gravity - General Relativity 2-Body Closed Form

Is there a closed form solution in general relativity to the 2-body orbit problem?

Answer

There is no general solution for the two body problem in general relativity.

But!

There are a few solutions for specific two body problems. These include the Curzon-Chazy metric (Two particles on a cylindrically symmetric axis)

$ds^2 = e^{-2\psi} dt^2 - e^{2(\psi - \gamma)} (d\rho^2 + dz^2) - e^{2\psi} \rho^2 d\phi^2$

and the Israel-Khan metric ("two black holes held in equilibrium by a strut"). Also of interest and related to the Israel-Khan metric :

"In a 1922 paper, Rudolf Bach and Hermann Weyl [3] discussed the superposition of two exterior Schwarzschild solutions in Weyl coordinates as a characteristic example for an equilibrium configuration consisting of two “sphere-like” bodies at rest. Bach noted that this static solution develops a singularity on the portion of the symmetry axis between the two bodies, which violates the elementary flatness on this interval. "

The Gott spacetime is constituted of two cosmic strings also, if that helps.

From "Exact Solutions of the Einstein Field Equations", by the way :

"In Einstein's theory, a two-body system in static equilibrium is impossible without such singularities - a very satisfactory feature of this non-linear theory."

Edit : I know those are all static two body solutions, and not orbit ones, but here lies the problem : orbit solutions are horrible. With a static 2 body soluion, you get to keep rotational symmetry and time symmetry. Once you go full orbit, you will basically lose all symmetries, and you will also get gravitational waves. That is when things become extremely non-linear, and hence hard to solve.