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?

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    $\begingroup$ 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? $\endgroup$ – Ben Crowell Jan 27 '18 at 1:40
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    $\begingroup$ 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? $\endgroup$ – probably_someone Jan 27 '18 at 3:46
  • $\begingroup$ Indeed, my question is about usually explicit analytical functions in terms of polynomial solution (but non infinite), trigonometric, exponential, etc. $\endgroup$ – Questar Jan 27 '18 at 10:26

It's both a mathematical issue (multiple nonlinear equations that relate to each other), and the underlying physical reality as to why that is so. The latter can be seen in when you consider an operational approach to solving the problem: body 1 (say a point source) creates a metric, and body 2 is moving in it. But body 2 also affects the metric, so body 1 gets affected and moves differently and so the metric it creates changes. Tathat in turn affects body 2, and so on ad infinitum. It's their effect on each other and that you cannot simplify it into an equivalent 1 body problem as you can in Netonian gravity because of the very nonlinear effects. You cannot define a center of mass like you can in Newtonian mechanics for that same reason.

So, your point that two produce additional effects is, two point sources, also an issue - as I described above. When you also consider that the two bodies may also have angular momentum it gets worse - the two body Kerr solution is also not known.

What it does not happen in Schwarzschild? You are right, there are still nonlinearities in the multiple equations for even one point source, but for a point source with no angular momentum you have a huge symmetry, spherical symmetry. Also, it is rest frame you consider it at rest, so is is also static. That reduces the set of equations to very few and makes it solvable.

The Kerr solution, for one point source but with angular momentum, took much longer to find. There's still a symmetry but it's axial symmetry rather than spherical, and instead of static and time-inversion symmetric you need to allow it to be time independent but not time-inversion symmetric so the time symmetry is weaker. I forget when it was discovered but it was decades after the Schwarzschild solution was, because the equations were more complex, and the physics had more possibilities with fewer symmetries and rotation.

  • $\begingroup$ In my point of view (probably wrong), due to the strict equality of the two bodies each body "fold" the space-time in the same way (even function) and therefore the mass center is just in the middle. $\endgroup$ – Questar Jan 27 '18 at 10:43
  • $\begingroup$ @Questar. You may be right in some sense if the masses are equal, but 1)not if not, and 2)much more importantly you still cannot solve the equations by taking that as the origin and getting a Schwarzschild solution with it as center, the nonlinearities still make it impossible to find a closed form solution. Still, there are some quantities that can be deduced to obtain good approximate solutions for black hole binary mergers gravitational wave frequencies, and good numerical approximations for the merger evolution. $\endgroup$ – Bob Bee Jan 27 '18 at 20:15

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