# Binary System in General Relativity: Analytical Metric?

My question is quite simple, why an analytical metric can't be found for a static binary system, even for a system under the Schwarschild condition : low field, in the vacuum between the bodies so $T_{\mu\nu} = 0$ and therefore $R_{\mu\nu} = 0$ , no rotation of the bodies itself and of the twin system, without electromagnetic forces, and far from the black hole conditions (e.g. a system based on two Sun like ours).

I read the discussion about the Binary black hole solution on Stack Exchange (link : Binary Black Hole Solution of General Relativity?). I understand well that in this case, motion of the two black holes produce additional effects.

The third-body issue is not relevant in this case, so where is the problem ? Just a question of "solving an equation" (ok, ok, a lot of equations) or is it more a physical issue ?

And a subsidiary question: why the same cause doesn't occurs in the Schwarschild approach?

• The basic issue is that the field equations are nonlinear. Are you asking for an exact solution, or an approximate one? It's not hard to find an approximate one. Are you satisfied with a solution in the form of an infinite sum, or a purely numerical computation? – Ben Crowell Jan 27 '18 at 1:40
• What do you mean by "analytical" here? Do you mean closed-form? Expressible in terms of elementary functions? If it's the latter, what do you include in your list of elementary functions? – probably_someone Jan 27 '18 at 3:46
• Indeed, my question is about usually explicit analytical functions in terms of polynomial solution (but non infinite), trigonometric, exponential, etc. – Questar Jan 27 '18 at 10:26