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

[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.

 

[PAllen]

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.

I just added this to my last post, that we need references. But I have recently looked at merger simulation papers, and what I see is that they excise singular regions altogether, thus they have nothing to say about this.

 

[PAllen]

I found the following quote on this from Abraham Loeb, the Harvard astrophysicist (discussing black hole mergers):

"Existing simulations cut out completely the region around the
singularities by postulating that this region will not have observable effects (and
justifiably so within General Relativity alone)"

This is from: https://arxiv.org/abs/1805.05865

 

[PeterDonis]

what I see is that they excise singular regions altogether

Hm, well that's rather inconvenient...

 

[timmdeeg]

Perhaps this is of interest:

What happens to apparent horizons in a binary black hole merger?​

We resolve the fate of the two original apparent horizons during the head-on merger of two non-spinning black holes. We show that following the appearance of the outer common horizon and subsequent interpenetration of the original horizons, they continue to exist for a finite period of time before they are individually annihilated by unstable MOTSs. The inner common horizon vanishes in a similar, though independent, way. This completes the understanding of the analogue of the event horizon’s "pair of pants’’ diagram for the apparent horizon. Our result is facilitated by a new method for locating marginally outer trapped surfaces (MOTSs) based on a generalized shooting method. We also discuss the role played by the MOTS stability operator in discerning which among a multitude of MOTSs should be considered as black hole boundaries.

 

[PAllen]

While I can’t find an arxiv version of that paper, here are a whole series of free papers by the same authors on this topic:

https://arxiv.org/search/?query=Poo...acts=show&order=-announced_date_first&size=50

 

[PAllen]

This suggests that my initial proposed picture of apparent horizon behavior for a coalescing BH cluster was correct; and that there is a well defined way to resolve the “outermost” problem (of course, I couldn’t solve it myself) and that you clearly do not need exotic matter to have an apparent horizon inside another.

They do not appear to address the issue of singularities at all, but for the purposes of this thread, their results on horizons are sufficient. It is possible and even routine to have a BH defined by an apparent horizon inside a larger horizon. This would be expected, among other cases, if a stellar mass BH merged with a supermassive BH.

Now the question of a BH forming from collapse within a supermassive BH is a separate issue. For that, my guess is that this is not possible without exotic matter. There would be no way for this to happen “fast enough” without exotic matter.

 

[PeterDonis]

While I can’t find an arxiv version of that paper

It appears to be in the list that you linked to:

https://arxiv.org/abs/2104.10265

 

[PeterDonis]

you clearly do not need exotic matter to have an apparent horizon inside another.

I have refreshed my memory about why you do need exotic matter: the Raychaudhuri equation. The best known context in which this reasoning appears is the Penrose singularity theorems. Once you have a 2-sphere on which the expansion of the outgoing null normals is negative (the trapped surface condition), the Raychaudhuri equation says that, provided the energy conditions hold (i.e., no exotic matter), the expansion gets more and more negative, going to minus infinity (i.e., reaching a singularity) in a finite amount of affine parameter along any timelike or null worldline pointing in the future direction from the initial trapped 2-sphere. This means it is impossible, provided the energy conditions hold, for the 2-spheres to "untrap" themselves (have the expansion of the outgoing null normals become positive again), which is what would be necessary to have one apparent horizon (marginally outer trapped surface) inside another.

Since this is a known mathematical theorem that makes no assumptions about symmetry, i.e., it holds for any curved spacetime provided the energy conditions are satisfied, if the paper appears to be saying something different, either the paper is wrong in that respect or something is being misinterpreted about what it says.

I haven't had a chance to read the paper in detail yet; I'll make further comments once I have.

 

[PAllen]

I think this paper https://arxiv.org/abs/1903.05626 describes their methodology well. One must distinguish between trapped surface, marginally outer trapped surface (MOTS), and outermost MOTS. The latter is the apparent horizon, and in BH merger, there is never more than one. What they argue is the utility and stability of MOTS, which allow identification and analysis of merger of BH within an apparent horizon. That was the problem I was trying to solve - how to identify the BH in a merging cluster after they are all inside both event and apparent horizon, but still far apart based on expectation from a little earlier. I believe this paper’s approach neatly solves that problem, allowing one to discuss a BH within a BH in a meaningful way.

