi=6M\pi/p$ as the leading order term. In a similar fashion, a series expansion of the exact Kepler orbit, represented by a Jacobi elliptic function, gives $u(\phi)=(1+e\cos\phi)/p$ to first order. The results are valid for $M/p<1/(2(3+e))$ or $r_s<p/(3+e)$.