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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.

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If you mean an analytical solution, the answer is clearly 'no'. – Johannes Jan 22 '11 at 17:19
There is something called the C-metric, which has a bunch of nonphysical anomalies (including a naked line singularity), but can be thought of as a spacetime containing two black holes. But I really don't think this is what you want. – Jerry Schirmer Jan 22 '11 at 22:54
Thanks everyone so far. I should point out that I was planning to question the general assumption in the repsonses that the existence of an analytical solution was "clearly no", when I discovered references to the "Double Kerr" solution. Now this is stationary and physically artificial (I think),but is still two black holes together in one analytical solution. However I accept that it is not the expected case of essentially corotating and converging BHs. It is claimed to have CTCs in some conditions. – Roy Simpson Jan 23 '11 at 20:24
up vote 5 down vote accepted

For a very recent authoritative review of the numerical approach, see Centrella et. al.
For the alternate parameterized post Newtonian approach, see Living Reviews of Relativity
and look for articles number 2007-2, 2006-4 and 2003-6.

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Thanks Jim, The NASA paper at 50 pages will take some time to read, but it will be interesting to learn why so many attempts failed due unexpected blowups (software failures) in the calculations! – Roy Simpson Jan 23 '11 at 20:42

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.

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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.

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Franz Pretorius has worked on this and developed animations.

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

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Thanks. I cannot access the paper directly but here is the central animation for anyone – Roy Simpson Jan 23 '11 at 20:28
@Roy: You can probably access the paper from a computer terminal in the library at whatever major university is near your home. If you have only small to medium sized colleges nearby, check their websites to see if they have a on-line subscription to Physics Review Letters. – dmckee Feb 1 '11 at 21:58

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.

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actually this result makes me think of a conjecture I have been forming recently that the behaviour of a two body system (approx equal mass) in General Relativity is essentially a Chaotic Problem (evolution depends precisely on initial conditions). – Roy Simpson Jan 23 '11 at 20:39
In a related manner I did some calculations related to this a couple of years ago. General relativity amplifies chaos. More to the point general relativity amplifies Lyapunov exponents. If there is a relativistic orbiting body, similar to Mercury, and another planet further out. The Newtonian case is chaotic, but with one of the planets general relativistic the chaotic dynamics is amplified. – Lawrence B. Crowell Jan 23 '11 at 23:31
@RoySimpson Equal mass binary black holes are simulated quite routinely and do not appear to be chaotic; i.e. running the same initial conditions on different processors, or slightly different initial conditions, does not give wildly different results. In the case of very different masses this is less clear. – AGML Mar 23 at 16:35

The basic problem is that the holes must radiate. The result is a completely asymmetrical spacetime that cannot be attacked analytically. There are a few ways around this.

You can use the post-Newtonian expansion. Here, GR is formulated as a series of corrections to Newtonian gravity in powers of $\frac{v}{c}$. The expansion is now known to very high order and remains integrable. It seems to give quite accurate results.

You can integrate the EFEs numerically. This is possible because, while entire binary spacetimes are very difficult to find analytically, it is still possible to find families of solutions modelling a single spacelike hypersurface of such spacetimes. In principle, since GR has a formulation as an initial value problem, it is then possible to integrate forward for as long as you want. Even if the initial slice isn't especially realistic, the no hair theorems give some comfort that after a quick relaxation period during which the unphysical deformations get released as "junk radiation", the actual inspiral becomes generic. Actually performing the simulations is very difficult for a number of reasons: the first binary black hole inspiral and merger was not successfully completed until 2006.

You can perturb around an exact black hole solution. This is the goal of the so-called "self-force" program. The idea is that the smaller black hole should deform the 'background' metric in a way analogous to the electromagnetic radiation reaction, or the QED self-energy. This turns out to be really hard to do in practice, although some progress has been made for Schwarzschild backgrounds.

You can construct a few highly unphysical solutions with multiple black holes. These typically have some kind of bizarre feature that somehow holds the holes in place. For example this paper; studies a pair of black holes held rigidly in place by an extremely pathological "cosmic string". Another possibility is to feed in gravitational radiation from infinity in such a way as to \emph{exactly balance} the radiation reaction, cancelling the inspiral. I'm not sure whether an exact solution exists for the latter case, but if the holes co-rotate you do get back at least one Killing vector (corresponding to a "helical" symmetry).

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How about stitching together two black hole solutions? You mentioned perturbing an exact black hole solution; how about solving the perturbation without evolving the black hole metric? E.g. adding a static perturbing field. – Otto Mar 23 at 3:21
@Otto Since GR is nonlinear, stitching together two black hole solutions is not trivial. You can't just superpose them, for example, like you can in EM theory. The basic problem is in coming up with a self-consistent history of the system: the geometry at a given moment depends on its entire past light cone. Similar problems obtain when adding a "static" perturbation: strictly speaking, this isn't consistent with GR. However you can do things like this in approximate limits. – AGML Mar 23 at 16:37
Thanks for your time. Exactly because GR is nonlinear I'm asking if it is possible to add a perturbation on top of it. I'm just curious since I haven't seen anyone superpose two black holes and then adding a perturbation to make first order corrections to the system $\overset{0}{g_{\mu\nu}}+\overset{1}{g_{\mu\nu}}+\epsilon h_{\mu \nu}$. I suppose that superposing two black hole solution works if they are infinitely far away right? I'm just wondering if there is some more fundamental problem with this vs the usual way to perturb black holes. – Otto Mar 24 at 1:16
@Otto Well, this is sort of how people generate initial data for binary black hole simulations in practice: you first simply add two Kerr-Schild metric together, and then optimize certain constraint equations to make things consistent/vacuum. I've never heard of someone trying this on a whole four-metric at once, though. – AGML Mar 24 at 16:52
I see, thanks AGML I know this is a bit off topic but would you happen to have a source I could use to read more about these things? – Otto Mar 27 at 10:00

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