On the singularity collision/merger issue, I cannot find any literature. I have a question posed to a world renowned GR expert who is willing to very occasionally answer my questions - I have not heard back yet.

 

[PeterDonis]

The latter is the apparent horizon, and in BH merger, there is never more than one.

As I read the paper, while this is true, the one apparent horizon (outermost MOTS) can have spacelike segments, which means it can appear to "go backward in time" in certain frames. So its shape can be more complicated than the simple "legs of trousers" image would suggest.

allowing one to discuss a BH within a BH in a meaningful way

If one accepts that the unstable ("negative eigenvalue", in the paper's terminology) MOTSs are physically meaningful, yes. I'm not sure what that would entail; in particular, I'm not sure how frame-dependent the unstable MOTSs are. (The fact that the paper claims an infinite number of them can be found in the interior of a Schwarzschild black hole indicates to me that they must be strongly frame-dependent in some way, since in all of the standard coordinate charts I'm aware of on this spacetime, the only MOTS is at the event horizon.)

 

[PeterDonis]

On the singularity collision/merger issue, I cannot find any literature. I have a question posed to a world renowned GR expert who is willing to very occasionally answer my questions - I have not heard back yet.

I'll be interested to hear if you get a response.

 

[PAllen]

If one accepts that the unstable ("negative eigenvalue", in the paper's terminology) MOTSs are physically meaningful, yes. I'm not sure what that would entail; in particular, I'm not sure how frame-dependent the unstable MOTSs are. (The fact that the paper claims an infinite number of them can be found in the interior of a Schwarzschild black hole indicates to me that they must be strongly frame-dependent in some way, since in all of the standard coordinate charts I'm aware of on this spacetime, the only MOTS is at the event horizon.)

That’s not my reading. They identify 3 stable MOTS, among possibly infinite unstable ones. They correspond naturally to the outer apparent horizon and two ‘generalized’ apparent horizon of the individual BH inside the outer one. See discussion on page 4 and fig. 3, especially. Also, other papers in this series give more detailed expositions, and provide more intuitive pictures.

 

[PeterDonis]

They identify 3 stable MOTS, among possibly infinite unstable ones. They correspond naturally to the outer apparent horizon and two ‘generalized’ apparent horizon of the individual BH inside the outer one.

As I read it, the "3" are really all part of one surface in spacetime, just with spacelike segments that "go back in time" in some frames. But I am still digesting the paper, and I have not looked at any of the others in the series, so I am probably missing a lot of details.

 

[timmdeeg]

Please allow me a few simple questions with regard to the meaning of "apparent horizon".

I understand that a apparent horizon is a trapped region. Does such a region have no singularity? Or only under certain conditions?

Let's consider the situation with the coalescing BH cluster inside an Outer Apparent Horizon.

Can photons between the cluster BH move in all directions including "upwards" to the OAH? In case yes, do they reach the OAH and are "frozen" then?

Now let's consider that all BH inside did coalesce. What emerges from that? A small BH with an apparent horizon inside the outer one?
Is there no scenario ending up with one big BH having one event horizon and one singularity?

 

[PAllen]

Please allow me a few simple questions with regard to the meaning of "apparent horizon".

I understand that a apparent horizon is a trapped region. Does such a region have no singularity? Or only under certain conditions?

Let's consider the situation with the coalescing BH cluster inside an Outer Apparent Horizon.

Can photons between the cluster BH move in all directions including "upwards" to the OAH? In case yes, do they reach the OAH and are "frozen" then?

Generally speaking, I think the answer is no, they can’t make progress towards the overall apparent horizon.

Now let's consider that all BH inside did coalesce. What emerges from that? A small BH with an apparent horizon inside the outer one?
Is there no scenario ending up with one big BH having one event horizon and one singularity?

I think one horizon left is definitely the end result. One of the dynamics discussed in the referenced papers is the role of unstable Mots annihilating interior stable Mots. So the end result is, indeed, one outermost Mots, the apparent horizon, and this will eventually coincide with the event horizon if nothing else happens to the big BH in its future.

