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

[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.