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

[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.

 

[PeterDonis]

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

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

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

 

[PAllen]

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

 

[PeterDonis]

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

I phrased my point poorly. Yes, of course you can define a thingie with 42 holes in it and call it a "spacelike surface" as long as its tangent vectors at every point are spacelike. I wasn't trying to claim that that is impossible.

Let me try to rephrase my point as as question: what do you think the topology of the overall manifold (spacetime) is in the "42 holes merge" case? I think it is still R2 x S2, the same as Schwarzschild spacetime. And I think that with that topology, there can only be one singularity, although we can of course choose spacelike surfaces (if we allow such surfaces to be geodesically incomplete) that are "punctured" by that one singularity multiple times, if we are willing to accept sufficient distortion in the implicit coordinate chart we are using.

If you agree that the topology of the manifold is still R2 x S2 in the "42 holes merge" case, then how can there be more than one singularity?

Or if, alternatively, you think the topology is something else, what do you think it is?

 

[PAllen]

I think there are two separate issues here.

I phrased my point poorly. Yes, of course you can define a thingie with 42 holes in it and call it a "spacelike surface" as long as its tangent vectors at every point are spacelike. I wasn't trying to claim that that is impossible.

Let me try to rephrase my point as as question: what do you think the topology of the overall manifold (spacetime) is in the "42 holes merge" case? I think it is still R2 x S2, the same as Schwarzschild spacetime. And I think that with that topology, there can only be one singularity, although we can of course choose spacelike surfaces (if we allow such surfaces to be geodesically incomplete) that are "punctured" by that one singularity multiple times, if we are willing to accept sufficient distortion in the implicit coordinate chart we are using.

I'll address the topology question next. Here I will say that I think slices that are punctured multiple times are the norm, in the following sense (and this I believe is true even in a model with one singularity - that it must bend deep into each leg). Consider an event on an external world line far in the future of the event where (all the singularities / the singularity - pick your model) stops being all in the causal future. Then, the only spacelike surfaces intersecting that world line at that point, that don't have multiple punctures, are ones that go very close to the past light cone. If one could pick a measure, I would say 'most' spacelike slices have multiple punctures.

If you agree that the topology of the manifold is still R2 x S2 in the "42 holes merge" case, then how can there be more than one singularity?

Or if, alternatively, you think the topology is something else, what do you think it is?

First, note that all three of the following have very different Penrose diagrams:

- Kruskal manifold
- 2 quadrants of the Kruskal manifold
- Oppenheimer Snyder collapse

On the first two, there seems little debate that you have R2 X S2 (depending a bit on definition of two quadrants).

On the last, there is at least a little dispute among experts due to an initial Cauchy surface being essentially indistinguishable from one leading to a neutron star, combined with a Geroch theorem that evolution from a Cauchy surface without violating energy conditions cannot change topology. Those favoring this argument claim that somehow, OS manifold must be R4. The majority argues there are technical loopholes in Geroch's theorem, and the OS manifold is R2 X S2. To me, this follows (but not at all obviously) from the most accepted Penrose diagram for OS manifold.

So for multiple BH, with or without merger, I claim the topology is certainly not R2 X S2. What it is, I have no real idea. As a simplistic guess, I can throw out R4 - L - L - L ... where L is a line. There are well known arguments for why R4 - L is the same as R2 X S2, but what you would call the multiple subtraction topology, I have no idea.

I still cannot find even one paper that discusses complete topology for even 2 BH merging. I did find a paper that gives a robust proof (weak assumptions) that the exterior topology of N BH is 'as simple as possible' given the excsision of the BH at their horizons. Interesting, but not the least bit surprising.

 

[PAllen]

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

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

 

[PeterDonis]

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

Hm, yes, I had forgotten about that issue.

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

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

 

[PeterDonis]

I did find the following from Hawking and Ellis

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

 

[PeterDonis]

I did find the following from Hawking and Ellis

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

 

[Nugatory]

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

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

 

[timmdeeg]

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

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