As to the singularity dynamics, I have not been able to find any literature on this. My intuition is that you get some topologically complicated singular region, while @PeterDonis is that there is really only ever one singularity in the future of everything inside all the BH. So far, neither of us has been able to justify our intuitions with references, though I have found references claiming that no one has simulated this yet.

 

[timmdeeg]

Generally speaking, I think the answer is no, they can’t make progress towards the overall apparent horizon.

Thanks. Which would mean that event horizon and apparent horizon have in common that inside r- and t-coordinate change roles. If in both cases the r-coordinate has the only choice to decrease then I wonder what $r=0$ would mean in case of the apparent horizon with a coalescing BH cluster inside.

Thanks for your thoughts regarding the other questions. I understand that questions which appear to be simple don't necessarily have simple answers as it is obvious in theses cases.

 

[PAllen]

Thanks. Which would mean that event horizon and apparent horizon have in common that inside r- and t-coordinate change roles. If in both cases the r-coordinate has the only choice to decrease then I wonder what $r=0$ would mean in case of the apparent horizon with a coalescing BH cluster inside.

Thanks for your thoughts regarding the other questions. I understand that questions which appear to be simple don't necessarily have simple answers as it is obvious in theses cases.

The phrase "r and t switching places" should be deleted from the internet and beyond. It is complete BS. Unfortunately, a more correct non-mathematical description requires more words and is more abstract. I'll try.

The exterior and interior Schwarzschild solutions are wholly separate coordinate patches, neither of which includes the horizon. For the interior patch, r and t are simply bad names for the coordinates. Better would be e.g. T and Z, respectively, with T being a timelike coordinate running from R to 0 (without ever reaching 0). Z is an axial coordinate, not a radial coordinate. While there are many very different ways to slice the interior into spatial slices by time, the ones used in the interior Schwarzschild patch have spatial hypersurfaces with topology of S2 X R. That is hyper-cylinders. Z is position along the axis of a hyper-cylinder. T, while a timelike coordinate, still gives the radius of the 2-sphere of the hyper-cylinder at time T (thus, the cylinders inherently shrink in radius as time advances). The "r=0" is not a point but a limit of vanishing radius hyper-cylinders of infinite extent.

The geometry of my proposed collapsing BH cluster, however, is much more complex than this. I could propose first that one consider a collapsing cluster of neutron stars. Then, while they are all still millions of miles apart, they are all within both an apparent horizon and an event horizon. In some vague, averaged way, the interior vacuum part would be like interior Schwarzschild, but the details would be very different. Then imagine replacing each neutron star with what was an (old) BH, and you have my scenario. My use of old is to suggest the the interior of each BH is vacuum for a Kruskal like spatial slice through the cluster as a whole. That is, it would not intersect any of the nonzero Ricci curvature present in the interior earlier in the history of each BH.

I am not sure how much this clears things up, or muddies the waters, but it is best I can do.

 

[PeterDonis]

I understand that a apparent horizon is a trapped region.

It's the boundary of a trapped region.

Does such a region have no singularity? Or only under certain conditions?

According to the Hawking-Penrose singularity theorems, if the energy conditions are satisfied in the trapped region, then it must contain a singularity. However, the singularity can be spacelike, like the one in Schwarzschild spacetime, so it can be to the future of everything else inside the trapped region, as in my description of the "pair of trousers" view of a black hole merger--in other words, there doesn't have to be a singularity inside each "leg" of the trousers, even though each "leg" is a trapped region. (As @PAllen has said, it does not appear that numerical simulations give any useful information about whether there actually is a singularity inside each "leg"--all we know from the singularity theorems is that there does not have to be, the singularity can be anywhere inside the entire trapped region.)

Can photons between the cluster BH move in all directions including "upwards" to the OAH?

Not if "the cluster BH" refers to the entire trapped region after the merger. Photons inside a trapped region can't move outward. That's what "trapped" means.

let's consider that all BH inside did coalesce. What emerges from that? A small BH with an apparent horizon inside the outer one?

