Are there physical models of spacetimes, which have bounded (four dimensional) h

Question

Are there physical models of spacetimes, which have bounded (four dimensional) holes in them?

And do the Einstein equations give restrictions to such phenomena?

Here by holes I mean constructions which are bounded in size in spacetime and which for example might be characterized by nontrivial higher fundamental groups.

On a related note:

Could relatively localized and special configurations of (classical) spacetime be interpreted as matter?

I.e. can field-like behavior emerge in characteristic structures of spacetime itself, like for example holes in the above sense, or localized areas of wild curvature? Gravitational waves certainly go in this direction, although they act on such a huge scale, that in their case we probably wouldn't recognize an organized, maybe even life-like behavior as such.

Explanation / Answer

The spacetime in general relativity does not contain "holes" in the sense of excized regions because of a physical argument--- if you can shoot a particle at the region, it should continue into the region. This is the reason that geodesic completeness is used instead of completeness in GR. The condition of geodesic completeness says that the manifold must not have places where geodesics stop for no reason.

Of course, the singularity theorems guarantee that geodesic completeness fails inside a black hole. But the failure in the case of time-like singularities is mild--- the singularity is only reachable by light rays.

The closest thing to an excized region is a black hole. The interior is excized in the sense that it is disconnected causally from the exterior. You can remove the interior and simulate the exterior only (classically) and you don't expect to run into too many troubles. Whether this is completely true in the quantum version is not clear to me.

As for other topological quantities, you can put them in by hand, but it is not clear if they can appear dynamically. There is the topological censorship conjecture, which states that you won't be able to see a topological transition in classical general relativity. I do not know the status (or even the precise statement) of this conjecture.

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