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

[PeterDonis]

these individual "black holes" -that have only apparent horizons- are still there, aren't they?

There can be multiple apparent horizons inside a single event horizon, yes.

The situation is reminiscent of a closed FLRW spacetime

Not really, since there are no horizons at all in a closed collapsing FRW spacetime.

in the Oppenheimer/ Snyder pressureless dust collapse model, one could imagine that these specs of dust are, themselves, collapsing objects with their apparent horizons

No, you can't, because in the O-S model the "dust" is continuous; it's not individual "specks" separated by vacuum, it's a single continuous region occupied by matter. "Dust" is a technical term in models like this and should not be interpreted literally.

There are also collapsing models where timelike singularities ( with their corresponding Cauchy horizons), are formed before the final spacelike singularity.

Can you give an example? If you are thinking of Kerr spacetime, no, there are no models similar to the O-S model for Kerr spacetime. Also, the presence of the Cauchy horizon at the inner horizon of Kerr spacetime makes most physicists believe that the region approaching and inside that horizon, which is the region where the timelike singularity is, is not physically reasonable and would not be expected to be present in a realistic model.

one can also have black holes "inside" other bhs in the cases of Bag of gold or Baby universe spacetimes.

The "bag of gold" model is simply a carefully chosen profile of matter inside the event horizon that causes additional apparent horizons to be present. So this is not an example of multiple event horizons.

The "baby universe" spacetime, AFAIK, is speculative only; nobody has ever actually produced a consistent model of it.

 

[Dimitris P]

Thank you for the answer!
I don't really disagree with most of the replies you gave in this thread, but i think that you're using a very restricted definition for a black hole spacetime.
a)In a closed, recollapsing universe ( like the k=1FLRW model), there is no future null infinity/ asymptotic flatness, so the "classic" definition of a black hole is not applicable. Yet, bhs can exist in such spacetimes.
The "problem" seems to be the notion of the Absolute Event Horizon.
By the way, in my 1st post I stressed the fact that the closed FLRW universes that end in a big crunch have the obvious difference that they have no exterior region or an event horizon !
b) About the OS model: One can consider it as a gluing of a FLRW region and a Schwarzschild exterior.
As in the case of a realistic model of a universe, the specs I was referred to is only an approximation, when you have a big number of them and they're approximately homogeneous.
The hypothetical example of a collapsing galaxy is not the same, of course, but it's related: If a whole region with lots of millions of black holes and lots of stars and other stuff collapses, for some time, an observer that falls freely inside could see other "individual" black holes inside this collapsing region! He could also jump inside one of them!
You're right that technically only one black hole is properly defined in that case, but l think that's a bit of nitpicking. Technically, our own galaxy's central Sagittarius A* supermassive black hole and Andromeda's one (and, possibly, lots of other smaller bhs that'll merge with them) are not "distinct" holes, because they share, probably, the same Event Horizon, after all!

 

[PeterDonis]

i think that you're using a very restricted definition for a black hole spacetime.

I'm using the definition that is the standard one in the GR literature. If you want to use a different definition, you should at least make clear what definition you are using. For example, if you want to use "black hole" to mean "apparent horizon and region inside it", then you should say so. And you should be prepared to deal with the limitations of that definition--for example, an apparent horizon is frame-dependent whereas an event horizon is not.

In a closed, recollapsing universe ( like the k=1FLRW model), there is no future null infinity/ asymptotic flatness, so the "classic" definition of a black hole is not applicable. Yet, bhs can exist in such spacetimes.

Please give a reference for this claim.

Technically, our own galaxy's central Sagittarius A* supermassive black hole and Andromeda's one (and, possibly, lots of other smaller bhs that'll merge with them) are not "distinct" holes, because they share, probably, the same Event Horizon, after all!

Only if they end up merging.

 

[Dimitris P]

Concerning your fourth objection: No, I was not thinking about Kerr. In the unperturbed, eternal Kerr spacetime only timelike singularities could exist (mathematically). And its well known, as you already pointed out, that due to blueshift/ mass inflation instabilities, instead of cauchy horizons, most probably null or spacelike singularities are happening there.
I had in mind some models of inhomogeneous collapse, where, temporarily, timelike singularities are developing, but the final endpoint of the collapse is a spacelike one, so there are both a Cauchy and an event horizon. I think Joshi described such models.
About bag of gold spacetimes:
An example is the gluing of a closed FLRW universe and a black hole spacetime, so my comment is similar to the previous ones.
As for the Baby Universe scenarios: I don't really think that all these models (e.g Frolov/ Novikov)are inconsistent.
They just need some underlying assumptions as every other model. Maybe you could call them controversial.