As far as I can tell from the references given so far, the overall apparent horizon in this case is a "trousers" with many legs instead of just two; but the "joins" between the various "legs" can be spacelike, so they can appear to "go back in time" in certain frames, meaning that "snapshots" taken at various times in those frames could indeed show one outer apparent horizon with many inner ones inside it. But there would come a time, in any frame, when that would no longer be the case: all that would be left would be the single outer apparent horizon that bounds the entire cluster.

Is there no scenario ending up with one big BH having one event horizon and one singularity?

We can't make any statements about an event horizon because we might not even be talking about a spacetime that can have one--the spacetime might not even have a future null infinity (FRW spacetime does not, for example, so neither does the spacetime of our actual universe in our best current model). That is why we have been talking about apparent horizons instead.

As has been noted already, we have not found any references so far that give any useful information about the status of singularities in the cases under discussion.

 

[PAllen]

...

We can't make any statements about an event horizon because we might not even be talking about a spacetime that can have one--the spacetime might not even have a future null infinity (FRW spacetime does not, for example, so neither does the spacetime of our actual universe in our best current model). That is why we have been talking about apparent horizons instead.

...

I've kind of switched to discussing this embedded in asymptotically flat spacetime, so we can validly consider event horizons.

 

[PeterDonis]

Which would mean that event horizon and apparent horizon have in common that inside r- and t-coordinate change roles.

I agree with @PAllen that this idea should be eradicated from the Internet and beyond. All it does is confuse people.

I wonder what $r=0$ would mean in case of the apparent horizon with a coalescing BH cluster inside.

If by $r = 0$ you just mean "wherever the singularity is", then we've already commented on what we don't know about singularities in these scenarios at this point.

If by $r = 0$ you are talking about an actual coordinate in an actual coordinate chart, remember we are not talking about spherically symmetric spacetimes here, so there is no "r-coordinate" in that sense. We don't know any exact solutions for these scenarios so we're depending on numerical simulations, and coordinate charts in those can be quite complicated.

 

[PeterDonis]

I've kind of switched to discussing this embedded in asymptotically flat spacetime

Looking at the paper you referenced, I see that it does use asymptotically flat initial data (the initial 3-surface is conformally flat and the conformal factor goes to $1$ as the distance from the finite region containing the initial holes goes to infinity).

However, I still think it's worth keeping in mind that asymptotic flatness might not apply in our actual universe.

 

[timmdeeg]

The phrase "r and t switching places" should be deleted from the internet and beyond. It is complete BS. Unfortunately, a more correct non-mathematical description requires more words and is more abstract. I'll try.

The exterior and interior Schwarzschild solutions are wholly separate coordinate patches, neither of which includes the horizon. For the interior patch, r and t are simply bad names for the coordinates. Better would be e.g. T and Z, respectively, with T being a timelike coordinate running from R to 0 (without ever reaching 0). Z is an axial coordinate, not a radial coordinate. While there are many very different ways to slice the interior into spatial slices by time, the ones used in the interior Schwarzschild patch have spatial hypersurfaces with topology of S2 X R. That is hyper-cylinders. Z is position along the axis of a hyper-cylinder. T, while a timelike coordinate, still gives the radius of the 2-sphere of the hyper-cylinder at time T (thus, the cylinders inherently shrink in radius as time advances). The "r=0" is not a point but a limit of vanishing radius hyper-cylinders of infinite extent.

The geometry of my proposed collapsing BH cluster, however, is much more complex than this. I could propose first that one consider a collapsing cluster of neutron stars. Then, while they are all still millions of miles apart, they are all within both an apparent horizon and an event horizon. In some vague, averaged way, the interior vacuum part would be like interior Schwarzschild, but the details would be very different. Then imagine replacing each neutron star with what was an (old) BH, and you have my scenario. My use of old is to suggest the the interior of each BH is vacuum for a Kruskal like spatial slice through the cluster as a whole. That is, it would not intersect any of the nonzero Ricci curvature present in the interior earlier in the history of each BH.

I am not sure how much this clears things up, or muddies the waters, but it is best I can do.

