# Graphical example of BH formation by PAllen

TrickyDicky
No, it isn't. Once again, you have things backwards. The EH is never "defined" in terms of its being a coordinate singularity in any chart. The fact of its being a coordinate singularity, as I said before, is *derived* from the underlying geometry plus the definition of the chart.
I'm not talking about any logical chain or causal issue, so it can't be backwards or not backwards, what I'm saying is independent of the "underlying geometry". Certainly in curved manifolds one can refer to the limits of a certain chart without reference to the specific underlying geometry, it is a fact that curved smooth manifolds in general can't be covered completely by a single chart.

I've never said it does. I've always said the EH "belongs" to the underlying geometry. Its existence and properties are independent of any chart.
As I said I'm not considering your fuzzy underlying geometry concept here
It just happens that charts are needed in differential geometry, at least differentiable manifolds are defined as those equipped with an equivalence class of atlases (collections of local charts) whose transition maps are all differentiable.

There is no coordinate singularity in K-S coordinates.
r=2GM is not defined in K-S coordinates, do you dispute that?

So what? That's a problem with the Schwarzschild coordinates, not with the EH.
No, I'm talking about the transition map between SC and K-S, so it is a problem also with the K-S space (the whole 4-regions) since they include the outside region.

Yes, there are. K-S coordinates do, so do ingoing Eddington-Finkelstein and Painleve. None of those charts are singular at the EH, so they "cover the transition" just fine.
See above.

You appear to believe that, if *any* chart is singular at the EH, *all* charts are "singular" there, because the transformation from the singular chart to any other chart must be singular there. That's wrong.
No, I don't believe that at all.

r=2GM is not defined in K-S coordinates, do you dispute that?
I certainly dispute it. Any point with coordinates U=V corresponds to r=2GM. There is nothing singular about the KS coords here, or metric expressed in KS coords here, or the curvature tensor expressed in KS coords here. The only false issue is that the tranform from KS to another chart (SC) that is singular here, is singular. The transform to SC is singular only because the SC coordinates are singular here.

This is exactly analogous to rectilinear coordinates versus polar coordinates in a flat 2-D Euclidean plane. Because polar coordinates have coordinate singlularity, transform between them and rectilinear coords at this point is also singular. Does this say there is something funny about the point on a plane you pick for the pole? Or is only a feature (not even a defect) of this particular coordinate choice.?

[edit: To be specific, any KS coordinates of form (V,U,θ,$\varphi$)=(k,k,θ,$\varphi$) corresponds to r=2GM, and the metric here is simply:

diag( -16 G^2 M^2/e, 16 G^2 M^2/e, 4G^2 M^2, 4 G^2 M^2 sin^2(θ))
]

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Mentor
r=2GM is not defined in K-S coordinates, do you dispute that?

http://en.wikipedia.org/wiki/Kruskal–Szekeres_coordinates

In K-S coordinates, r is a function of the K-S U and V (those are the Wiki page names for the spacelike and timelike non-angular K-S coordinates), given implicitly by

$$V^2 - U^2 = \left( 1 - \frac{r}{2M} \right) e^{r / 2M}$$

Which makes it obvious that if r = 2M, V = +/- U, so the full "horizon" in the maximally extended spacetime is two intersecting 45-degree lines in the standard K-S diagram. The horizon we've been talking about, the future horizon for observers in Region I (the "right wedge" of the diagram) is the line V = U with U, V > 0. No coordinate singularity anywhere.

No, I don't believe that at all.

Then why do you consider the coordinate singularity at the horizon in Schwarzschild coordinates to somehow indicate an issue with other charts which are not singular at the horizon?

Mentor
As I said I'm not considering your fuzzy underlying geometry concept here

So you consider the concept of spacetime having a geometry to be "fuzzy"? Hmm.

TrickyDicky
Yes.*

You are right, sorry, I managed to confuse myself, and typing in a rush didn't help.
I was thinking about the transformation from KS to SC.