 

[PeterDonis]

I had in mind some models of inhomogeneous collapse, where, temporarily, timelike singularities are developing, but the final endpoint of the collapse is a spacelike one, so there are both a Cauchy and an event horizon. I think Joshi described such models.

If you can find a reference, that would be helpful. These models sound interesting but I'm not familiar with them.

About bag of gold spacetimes:
An example is the gluing of a closed FLRW universe and a black hole spacetime

Please give a reference. The only such model I'm aware of is the Oppenheimer-Snyder model, which is not a "bag of gold" spacetime as I understand that term.

As for the Baby Universe scenarios: I don't really think that all these models (e.g Frolov/ Novikov)are inconsistent.

Again, please give a reference. We need some specific models as a basis for discussion.

 

[Dimitris P]

The collapse of a closed FLRW universe is very similar, in fact, with a merging of lots of individual black holes in a realistic case!
As for a reference: Roger Penrose's lectures about cosmology, gravitational entropy etc.(there are lots of them).
About the coalescence of S A* with Andromeda's supermassive bh: Come on, it's more than obvious that I was hypothesizing about a future merging!

 

[Dimitris P]

I'll search for references, maybe tomorrow. I have a lot of work to do now!
Anyway, thanks for the intriguing conversation.

 

[PeterDonis]

The collapse of a closed FLRW universe is very similar, in fact, with a merging of lots of individual black holes in a realistic case!

No, it isn't, because, as I have already commented, there are no event horizons in a closed collapsing FRW universe. The only similarity is that the interior of the Oppenheimer-Snyder solution is a portion of a closed collapsing FRW universe, but that is a collapse of one black hole, not multiple black holes merging.

As for a reference: Roger Penrose's lectures about cosmology, gravitational entropy etc.(there are lots of them).

We need a specific reference, not a vague gesture in the direction of "lots of them".

it's more than obvious that I was hypothesizing about a future merging!

If you're hypothesizing, you should make the hypothesis clear: for example, "If the two holes merge in the future, they will share a single singularity". Don't assume that what is "more than obvious" to you--because you are the one thinking up the scenario--will be obvious to others.

 

[Dimitris P]

Your comment about the possible future merging of the supermassive black holes is not at all relevant to the point I've made! It's totally irrelevant if these holes will actually merge or not.
It is possible that such a thing will happen, and that's enough!
The point is that when everybody today uses the term "black hole", he/she's not referring necessarily to the definition that you have in mind.
This well known definition presupposes the notion of future null infinity etc (actually I've already mention that), so it's suitable for rigorous mathematical proofs, but not for complicated situations.
It is well known that there is no strict definition for a black hole that could be applied everywhere:
A Kerr/ Ads spacetime is very different from a Kerr/ de Sitter one. They have something common.
Call it black hole, or whatever you want ! It doesn't matter at all.
The exact GR solutions are there, and that's all we need!
There are also even more weird cases with planar black holes, or( in d>4) black rings and more...
Certain conditions are also necessary for the definition of mass in GR. Does that mean that one cannot have a definition for mass in Schwarzschild/dS, for example? I don't see why one can't use different definitions for each case.
Of course there are some philosophers that are bothered much about strict definitions, but GR has a very rich structure to be confined to our intuitions about simplicity and rigour.
A Kerr black hole with a/m= 0.99999 is totally different from one that has exactly a/m= 1. Does it really matter how you call the extremal horizon?
Is it a "degenerate", or a Cauchy horizon? Who cares?

 

[Dimitris P]

About the case of a collapsing FLRW universe:
A physically realistic collapsing universe is a huge mess, and that's exactly what R. Penrose stresses in many of his lectures about cosmology:
Due to unavoidable inhomogeneities (one needs extreme fine tuning for an idealised homogeneous collapse), the final stages of the collapsing phase are, actually, a coalescence of huge supermassive black holes (yes, I call them black holes without problem. You can call them apparent (or whatever) black holes, if you like.).
Penrose shows how different a realistic time reversed version of an expanding universe actually is, compared with the naive expectations, due to the 2nd law of thermodynamics (but that's a different story).
This is actually old news: Everybody that attended one of his cosmology lectures remembers his arguments (and his characteristic drawings).