Thanks for this clarification.
When I said "inside r- and t-coordinate change roles" I had in mind "Exploring Black Holes", Taylor&Wheeler Page 3-10, where they say "Inside there is an interchange of the character of the t-coordinate and r-coordinate." And indeed looking at the metric the signs of radial- and time-part are changing. I understand that such that the r-coordinate behaves timelike in the sense that is has only one direction, that to the future which means towards the singularity. Admittedly this reasoning is very simple and thus no alternative to your more technical explanation.

You mentioned (old) BH probably in the sense of eternal BH. Would your scenario be much different in case we talk about BH due to gravitational collapse whereby it is assumed that there is stress-energy (as a consequence of avoiding the singularity)?

 

[timmdeeg]

According to the Hawking-Penrose singularity theorems, if the energy conditions are satisfied in the trapped region, then it must contain a singularity. However, the singularity can be spacelike, like the one in Schwarzschild spacetime, so it can be to the future of everything else inside the trapped region, as in my description of the "pair of trousers" view of a black hole merger--in other words, there doesn't have to be a singularity inside each "leg" of the trousers, even though each "leg" is a trapped region. (As @PAllen has said, it does not appear that numerical simulations give any useful information about whether there actually is a singularity inside each "leg"--all we know from the singularity theorems is that there does not have to be, the singularity can be anywhere inside the entire trapped region.)

Ok. It's interesting that the question "is a singularity inside each "leg"" isn't simply answerable by "yes". Intuitively one could think that anything inside the BH "leg" falls towards its singularity and reaches it much before the leg reaches the "waist" in the far future.

Not if "the cluster BH" refers to the entire trapped region after the merger. Photons inside a trapped region can't move outward. That's what "trapped" means.

Ok, understand.

As far as I can tell from the references given so far, the overall apparent horizon in this case is a "trousers" with many legs instead of just two; but the "joins" between the various "legs" can be spacelike, so they can appear to "go back in time" in certain frames, meaning that "snapshots" taken at various times in those frames could indeed show one outer apparent horizon with many inner ones inside it. But there would come a time, in any frame, when that would no longer be the case: all that would be left would be the single outer apparent horizon that bounds the entire cluster.We can't make any statements about an event horizon because we might not even be talking about a spacetime that can have one--the spacetime might not even have a future null infinity (FRW spacetime does not, for example, so neither does the spacetime of our actual universe in our best current model). That is why we have been talking about apparent horizons instead.

As has been noted already, we have not found any references so far that give any useful information about the status of singularities in the cases under discussion.

Thank you so much for your explanations.

 

[timmdeeg]

I agree with @PAllen that this idea should be eradicated from the Internet and beyond. All it does is confuse people.

Please see my comment in #92

If by $r = 0$ you are talking about an actual coordinate in an actual coordinate chart, remember we are not talking about spherically symmetric spacetimes here, so there is no "r-coordinate" in that sense. We don't know any exact solutions for these scenarios so we're depending on numerical simulations, and coordinate charts in those can be quite complicated.

Yes, understand.

 

[PeterDonis]

When I said "inside r- and t-coordinate change roles" I had in mind "Exploring Black Holes", Taylor&Wheeler Page 3-10, where they say "Inside there is an interchange of the character of the t-coordinate and r-coordinate."

As this is stated, it is an artifact of the particular coordinate chart they are using, Schwarzschild coordinates. There are other charts, such as Painleve or Eddington-Finkelstein, where $r$ as a coordinate remains spacelike inside the horizon.

An invariant way of stating what Taylor & Wheeler probably have in mind is that, inside the horizon, all timelike worldlines have $r$ (the areal radius, not the coordinate) decreasing with proper time along the worldline.

BH due to gravitational collapse whereby it is assumed that there is stress-energy (as a consequence of avoiding the singularity)?

If by "BH due to gravitational collapse" you mean a model like the Oppenheimer-Snyder model, where there is a region containing stress-energy (describing the collapsing object that forms the hole), there is still a singularity; having collapsing matter does not "avoid" it. I'm not aware of any classical GR model containing a black hole that does not have a singularity (although as @PAllen has pointed out, numerical simulations tend to leave out the part of the spacetime that contains the singularity).

Intuitively one could think that anything inside the BH "leg" falls towards its singularity and reaches it much before the leg reaches the "waist" in the far future.