Then why do you consider the coordinate singularity at the horizon in Schwarzschild coordinates to somehow indicate an issue with other charts which are not singular at the horizon?
In KS coordinates I agree there is no coordinate singularity, because the true singularity allows us to cover the whole spacetime with one chart of coordinates.
But again, there seems to be certain agreement on considering this space unphysical, in which case it shouldn't be used to demonstrate the physicality-coordinate-independence of EHs.

TrickyDicky
PAllen said:
Because polar coordinates have coordinate singlularity, transform between them and rectilinear coords at this point is also singular. Does this say there is something funny about the point on a plane you pick for the pole? Or is only a feature (not even a defect) of this particular coordinate choice.?
That has been my point the whole time wrt EHs, that its only a coordinate dependent feature, and there's nothing that differentiates it from other points.

That has been my point the whole time wrt EHs, that its only a coordinate dependent feature, and there's nothing that differentiates it from other points.

But this I disagree with. A singular point in some coordinates is only a feature of those coordinates. An EH can be defined without reference to any coordinates, and computed in any coordinates including those with no coordinate singularity there. The definition of EH is purely in terms of the boundary between events from which null paths escape to future null infinity and those from which they don't. I emphasize: no coordinates at all are needed to apply this definition.

[Edit: To clarify this in relation to my discussion with Peter Donis: Peter was emphasizing that this distinguishes BH Event Horizons from Rindler Horizon, in that the latter have no definition as a function of spacetime as a whole - they are defined in reference to a particular world line. I was emphasizing that any horizon (EH or Rindler) constitutes a causal boundary only for some specific class of observers (observer being defined by a timelike world line). As I see it, we had no real disagreement; we were emphasizing different aspects of horizons.]

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Mentor
In KS coordinates I agree there is no coordinate singularity

Ok, good.

But again, there seems to be certain agreement on considering this space unphysical, in which case it shouldn't be used to demonstrate the physicality-coordinate-independence of EHs.

The *entire* manifold covered by the K-S chart is unphysical, yes, because it contains the "white hole" region and a second asymptotically flat region (Regions III and IV as they are usually labeled on the K-S chart). But there is nothing physically unreasonable about using a *portion* of that manifold in a more realistic model. That's what the Oppenheimer-Snyder model does: it uses a portion of Regions I and II of the maximally extended Schwarzschild spacetime, joined to a collapsing FRW spacetime. And this model still contains an event horizon--a portion of the future horizon that forms the boundary between Regions I and II in the vacuum portion of the spacetime. The "collapsing star cluster" model PAllen has been discussing is the same kind of thing.

Gold Member
You can say it, but it would be false. Let's say (at time t0 in some reasonable chosen coordinates) the shell is at r=(1+δ)R, R being the SC radius. I claim there is some r0 > 0 and r0 < R, such that light emitted at (t0,r0) is never received by a distant observer, while light emitted at (t0,r0+ε) is eventually received by a distant observer. This is or isn't true. I claim it is. If it is, how would you describe this other than saying at t0 the event horizon is a 2-sphere at r0 plus a 2-sphere around each little BH on the shell?
Light can be blocked say by a rock. So in this case I say that light does not reach observer because it is blocked at later time and not because it is stuck (or goes backwards) right at r0.

Light can be blocked say by a rock. So in this case I say that light does not reach observer because it is blocked at later time and not because it is stuck (or goes backwards) right at r0.

And who arranges the rocks? Why not just admit:

1) You think what GR predicts violates your sense of plausibility.

2) As a result you think the GR is incorrect in this scenario.

That would be honorable rather than claiming that the top experts in GR (not me, others who I study) are misinterpreting its equations. Further, except for the details of where it breaks down, you would then be in good company - many serious physicists think GR breaks down in the vicinity of horizons, and many more near the singularity. But that is different from disputing what GR predicts.

(Personally, I think, at least macroscopically, GR only breaks down near the singularity, and the horizon behavior you don't like actually occurs. I think, microscopically, a horizon may not be a true horizon, but macroscopically it behaves as GR predicts. Horizon behavior is likely to be subject to observational test within the next decades. )

Max™
Fascinating thought experiment, but my head jumped to a somewhat altered version.

