# Two body problem

## Main Question or Discussion Point

Do you think it is (in principle) possible to write down an exact solution, using the current geometric model of space-time, describing two idealized bodies (say two black holes or two stars) of a different mass orbiting each other?

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pervect
Staff Emeritus
Do you think it is (in principle) possible to write down an exact solution, using the current geometric model of space-time, describing two idealized bodies (say two black holes or two stars) of a different mass orbiting each other?
If you add in appropriate boundary conditions (such as asymptotic flatness), I'm sure such a time-varying solution exists. Gravitational radiation will prevent a non-timevarying solution, even in an initially corating coordinate system.

Actually, there should be more than one solution - there should be a set of equivalent solutions, an "equivalence class". Example: for the simpler problem of a single body, we have equivalent Schwarzschild and isotropic solutions.

However, I don't think it's known whether or not your general solution can be "written down" using standard functions. I would suspect it probably can't but I can't make any definite statement.

Wallace
I think the best we can do at present for two bodies are numerical solutions rather than an analytic form. It's surprising really that we still can't properly solve GR for really simple situations like this!

I would guess that in principle is should be possible to do though, I can't think of a reason why a solution couldn't be derived, though I could be wrong.

However, I don't think it's known whether or not your general solution can be "written down" using standard functions.
"standard functions"? But therein lies the trick with "analytic solutions".

We can write the full set of equations and boundary conditions such that their solution is the expression you're interested in. You can then numerically calculate your solution to an arbitrary degree of precision, but you most likely cannot write down the exact solution in terms of the usual standard functions like "cos".

But so what if you could? You can't exactly write the cosine of most numbers anyway: instead you still need to numerically approximate it by estimating the solution to the system of equations that the cosine itself satisfies (and is described by). Why so much fuss over whether your solution can be conveniently expressed by relations to the movement of an ideal pendulum?

In many EM problems you can't write down the field even in terms of the trigonometric functions, but often the solution in one problem can be related clearly to the solution in another problem.. so new functions (Bezel, Airy, etc) were given standard names. By using these new definitions (that is, by relating to simpler EM problems rather than just ancient pendula etc) the more complex solutions can also be "expressed analytically".

I figure in time we'll figure out relationships between enough different metrics that useful new standard functions (defined only as the solutions of particular previously-"non-analytic" problems from GR) will be chosen, then we'll write everything else down in terms of those functions. And then (despite needing the exact same algorithms to actually estimate any numerical value), we'll call them "analytic solutions".