Intuition is a very poor guide in cases like this. In the "trousers" model, "time" inside the trousers is extremely distorted, in the sense that the "length" of worldlines in the model can be much, much longer than the proper time elapsed along them. So, for example, a timelike worldline that crosses the horizon far down one of the "legs" can still end on the singularity up at the top of the trousers, even though very little proper time elapses along the worldline between those two points, and even though much more proper time elapses along a worldline that stays outside the trousers between the "heights" in the model at which the first worldline falls into the "leg" and when that "leg" meets the "waist" of the trousers.

Note also that, even in a single "cylinder" diagram of a single black hole (for example, a diagram drawn in Eddington-Finkelstein coordinates), where it seems like the singularity is "at the center of the cylinder", the singularity is still spacelike and is to the future of everything inside the horizon, so two timelike worldlines that fall through the horizon at very different "outside" times still hit the singularity "at the same time" viewed from inside the horizon. The fact that the singularity appears as a vertical line at the center of the cylinder in this diagram is, in that respect, highly misleading, since the singularity is not a place in space but a moment of time. A Kruskal or Penrose diagram gives a much less misleading picture of what is actually going on.

 

[timmdeeg]

As this is stated, it is an artifact of the particular coordinate chart they are using, Schwarzschild coordinates. There are other charts, such as Painleve or Eddington-Finkelstein, where $r$ as a coordinate remains spacelike inside the horizon.

Yes the whole book (Copyright 2000) is about Schwarzschild coordinates and ends with a very short chapter about the Friedmann Universe. I like it very much.

An invariant way of stating what Taylor & Wheeler probably have in mind is that, inside the horizon, all timelike worldlines have $r$ (the areal radius, not the coordinate) decreasing with proper time along the worldline.

Thanks, very helpful, this confirms that invariant descriptions should be preferred. Shouldn't that include Null worldlines too?

If by "BH due to gravitational collapse" you mean a model like the Oppenheimer-Snyder model, where there is a region containing stress-energy (describing the collapsing object that forms the hole), there is still a singularity; having collapsing matter does not "avoid" it. I'm not aware of any classical GR model containing a black hole that does not have a singularity (although as @PAllen has pointed out, numerical simulations tend to leave out the part of the spacetime that contains the singularity).

Yes I meant the Oppenheimer-Snyder model. Thanks for clarifying the singularity issue.

Intuition is a very poor guide in cases like this. ... So, for example, a timelike worldline that crosses the horizon far down one of the "legs" can still end on the singularity up at the top of the trousers, even though very little proper time elapses along the worldline between those two points, and even though much more proper time elapses along a worldline that stays outside the trousers between the "heights" in the model at which the first worldline falls into the "leg" and when that "leg" meets the "waist" of the trousers.

Ahh this is surprising and good to know, then your earlier comments make sense.

Note also that, even in a single "cylinder" diagram of a single black hole (for example, a diagram drawn in Eddington-Finkelstein coordinates), where it seems like the singularity is "at the center of the cylinder", the singularity is still spacelike and is to the future of everything inside the horizon, so two timelike worldlines that fall through the horizon at very different "outside" times still hit the singularity "at the same time" viewed from inside the horizon. The fact that the singularity appears as a vertical line at the center of the cylinder in this diagram is, in that respect, highly misleading, since the singularity is not a place in space but a moment of time. A Kruskal or Penrose diagram gives a much less misleading picture of what is actually going on.

Regarding the spacelike singularity I might have a misconception. What does that really mean? To my understanding two points in space are spacelike separated in case one is not within the past light cone of the other (regarding the future light cone vice versa). But how can a point in time (the singularity) which (as I read sometimes) is not even part of the manifold be spacelike?

As to the "the singularity is "at the center of the cylinder"" are the points on this vertical line in Eddington-Finkelstein diagrams representing the points in time at which infalling objects are reaching the singularity?

 

[PAllen]

You mentioned (old) BH probably in the sense of eternal BH. Would your scenario be much different in case we talk about BH due to gravitational collapse whereby it is assumed that there is stress-energy (as a consequence of avoiding the singularity)?