Set up the same shell of stars, but instead of a slow collapse, have them rotating fast enough to prevent the formation of a horizon, and then apply the brakes until a horizon forms.

Aimless
(Personally, I think, at least macroscopically, GR only breaks down near the singularity, and the horizon behavior you don't like actually occurs. I think, microscopically, a horizon may not be a true horizon, but macroscopically it behaves as GR predicts. Horizon behavior is likely to be subject to observational test within the next decades. )

Event horizons are mathematically nice in that they can be described independent of choice of coordinate chart or observer, but in my opinion I think it's wrong to focus on the concept of the event horizon for asking whether or not GR breaks down in those regions. There are a couple of reasons for this: the first is that the true event horizon can only be observed at infinity, so any physical observer will never (by definition) see one. The second is that if all physical black holes eventually decay then there are no event horizons.

Based on this, then, I feel that it is more physically appropriate to focus on apparent horizons rather than event horizons. However, there is nothing intrinsically different about the apparent horizon of an observer due to a black hole spacetime versus, say, a Rindler horizon. Nor is there a reason to suspect that GR breaks down in the vicinity of such a horizon. (The singularity is another matter entirely.)

That said, I am highly interested in how microscale physics affects the behavior of horizons; specifically, the idea that fluctuations might give rise to a "quantum width" for observed horizons (although this is unlikely to be measurable for any macro-BH).

Mentor
The second is that if all physical black holes eventually decay then there are no event horizons.

This is not quite true; even if all BHs eventually evaporate, they still have interior regions while they exist. That is, there are still regions of spacetime that can't send light signals to future null infinity--the interiors of BHs between the time that they form and the time that they finally evaporate. So the spacetime still contains EHs, the boundaries of these regions.

The difference with evaporating BHs is that their EHs do not connect to future timelike infinity, whereas if BHs last forever, they do.

Here are a couple of Penrose diagrams illustrating what I have said. First, one of a BH that doesn't evaporate:

The "corner" at the top, where the horizon, r = 0, and r = infinity lines all meet, is future timelike infinity. Second, a BH that evaporates away completely:

http://en.wikipedia.org/wiki/File:Black_hole_Penrose.png

Notice that here the wavy horizontal line, indicating the horizon, does *not* reach the "corner" at the very top, which is future timelike infinity; instead there is a vertical segment above where the horizon ends, which is the r = 0 worldline to the future of the BH's final evaporation.

Notice also that in both cases, there is still a region inside a horizon, which can't send light signals to future null infinity (signals sent in this region hit the singularity instead).

Aimless
This is not quite true; even if all BHs eventually evaporate, they still have interior regions while they exist. That is, there are still regions of spacetime that can't send light signals to future null infinity--the interiors of BHs between the time that they form and the time that they finally evaporate. So the spacetime still contains EHs, the boundaries of these regions.

The difference with evaporating BHs is that their EHs do not connect to future timelike infinity, whereas if BHs last forever, they do.

From a strictly GR standpoint, you are correct; this is the crux of the information paradox. However, in this case the existence of the event horizon is dependent on the existence of the singularity.

From the perspective of an outside observer, they will see an infalling object remain just above their own apparent horizon while the black hole evaporates, and the object will appear to cross the horizon at the exact moment the horizon disappears. While the object's worldline would have intersected the singularity, any information about what occurred in the interior region is lost and an outside observer see only a discontinuity in the worldline of the infalling object at the moment of the disappearance of the black hole (well, presumably the object was destroyed at the singularity).

But, this illustrates somewhat the point I was trying to make: that discontinuity is due to the object hitting the singularity. If quantum gravity effects prevent the formation of an actual singularity then there's no reason to think there would be a discontinuity in the worldline of the object. Presumably, it would instead get frozen in some region near the singularity, waiting for the trapped region to shrink enough for it to escape; likewise, light emitted from the object would be similarly preserved, and eventually reach null infinity.

Everyone agrees that GR has to break down at the singularity, but effects like the information paradox are due to the presence of the singularity itself. This is why I feel that it's a mistake to focus on event horizons rather than apparent horizons.