 

[PeterDonis]

Your comment about the possible future merging of the supermassive black holes is not at all relevant to the point I've made! It's totally irrelevant if these holes will actually merge or not.
It is possible that such a thing will happen, and that's enough!

It is if you are claiming that the holes share one singularity. That's not true if they never merge. Just having it be "possible" that they will merge is not enough. They have to actually merge some time in the future.

The point is that when everybody today uses the term "black hole", he/she's not referring necessarily to the definition that you have in mind.

In this thread, we are using the standard GR definition that I gave (but see further comments below for a clarification). If you want to talk about other kinds of spacetime geometric phenomena, please start a new thread.

A Kerr/ Ads spacetime is very different from a Kerr/ de Sitter one. They have something common.

For Kerr-de Sitter spacetime, the black hole definition I gave is easily generalized so that the cosmological horizon plays the role of future null infinity; i.e., the event horizon in this spacetime is the boundary of the region that cannot send light signals to the cosmological horizon.

I have not seen a similar generalization made for Kerr-Anti de Sitter spacetime, but I suppose one could be made.

In all of these cases, there is still an event horizon, bounding some region from which light signals cannot escape, and a singularity inside it. So if you want to treat that as the generalized definition of a "black hole" for this thread, that's fine. It still doesn't include cases like closed FRW universes. So once again, if you want to talk about cases other than the ones that fall into the general category I just described, please start a new thread.

 

[PeterDonis]

Penrose shows how different a realistic time reversed version of an expanding universe actually is, compared with the naive expectations, due to the 2nd law of thermodynamics (but that's a different story).

Yes, Penrose shows how we would expect a closed FRW universe that recollapses to recollapse to a "Big Crunch" in which the Weyl tensor increases without bound, whereas in the initial "Big Bang", the Weyl tensor vanishes. That doesn't change the fact that there are no event horizons in a closed collapsing FRW universe. So, once again, it doesn't fall into the "black hole" category that we are discussing in this thread. So if you want to talk about it, please start a new thread.

 

[Dimitris P]

Once again:
The point that I've made with the supermassive black hole coalescence is that, according to the typical definition that you're using, these cannot be considered "Different" black holes in the case that they will merge in the future, and I already told you, more than once, that, typically, I agree! Call them what you like.
That does not change anything, not in this case, neither in the case of a galaxy that's collapsing inside its Schwarzschild radius or in the case of a closed universe!
General Relativity gives as an accurate (as far as we know) description of what's happening in all these cases and that's sufficient enough.
Actually, if I wanted to argue about just words , I could start a thread in some other kind of forum.
That's not my intention.
Concerning R. Penrose's lectures: So, if you're aware of them, why did you ask for references?
I'm sure, also, that you certainly remember that he called them black holes! (Heresy!).

.

 

[PeterDonis]

That does not change anything, not in this case, neither in the case of a galaxy that's collapsing inside its Schwarzschild radius or in the case of a closed universe!

You have already been corrected more than once about your claims regarding a closed universe. A closed universe simply does not work the same as any of the other examples we have discussed.

Concerning R. Penrose's lectures: So, if you're aware of them, why did you ask for references?

I asked you for references regarding your claims, not Penrose's. His claims are not the same as yours.

I'm sure, also, that you certainly remember that he called them black holes!

He did use the term, yes. But not with any of the definitions I have said are on topic in this thread. And as I've already told you more than once, if you want to discuss other definitions, you need to start a separate thread.

Enough is enough. You are now banned from further posting in this thread.

 

[PAllen]

No. The definition of a black hole is a region of spacetime that cannot send light signals to infinity. Once you are inside such a region, you're inside it. The idea of having a second such region inside the first doesn't even make sense.

I would like to find a way to ask about the original intent of the thread in a hopefully meaningful way. I think it is an interesting question.

Let us say we have a definition of BH for cosmological spacetimes (FLRW in the continuous limit, but this is clearly an approximation). There are clearly things we treat as BH in a universe that is only approximately FLRW.

