Exact Results for the Kepler Problem in General Relativity
Abstract
Description
Exact results are derived, specifically the perihelion shift and the Kepler orbit, for a bound test particle in the Schwarzschild metric with cosmological constant $Λ=0$. A series expansion, of $Δϕ= 2(2(1-2M/p(3-e))^{-1/2} K((4eM/p)/(1-2M/p(3-e)))-π)$, the exact perihelion shift, admits the standard approximation $Δϕ=6Mπ/p$ as the leading order term. In a similar fashion, a series expansion of the exact Kepler orbit, represented by a Jacobi elliptic function, gives $u(ϕ)=(1+e\cosϕ)/p$ to first order. The results are valid for $M/p<1/(2(3+e))$ or $r_s<p/(3+e)$.
7 pages, 0 figures
7 pages, 0 figures