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

[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.

 

[PeterDonis]

1) The singularity is no linger in the causal future of an external observer. Thus a spacelike slice can reach the singularity.

This criterion as you state it doesn't work, because the singularity is not a single point but a spacelike line, which, as you note, is infinitely long. That means that, at any event whatever on the external observer's worldline, some portion of the singularity will be within the future light cone, and the rest of the singularity will be spacelike separated. The only thing there won't be is any portion of the singularity in the past light cone of any event on the external observer's worldline.

I think the criterion you meant to state is that the event at which the surface of the collapsing object that formed the black hole hits the singularity is no longer in the causal future of an external observer. That criterion makes sense since that particular event on the singularity will have a well-defined past light cone, and any external observer's worldline will exit that past light cone at some point. There will still be points on the singularity that are within the external observer's future light cone after that, but none of them will be events where any of the collapsing matter that formed the hole hits the singularity.

 

[PAllen]

I said
"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)."

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.

My wording was poor. My intuition is the opposite of yours. I should have said either:

"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)."

or

"To my intuition, this makes it hard to imagine that well before coalescence of the BH cluster, a spacelike slice will not hit a singularity in each separate BH (leg)."

I have edited my original post to clarify this.

 

[PAllen]

This criterion as you state it doesn't work, because the singularity is not a single point but a spacelike line, which, as you note, is infinitely long. That means that, at any event whatever on the external observer's worldline, some portion of the singularity will be within the future light cone, and the rest of the singularity will be spacelike separated. The only thing there won't be is any portion of the singularity in the past light cone of any event on the external observer's worldline.

This is just a minor imprecision. If you track a stationary external observer, all history to the past some event will have all of the singularity in their future light cone (for a collapse singularity; this is not true for a Kruskal singularity). After this time, more and more of the singularity will be outside the causal future.

Of course, what you say below is also true, by my intended emphasis is captured by the above statement. This is what makes my intuition suggest a spacelike surface through all the legs before merger can, indeed, reach a singularity in each leg.

I think the criterion you meant to state is that the event at which the surface of the collapsing object that formed the black hole hits the singularity is no longer in the causal future of an external observer. That criterion makes sense since that particular event on the singularity will have a well-defined past light cone, and any external observer's worldline will exit that past light cone at some point. There will still be points on the singularity that are within the external observer's future light cone after that, but none of them will be events where any of the collapsing matter that formed the hole hits the singularity.

 

[PeterDonis]

My intuition is the opposite of yours.

Ah, ok. Thanks for clarifying.

 

[PeterDonis]

If you track a stationary external observer, all history to the past some event will have all of the singularity in their future light cone

Yes, and the point at which this stops being true is the point at which the external observer's worldline exits the past light cone of the event where the surface of the collapsing matter reaches the singularity. The latter event is the "corner" where the $r = 0$ locus stops being timelike (the center of the collapsing matter) and starts being spacelike (the singularity).

(for a collapse singularity; this is not true for a Kruskal singularity).

Yes, my previous comment about some portion of the singularity always being spacelike separated from the observer was only true for a Kruskal singularity, but you're right, we're talking about the collapse case here.