Then, imagine that in some region, by quirk of ejection from galaxies, or just wildly improbable initial conditions, we have a region of spacetime with 42 BH with minimal mutual relative motion. Over time, they will coalesce. In their initial state, each has a separate, observable, near horizon region that produces characteristic optical distortions of the distant stars. For all practical purposes, there are 42 separate horizons, of observable angular size for some moderately distant observer.

As they coalesce, long before their initially observed horizons would be expected to meet, they are collectively within the Schwarzschild radius associated with their aggregate distantly measured mass (e.g. by orbits of test bodies around the cluster of BH). By the hoop conjecture, they must now all be within the event horizon of aggregate BH. To the distant observer, they all will have effectively vanished, and only a giant ultrablack region will be 'seen'. Yet, internally, the quasilocal physics around each BH cannot have radically changed.

What to call the state after the supermassive horizon has formed, but each formerly separate BH is still expected to be many millions of kilometers from is neighbors? Physically, it seems 42 BH within a large BH is the most natural description, even if this doesn't fit the formal mathematical definition.

 

[timmdeeg]

Just for clarification, would an observer far away recognize the formation of the supermassiv horizon as a sudden event by observing the stars behind the BH cluster?

 

[PAllen]

Just for clarification, would an observer far away recognize the formation of the supermassiv horizon as a sudden event by observing the stars behind the BH cluster?

Fast, but not sudden. I believe the cluster would be observed to rapidly blacken, with more and more extreme optical distortion around near its outer region. Quickly, it would be spherical black region with extreme optical distortion around its edge. This follows from the "no hair theorem" for BH.

 

[PeterDonis]

Let us say we have a definition of BH for cosmological spacetimes

The only really workable definition for spacetimes that do not have a future null infinity, such as FLRW, is that a "black hole" is a region bounded by an apparent horizon. Since apparent horizons are frame-dependent, this definition has significant limitations, but there simply is no invariant that we can use to define a "black hole" in these spacetimes.

With the "apparent horizon" definition, as I believe I mentioned much earlier in this thread, it is possible to have a "black hole" inside another "black hole"--i.e., an apparent horizon, a region inside it, and then another apparent horizon inside that region--but it requires fairly exotic conditions (for example, I believe violations of energy conditions are required).

As they coalesce, long before their initially observed horizons would be expected to meet, they are collectively within the Schwarzschild radius associated with their aggregate distantly measured mass (e.g. by orbits of test bodies around the cluster of BH). By the hoop conjecture, they must now all be within the event horizon of aggregate BH

The hoop conjecture doesn't apply to spacetimes with no future null infinity--at least not in the "event horizon" form that you have stated here. There might be another similar conjecture that involves apparent horizons, but I'm not aware of one.

To the distant observer, they all will have effectively vanished

Note that this can happen with an apparent horizon: outgoing light at an apparent horizon no longer moves outward locally, so it is "stuck" there, and can remain "stuck" there for very long periods of time, so to distant observers, it looks the same as an event horizon would look (if an event horizon were possible in their spacetime). So from observational evidence alone we cannot infer the presence of event horizons. This should not be surprising since the event horizon is teleological--where it is depends, not just on the local physics, but on the entire global future of the spacetime. And of course we can't know that.

 

[PeterDonis]

I believe the cluster would be observed to rapidly blacken, with more and more extreme optical distortion around near its outer region. Quickly, it would be spherical black region with extreme optical distortion around its edge. This follows from the "no hair theorem" for BH.

The "no hair" theorem also does not apply in spacetimes with no future null infinity and therefore no event horizons. When stated properly, the "no hair" theorem is just the fact that the Kerr-Newman family of spacetime geometries contains all possible black holes, where "black hole" is defined using the standard "event horizon" definition that requires the presence of a future null infinity. There is no "no hair" theorem that I'm aware of for any other family of spacetimes, including FRW spacetimes with apparent horizons in them.

 

[PeterDonis]

would an observer far away recognize the formation of the supermassiv horizon as a sudden event by observing the stars behind the BH cluster?

From far away, the formation of an apparent horizon around the entire cluster would look much as @PAllen described, but not for the reason he gave (the "no hair" theorem), since, as I pointed out just now, that theorem doesn't apply in spacetimes that don't have a future null infinity. The correct way to analyze such a case (although it would require numerical simulation for anything quantitative) would be to analyze the implications of an apparent horizon forming around the entire cluster, with the individual apparent horizons of each individual BH inside it.