By old, I definitely do not mean eternal. I am only interested in BH from collapse, which lack the white hole region and also the wormhole to another universal sheet (which is part of the full Kruskal geometry, but is not present at all in a BH from collapse).

From an external observer point of view, there are several 'age' criteria that can be applied to a BH.

1) The singularity is no linger in the causal future of an external observer. Thus a spacelike slice can reach the singularity. This is a fully classical criterion.
2) The last (non-hawking) photon, of any frequency whatsoever, that will ever be received from the BH is received (per a thermodynamic emission model discussed on pp. 872-3 of MTW). This is 'almost classical'.

Both of these times arrive quite fast for an external observer (milliseconds to days at most). Of interest for quantum treatments are the scramble time and the Page time. These arrive much later. I will not define these here, except to say I am thinking of a BH around this old, without necessarily caring whether the quantum modeling behind their definitions is actually true. My goal is a state where the separate BH have had their singularity outside of exterior observer causal future for a very long time.

To my intuition, this makes it hard to imagine anything other than that well before coalescence of the BH cluster, a spacelike slice will hit a singularity in each separate BH (leg). [edited for unintended interpretation pointed out by @PeterDonis ]

@PeterDonis has a different intuition. He correctly notes that because there is only one absolute event horizon (of multi-legged pants shape), the singularity theorems only guarantee one singularity. However, they say almost nothing about its nature or nearby geometry or topology, nor do they preclude multiple singularities. They just say there must be at least one of some type.

So without some very careful simulation or detailed analysis (which neither of us can find), questions about the singularities appear unresolvable.

 

[PAllen]

Regarding the spacelike singularity I might have a misconception. What does that really mean? To my understanding two points in space are spacelike separated in case one is not within the past light cone of the other (regarding the future light cone vice versa). But how can a point in time (the singularity) which (as I read sometimes) is not even part of the manifold be spacelike?

It's not a 'point' in time. For an ideal Schwarzschild BH, it is a missing spacelike line bounded all around by ever decreasing radius hypercylinders. The proper length of these asymptotically bounding hypercylinders is infinite.

Also note that for any 3x1 coordinates system, a 'moment in time' is a spacelike hypersurface.

 

[PeterDonis]

Shouldn't that include Null worldlines too?

Yes.

What does that really mean?

It means the singularity is not a place in space, it's a moment of time.

To my understanding two points in space are spacelike separated in case one is not within the past light cone of the other

Spacetime, not space. "Space" doesn't even have any light cones.

how can a point in time (the singularity) which (as I read sometimes) is not even part of the manifold be spacelike?

Strictly speaking, yes, the singularity is not part of the manifold; but it can still be viewed as the limiting case of surfaces of constant $r$ as $r \rightarrow 0$. Since surfaces of constant $r$ inside the horizon are spacelike, so is their limit as $r \rightarrow 0$.

are the points on this vertical line in Eddington-Finkelstein diagrams representing the points in time at which infalling objects are reaching the singularity?

No. Inside the cylinder, vertical lines are spacelike, not timelike; they represent moments of time, not places in space. So all points on the singularity line at the center are at the same moment of time, not different ones.

Different points on the vertical lines inside the horizon represent different points in space inside the horizon, at those moments of time. So different points on the singularity line at the center represent arriving at the singularity (moment of time) at different points in space.

 

[PeterDonis]

To my intuition, this makes it hard to imagine that well before coalescence of the BH cluster, a spacelike slice will hit a singularity in each separate BH (leg).

@PeterDonis has a different intuition.

My personal intuition is actually similar to what you describe here as regards the singularity; I think I have already said that, in my view, the singularity is "at the top of the trousers" and that there isn't one inside each leg, or even inside the "waist" of the trousers after the legs join, until you get to the very top of the trousers.

He correctly notes that because there is only one absolute event horizon (of multi-legged pants shape), the singularity theorems only guarantee one singularity. However, they say almost nothing about its nature or nearby geometry or topology, nor do they preclude multiple singularities. They just say there must be at least one of some type.

My point with those statements was simply that, whatever our intuitions may say, we don't actually know much at all because of the lack of exact solutions, the limitations of the current numerical simulations, and the limited nature of what the singularity theorems tell us.