Black hole inside of a black hole.... can it be done?

[PeterDonis]

This is what makes my intuition suggest a spacelike surface through all the legs before merger can, indeed, reach a singularity in each leg.

But such a "spacelike surface" would in fact not be one, because it would have a "hole" at each singularity it intersects (since singularities are not part of the manifold). A similar remark would be true of a "spacelike surface" in a spacetime with one black hole and one singularity, that cut through the cylinder (the hole) "above" the point where the collapsing matter reaches the center (since that point is where the singularity begins). To make valid spacelike surfaces that do not have holes, you would need to "push" the central portion of the surface (the part inside each leg/cylinder) down below the point where the singularity begins (inside each leg/cylinder).

Once you've done that, it seems to me that there are no longer really two distinguishable alternatives: the "one singularity at the top of the trousers" case and the "multiple singularities branching down from the top of the trousers into each leg" case are actually the same, as far as anything inside the manifold is concerned.

 

[PAllen]

I don’t buy that. A spacelike surface can be geodesically incomplete just like the overall manifold - it is just a submanifold. So my intuition strongly suggests there would exist a geodesically incomplete achronal surface with 42 missing points before the merger.

 

[PeterDonis]

A spacelike surface can be geodesically incomplete just like the overall manifold - it is just a submanifold. So my intuition strongly suggests there would exist a geodesically incomplete achronal surface with 42 missing points before the merger.

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.

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?

 

[PAllen]

I think there are two separate issues here.

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.

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.

 

[PAllen]

I did find the following from Hawking and Ellis, which I take as favoring much of my viewpoint:

https://www.semanticscholar.org/pap...d89b35644260ef9428d2275f47b047de7aa/figure/54

 

[PeterDonis]

Those favoring this argument claim that somehow, OS manifold must be R4.

Hm, yes, I had forgotten about that issue.

To me, this follows (but not at all obviously) from the most accepted Penrose diagram for OS manifold.

Since you admit it's not at all obvious , can you briefly describe the argument for this?

 

[PeterDonis]

I did find the following from Hawking and Ellis

I'll take a look when I have a chance; I had forgotten that Hawking & Ellis treat this very subject in some detail. Shows how long it's been since I've looked at Hawking & Ellis.

 

[PeterDonis]

I did find the following from Hawking and Ellis

Looking at that figure (Fig. 60 in Hawking & Ellis), it could be similar to what has been described previously in this thread, just with a part at the top not shown in the figure. Basically, in terms of the apparent horizons and trapped regions, we start out with two trapped regions bounded by apparent horizons, and each one is just a simple tube. Then, after the merger, a third trapped region forms outside them, but this region is a ring, and gradually expands towards the two tubes inside. If the ring eventually meets the tubes (somewhere up above the top of the diagram), the three trapped regions would merge into one. So topologically the overall trapped region in the spacetime as a whole would still be something like trousers with legs, just with a much weirder shape at the "join" of the legs (and a much weirder looking "waist" region) than the event horizon shown in the figure.

 

[Nugatory]

Taylor&Wheeler Page 3-10, where they say "Inside there is an interchange of the character of the t-coordinate and r-coordinate."

That is an unfortunate way of phrasing it; better might be to use letters other than $r$ and $t$ to label the timelike and spacelike coordinates inside the horizon. See, for example, remark 1 on page 5 of https://arxiv.org/abs/0804.3619

 

[timmdeeg]

That is an unfortunate way of phrasing it; better might be to use letters other than $r$ and $t$ to label the timelike and spacelike coordinates inside the horizon. See, for example, remark 1 on page 5 of https://arxiv.org/abs/0804.3619

Yes agreed, also @PAllen and @PeterDonis have pointed that out already. I think that the invariant way to describe this situation inside the BH mentioned by @PeterDonis in #95 is much preferable.