 

[PAllen]

With the "apparent horizon" definition, as I believe I mentioned much earlier in this thread, it is possible to have a "black hole" inside another "black hole"--i.e., an apparent horizon, a region inside it, and then another apparent horizon inside that region--but it requires fairly exotic conditions (for example, I believe violations of energy conditions are required).

Note that in my example, there is no matter at all, just evolution of Weyl curvature. Clearly no energy condition violations. Of course, the situation is implausible for other reasons.

 

[PeterDonis]

Note that in my example, there is no matter at all, just evolution of Weyl curvature.

If that's all that's present, I don't think it's possible to have one apparent horizon inside another; I think all you can get with pure Weyl curvature is one apparent horizon (possibly with multiple "legs" that merge as multiple individual objects coalesce). I think you need a nonzero stress-energy tensor that violates energy conditions to get multiple apparent horizons inside one big one.

 

[PAllen]

If that's all that's present, I don't think it's possible to have one apparent horizon inside another; I think all you can get with pure Weyl curvature is one apparent horizon (possibly with multiple "legs" that merge as multiple individual objects coalesce). I think you need a nonzero stress-energy tensor that violates energy conditions to get multiple apparent horizons inside one big one.

That doesn’t make sense for my scenario. All the apparent horizon BH can be old, with no matter inside. There is no plausible way for them to coalesce without producing a boundIng apparent horizon containing the original ones, way before any mergers. You can take my scenario as taking place in asymptotically flat spacetime. Then the hoop conjecture and no hair theorem do apply.

 

[PeterDonis]

That doesn’t make sense for my scenario.

Then I don't think your scenario is possible if it involves multiple small apparent horizons inside one big one, with no matter anywhere.

You can take my scenario as taking place in asymptotically flat spacetime.

In an asymptotically flat spacetime, the event horizon for the case you describe would be outside the apparent horizon, and would be a single surface in spacetime with multiple "legs" to the past and one big region to the future. There would still not be multiple apparent horizons inside one big one; there might be an apparent horizon with multiple "legs" (similar to the overall shape of the event horizon), but it would be inside the event horizon, as I noted just now.

Then the hoop conjecture and no hair theorem do apply.

Yes, but all they say then is that there would be an event horizon with the shape I described above.

 

[PeterDonis]

There is no plausible way for them to coalesce without producing a boundIng apparent horizon containing the original ones, way before any mergers.

Bear in mind that, unlike event horizons (which are always null surfaces), apparent horizons can be spacelike surfaces (or timelike, but that case doesn't concern us here--it comes into play in cases like black hole evaporation by Hawking radiation). Heuristically, in a coalescence such as you describe, at the point where, according to your description, all of the mass of the cluster is just inside the Schwarzschild radius corresponding to that mass, the apparent horizons would all "jump" outward along spacelike surfaces to become one big apparent horizon. You would not have multiple small apparent horizons inside one big one.

 

[PAllen]

Responding overall to @PeterDonis recent posts, I think the key point is that any type of horizon is not really a locally detectable physical phenomenon. Event horizons require information on the whole future of the universe. Apparent horizons are coordinate dependent and undetectable to a local inertial frame (as are event horizons). Further, the key point I missed was that the definition of an apparent horizon simply doesn't normally allow for one inside another. An apparent horizon is the outermost light trapping surface. Once my 42 BH are within a common apparent horizon, the whole interior is light trapping, but only the outermost surface qualifies as an apparent horizon.

Thus, the real difficulty is simply the inability to define a BH boundary for a BH within a BH. After a good bit of thought on this, I am unable to come up with a reasonable way to do this. I suspect it is simply not possible.

The alternative is to give up on any notion of horizon within a horizon for discussing a BH within a larger BH horizon. Instead, focus on unbounded curvature invariants. In this case, I claim that a reasonable spacelike slice through a collapsing collection of Schwarzschild BH (each of which is 'old'), shortly after their event horizons have merged (i.e. through the 'waist' of 42 legged pants just above the legs), will have 42 areas where curvature invariants are unbounded, separated by regions of well behaved curvature invariants. The interior topology will be complicated, because each curvature singularity is bounded by S2 X R, with vanishing S2 radius. A much later slice should have a large region regular curvature invariants, and a single presumably extremely complicated (topologically) region with unbounded curvature invariants.

