Kepler problem in general relativity

Remarks

The problem is solved. By spherical symmetry each geodesic stays in a plane, which can be taken to be $\theta = \pi/2$. The Killing vector fields $\partial_t$ and $\partial_\varphi$ give two conserved quantities, the energy and the angular momentum per unit mass. With $u = 1/r$, $h = r^2 \, d\varphi/d\tau$ and $\tau$ the proper time, a timelike geodesic obeys the orbit equation $$\frac{d^2 u}{d\varphi^2} + u = \frac{GM}{h^2} + \frac{3GM}{c^2} u^2.$$ The last term is absent in the Newtonian problem. The orbit equation can be integrated in terms of elliptic functions. Unlike Newtonian bound orbits, bound relativistic orbits are in general not closed ellipses: the perihelion advances from one orbit to the next. There is no stable circular orbit with $r < 3 r_s$, and circular photon orbits exist only at $r = 3r_s/2$. Einstein treated planetary motion in 1915 with an approximate solution, and Schwarzschild found the exact metric in 1916.