I think there are two separate issues here.
PeterDonis said:
I phrased my point poorly. Yes, of course you can define a thingie with 42 holes in it and call it a "spacelike surface" as long as its tangent vectors at every point are spacelike. I wasn't trying to claim that that is impossible.
Let me try to rephrase my point as as question: what do you think the topology of the overall manifold (spacetime) is in the "42 holes merge" case? I think it is still R2 x S2, the same as Schwarzschild spacetime. And I think that with that topology, there can only be one singularity, although we can of course choose spacelike surfaces (if we allow such surfaces to be geodesically incomplete) that are "punctured" by that one singularity multiple times, if we are willing to accept sufficient distortion in the implicit coordinate chart we are using.
I'll address the topology question next. Here I will say that I think slices that are punctured multiple times are the norm, in the following sense (and this I believe is true even in a model with one singularity - that it must bend deep into each leg). Consider an event on an external world line far in the future of the event where (all the singularities / the singularity - pick your model) stops being all in the causal future. Then, the only spacelike surfaces intersecting that world line at that point, that don't have multiple punctures, are ones that go very close to the past light cone. If one could pick a measure, I would say 'most' spacelike slices have multiple punctures.
PeterDonis said:
If you agree that the topology of the manifold is still R2 x S2 in the "42 holes merge" case, then how can there be more than one singularity?
Or if, alternatively, you think the topology is something else, what do you think it is?
First, note that all three of the following have very different Penrose diagrams:
- Kruskal manifold
- 2 quadrants of the Kruskal manifold
- Oppenheimer Snyder collapse
On the first two, there seems little debate that you have R2 X S2 (depending a bit on definition of two quadrants).
On the last, there is at least a little dispute among experts due to an initial Cauchy surface being essentially indistinguishable from one leading to a neutron star, combined with a Geroch theorem that evolution from a Cauchy surface without violating energy conditions cannot change topology. Those favoring this argument claim that somehow, OS manifold must be R4. The majority argues there are technical loopholes in Geroch's theorem, and the OS manifold is R2 X S2. To me, this follows (but not at all obviously) from the most accepted Penrose diagram for OS manifold.
So for multiple BH, with or without merger, I claim the topology is certainly
not R2 X S2. What it is, I have no real idea. As a simplistic guess, I can throw out R4 - L - L - L ... where L is a line. There are well known arguments for why R4 - L is the same as R2 X S2, but what you would call the multiple subtraction topology, I have no idea.
I still cannot find even one paper that discusses complete topology for even 2 BH merging. I did find a paper that gives a robust proof (weak assumptions) that the exterior topology of N BH is 'as simple as possible' given the excsision of the BH at their horizons. Interesting, but not the least bit surprising.