Binary Black Hole Solution of General Relativity?

Question

This is rather a technical question for experts in General Relativity. An accessible link would be an accepable answer, although any additional discussion is welcome.

GR has well known solutions relating to single Black Holes: Schwarzchild, Rotating & Rotating with Charge. These solutions demonstrate some non-trivial GR behaviour. However do there exist any (corresponding) binary star/black hole solutions? Because of the non-linearity of GR such a solution could well demonstrate additional properties to a "solution" that simply consisted of a pair of "distant" Schwarzchild solutions glued together.

Answer 1

For a very recent authoritative review of the numerical approach, see Centrella et. al. http://arxiv.org/abs/1010.5260
For the alternate parameterized post Newtonian approach, see Living Reviews of Relativity http://relativity.livingreviews.org/Articles/subject.html
and look for articles number 2007-2, 2006-4 and 2003-6.

Answer 2

There are no exact solutions, only approximations and numerical solutions.

Don't forget that orbiting black holes will radiate gravitational waves so any solution would have to include those and the corresponding decay of the orbit until the black holes coalesce.

Answer 3

Franz Pretorius has worked on this and developed animations.

http://prl.aps.org/abstract/PRL/v95/i12/e121101

The field is numerical relativity. Matthew Choptuik also, I believe, has done work on this.

Answer 4

According to general relativity, a pair of massive bodies that orbit each other emits gravitational waves - for analogous reasons to the reasons why accelerating charges in electrodynamics emit electromagnetic waves.

So there can't be any static solutions resembling binary stars or binary black holes. The solutions have to be non-static and a complicated system of two orbiting bodies that emits gravitational waves - and eventually collapses into one object - clearly can't be solved analytically.

These things are usually discussed numerically, see other answers. In particular, the 1993 Nobel physics prize was given for an observation of a pulsar whose frequency changes in time exactly in the right way to be explained as the loss of energy caused by the emission of gravitational waves as predicted and calculated by general relativity.

Answer 5

A way of physically thinking about this is that a two body problem in general relativity does not generally have closed orbits. If one of the bodies is very large and the other a small satellite the problem is integrable. The periapsis (perihelion) advance of the small satellite is repeated with each orbit, which makes the problem integrable. If the two bodies are of comparable mass the orbits of the two are perturbed in a manner which emulates a third orbiting body in Newtonian mechanics. The three body problem is not integrable in general. The periapsis advance of either mass adjusts to the changing position of the other mass, which “emulates” the presence of a third body. Curiously, before Einstein people thought there was another planet near the sun which perturbed Mercury, what they called planet Vulcan. If the two bodies are close enough and they are in an orbit with a quadrupole moment (Keplerian orbit, ellipses etc) there is the emission of gravity waves. Gravity waves are mass-energy and contribute to the gravity field. So a two body system in effect generates what might be thought of as a third body, or N-bodies.