Einstein-Strauss solution with one or two bubbles

[smallphi]

The Einstein-Strauss solution of Einstein equations is used to model a gravitationally bound system in otherwise homogeneuous FRW universe.

It is a black hole at the center of an otherwise empty spherical bubble surrounded by expanding FRW matter dominated universe. The mass of the black hole is set equal to the mass that would be in the bubble if it the solution was FRW everywhere. With that matching, the FRW solution outside the bubble doesn't 'feel' the presense of the bubble and expands as if the bubble doesn't exist.

The metric inside the bubble is Shwartzschild and doesn't 'feel' the expanding universe surrounding it. Can we say that this is a result of some version of the Birkhoff theorem: since the spacetime considered is spherically symmetric, the metric inside the bubble should depend only on the mass inside i.e. on the black hole only?

The opinion in the field is that nothing changes in the case of two non-overlapping bubbles with black holes at their centers immersed in expanding matter dominated FRW (masses of black holes matched to FRW mass as usual). The metric inside each bubble is still Schwarzschild corresponding to the mass of the black hole and doesn't 'feel' the expanding universe outside.

Birkhoff theorem cannot be evoked in this case since the spherical symmetry of spacetime is broken by the presense of two bubbles. 'Principle of superposition' i.e. slap the second bubble on the spacetime of one bubble in expanding FRW and nothing will change, cannot be used either since Einstein equations are not linear.

What would be the rigorous reasoning to justify the conclusion that the metric inside both bubbles is still schwarzshild despite the fact the spacetime is no longer spherically symmetric?

All I can think about is: guess the solution and just plug in Einstein eq. to prove its consistent. Is there more elegant argument?

 

[Dale]

What would be the rigorous reasoning to justify the conclusion that the metric inside both bubbles is still schwarzshild despite the fact the spacetime is no longer spherically symmetric?

Around each bubble you can make a spherically symmetric boundary. Since within that boundary the source is spherically symmetric and since on the boundary the fields are spherically symmetric you can still use arguments of spherical symmetry to get the solution within that region.

Note, this is not a superposition argument. You are correct that the equations are non linear, but although they are nonlinear the symmetry arguments still apply as long as both the matter and the boundary conditions reflect the same symmetry.