Any more precise description would require simulation which I suspect is still beyond current capability. Numerical simulations of BH mergers excise singular regions for tractability.

 

[PeterDonis]

Apparent horizons are coordinate dependent and undetectable to a local inertial frame (as are event horizons).

It's worth noting, though, that apparent horizons are much closer, in a sense, to being locally detectable than event horizons are. The definition of an apparent horizon is a surface foliated by marginally outer trapped 2-spheres, i.e., 2-spheres for which the expansion of the congruence of radially outgoing null normals is zero. The coordinate dependence of this definition comes from the coordinate dependence of the definition of "radially outgoing null normals"--roughly speaking, I can change which particular null vectors are "normal" to the 2-sphere by changing which spacelike 3-surface I consider to be a "surface of constant time" in which the 2-sphere is embedded. (My understanding of this technical point is rough, so I may be leaving things out.)

However, the effect of this coordinate dependence is not as problematic as one might think. For the case we are considering, it is extremely likely that any reasonable choice of "null normals" for an observer free-falling inwards will lead to the same congruence of null worldlines being identified, and since the expansion of any given congruence is invariant, this means that it is extremely likely that any observer free-falling inward will identify the same apparent horizon as any other. The only "non-local" element is that, in order to measure the expansion of the congruence of radially outgoing null normals, one has to be able to sample enough of the 2-sphere, so to speak, which can't be done locally by a single observer; but it could be done, reasonably, by a fairly small family of observers falling inward along slightly different radial lines and comparing their measurements. While not precisely "local", this is still a lot closer to being "local" than having to know the entire global future of the spacetime, which is what would be required to know the location of the event horizon (for spacetimes where that concept makes sense).

 

[PeterDonis]

I claim that a reasonable spacelike slice through a collapsing collection of Schwarzschild BH (each of which is 'old'), shortly after their event horizons have merged (i.e. through the 'waist' of 42 legged pants just above the legs), will have 42 areas where curvature invariants are unbounded, separated by regions of well behaved curvature invariants.

I think "unbounded" is too strong here. I think there will be considerable variation in curvature invariants along such a spacelike surface, with the regions showing the highest values being the ones coming from the 42 "legs". But I don't think curvature invariants will be unbounded in these regions, at least not for a "reasonable" spacelike slice.

The heuristic that seems to be underlying your view here is that, "inside" each of the legs before they merge, curvature invariants are already unbounded. But I don't think that's true. Curvature invariants don't become unbounded until you approach the singularity, and there is not a singularity inside each of the 42 "legs". There is only one singularity, and it is up at the top, at the "waist" of the trousers. It doesn't "dip" downwards in the regions above the legs.

 

[PeterDonis]

The interior topology will be complicated, because each curvature singularity is bounded by S2 X R, with vanishing S2 radius.

I don't think this is correct; I think the same observation I made in the last paragraph of my previous post applies to this as well.

 

[PAllen]

I think "unbounded" is too strong here. I think there will be considerable variation in curvature invariants along such a spacelike surface, with the regions showing the highest values being the ones coming from the 42 "legs". But I don't think curvature invariants will be unbounded in these regions, at least not for a "reasonable" spacelike slice.

The heuristic that seems to be underlying your view here is that, "inside" each of the legs before they merge, curvature invariants are already unbounded. But I don't think that's true. Curvature invariants don't become unbounded until you approach the singularity, and there is not a singularity inside each of the 42 "legs". There is only one singularity, and it is up at the top, at the "waist" of the trousers. It doesn't "dip" downwards in the regions above the legs.

I disagree with this. If I have two BH in distant mutual orbit, I claim there are two separate S2 X R singular regions. What on Earth stops one from glueing two regions of Kruskal geometry together, each of which is an exterior quadrant plus the part of one interior quadrant that would present in a collapse BH? The glueing would produce a shared exterior region, with two wholly separate interiors.

