If you take Einsteins field equation with a homogenous (and isotropic) mass density, no pressure, no cosmological constant and a flat, non-expanding spacetime, the result is a collapsing space-time (right?).
If you consider a system of two (relative to each other) still-standing bodies with no non-gravitational forces (thus, no pressure) involved, they would collapse by their gravitational attraction. A stable, non-static solution exists though, by (instead of standing still) simply letting the two bodies orbit around each other.
The orbiting bodies system is stable in a Newtonian context. Is it also stable according to GR in an otherwise flat space-time?
Can this setting be generalized to arbitrary many bodies, maintaining its stability?
Can this setting be generalized to infinitely many (discrete) bodies?
Does that provide for a stable, infinite and non-expanding universe? If so, which physical observations or first principles does it contradict to? (After all, we assume the universe to be expanding) If not, why?
Or, to turn the question around, can you mathematically prove that there is no stable solution to the EFE in a flat, non-expanding universe?
PS: If the answer involves any formulas, I'd very much appreciate a coordinate-free formulation or (if possible, even better) a Gravitoelectromagnetic approximation. If too many indices and primes are involved, my brain crashes with a stack overflow. :)
PPS: Of course, any $n$-body system with $n > 2$ will be chaotic and not analytically solvable. That doesn't necessarily mean it's unstable or that you cannot argue about its stability, though.
