Exact Results for the Kepler Problem in General Relativity

TL;DR

This paper derives exact analytical expressions for the perihelion shift and Kepler orbit of a test particle in Schwarzschild spacetime, providing precise formulas and their approximations within general relativity.

Contribution

It presents the first exact solutions for perihelion shift and Kepler orbit in Schwarzschild spacetime, including series expansions and conditions of validity.

Findings

Exact perihelion shift formula derived

Kepler orbit expressed via Jacobi elliptic functions

Standard approximation recovered as leading order

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 $r_s<p/(3+e)$.