 

[PAllen]

It's worth noting, though, that apparent horizons are much closer, in a sense, to being locally detectable than event horizons are. The definition of an apparent horizon is a surface foliated by marginally outer trapped 2-spheres, i.e., 2-spheres for which the expansion of the congruence of radially outgoing null normals is zero. The coordinate dependence of this definition comes from the coordinate dependence of the definition of "radially outgoing null normals"--roughly speaking, I can change which particular null vectors are "normal" to the 2-sphere by changing which spacelike 3-surface I consider to be a "surface of constant time" in which the 2-sphere is embedded. (My understanding of this technical point is rough, so I may be leaving things out.)

However, the effect of this coordinate dependence is not as problematic as one might think. For the case we are considering, it is extremely likely that any reasonable choice of "null normals" for an observer free-falling inwards will lead to the same congruence of null worldlines being identified, and since the expansion of any given congruence is invariant, this means that it is extremely likely that any observer free-falling inward will identify the same apparent horizon as any other. The only "non-local" element is that, in order to measure the expansion of the congruence of radially outgoing null normals, one has to be able to sample enough of the 2-sphere, so to speak, which can't be done locally by a single observer; but it could be done, reasonably, by a fairly small family of observers falling inward along slightly different radial lines and comparing their measurements. While not precisely "local", this is still a lot closer to being "local" than having to know the entire global future of the spacetime, which is what would be required to know the location of the event horizon (for spacetimes where that concept makes sense).

But the problem is that given a simple BH, for example, any two sphere inside the apparent horizon is trapping surface. The apparent horizon is the outermost one. There does not seem to be any way to make the 'outermost' part of this definition work for a an apparent horizon inside another. The purported interior one is just another of infinitely many trapping surfaces that are not outermost.

 

[PeterDonis]

What on Earth stops one from glueing two regions of Kruskal geometry together, each of which is an exterior quadrant plus the part of one interior quadrant that would present in a collapse BH?

Because you can't glue two exteriors together this way. The exterior is "one-sided"--only one side can join to an interior through a horizon, the other side has to go out to infinity. Even if you wave your hands and say we're talking about some "Kruskal-like" geometry in a spacetime that doesn't have a conformal infinity, like FRW, then you have the problem of how to separate the interiors--because the interior of Kruskal doesn't stop at any finite point, it extends all the way to infinity in Kruskal coordinates. So even if you try to "glue" two finite pieces of an exterior together, you can't separate the two interiors.

 

[PeterDonis]

There does not seem to be any way to make the 'outermost' part of this definition work for a an apparent horizon inside another.

Yes, I agree that this is why the "black hole inside another black hole" idea doesn't work even if we define "black hole" using apparent horizons instead of event horizons. I was just pointing out that, even though the concept of an apparent horizon is, strictly speaking, coordinate-dependent and not precisely "local", it still can be used in many cases as a workable "reasonably close to local" criterion for the boundary of a "black hole" if one wants to avoid the "event horizon" definition.

 

[PAllen]

Because you can't glue two exteriors together this way. The exterior is "one-sided"--only one side can join to an interior through a horizon, the other side has to go out to infinity. Even if you wave your hands and say we're talking about some "Kruskal-like" geometry in a spacetime that doesn't have a conformal infinity, like FRW, then you have the problem of how to separate the interiors--because the interior of Kruskal doesn't stop at any finite point, it extends all the way to infinity in Kruskal coordinates. So even if you try to "glue" two finite pieces of an exterior together, you can't separate the two interiors.

I don't see this. Take an exterior constant r hyperbola in a Kruskal and identify it with a similar constant r hyperbola in another Kruskal. The fact that you can't draw this on a flat piece of paper is irrelevant. One Kruskal is effectively only the 'right half', the other is the 'left half'. Also, ignore the white hole regions.

[edit: it might help if we could locate a professional reference on this. I cannot find any discussing this issue at all, let alone supporting the view that orbiting BH have only one singularity]

 

[PeterDonis]

Take an exterior constant r hyperbola in a Kruskal and identify it with a similar constant r hyperbola in another Kruskal.

Even if I accept for the sake of argument that this works for two separate BHs that are both "eternal" and never merge, I think it still doesn't work if they merge, and the merger case is the one we have been discussing.

I think we probably need some references that give results of appropriate numerical simulations (since there are no known exact solutions for what we're discussing). AFAIK what I have been describing is what numerical simulations say about mergers, but it's been quite a while since I looked at this.