From a strictly GR standpoint, you are correct; this is the crux of the information paradox. However, in this case the existence of the event horizon is dependent on the existence of the singularity.

This is false. In classical GR, the event horizon forms before the singularity, and there exist event horizons not associated with singularities.

From the perspective of an outside observer, they will see an infalling object remain just above their own apparent horizon while the black hole evaporates, and the object will appear to cross the horizon at the exact moment the horizon disappears. While the object's worldline would have intersected the singularity, any information about what occurred in the interior region is lost and an outside observer see only a discontinuity in the worldline of the infalling object at the moment of the disappearance of the black hole (well, presumably the object was destroyed at the singularity).

But what about a collapse as described in this thread? Then, an outside observer sees a central star darken, and disappear, inside a black region. Why don't you read this whole thread before repeating thing refuted at the beginning of this thread.

Austin0
From the perspective of an outside observer, they will see an infalling object remain just above their own apparent horizon while the black hole evaporates, and the object will appear to cross the horizon at the exact moment the horizon disappears. While the object's worldline would have intersected the singularity, any information about what occurred in the interior region is lost and an outside observer see only a discontinuity in the worldline of the infalling object at the moment of the disappearance of the black hole (well, presumably the object was destroyed at the singularity).

What do you mean when you say an outside observer will "see" an infalling object remain just above the horizon??
Doppler shift aside ,even if the light wasn't shifted out of visibility there would not be any remaining image of the object hovering on the horizon. From the outside it certainly seems like the object would simply totally disappear. Period.

Aimless
But what about a collapse as described in this thread? Then, an outside observer sees a central star darken, and disappear, inside a black region. Why don't you read this whole thread before repeating thing refuted at the beginning of this thread.

I fail to see how my point was refuted at the beginning of the thread. In the gravitational collapse scenario, assuming the black hole is permanent, then yes, of course an event horizon forms, and forms before the singularity.

My claim is the following: given the following two assumptions, 1), that all black holes eventually evaporate due to Hawking radiation, and 2), based on whatever unknown quantum gravity effects might exist, there are no spacetime singularities and there is some resolution to the information paradox, then event horizons don't exist.

As an example, consider a spacetime containing a smooth spherically symmetric time varying matter density such that at early times there are no horizons, at intermediate times a collapse occurs such that a trapped region forms, and at late times (for whatever reason) the collapse reverses and the trapped region disappears. What happens to information from events inside the trapped region?

It must either exit to the untrapped region (in which case the trapping surface isn't an event horizon) or it must be destroyed. If Hawking radiation is completely thermal then that suggests that information from those events is destroyed, but that view seems to be falling out of favor. If so, if the information persists in some way, then I do not see how it is possible to call the surface bounding the trapped region an event horizon.

My original point was that quantum effects seem to imply that event horizons are not impermeable; thus, I feel that apparent horizons are more physically relevant and interesting. I stated this poorly (and incorrectly) above, and you were right to call me on it; my apologies.

Aimless
Doppler shift aside ,even if the light wasn't shifted out of visibility there would not be any remaining image of the object hovering on the horizon.

Huh? Of course there would. To an outside observer the object appears to freeze just outside of the horizon while the light from that object redshifts away. Sure, it'll very quickly become colder than the CMB, but it'll never become completely black.

I fail to see how my point was refuted at the beginning of the thread. In the gravitational collapse scenario, assuming the black hole is permanent, then yes, of course an event horizon forms, and forms before the singularity.

No, even for an impermanent black hole formed from collapse that decays, the event horizon (semiclassically defined) precedes the singularity.
My claim is the following: given the following two assumptions, 1), that all black holes eventually evaporate due to Hawking radiation, and 2), based on whatever unknown quantum gravity effects might exist, there are no spacetime singularities and there is some resolution to the information paradox, then event horizons don't exist.

This is true only if you insist on a strictly classical definition of event horizon while using quantum definitions elsewhere. Note that the original semiclassical derivation of Hawking radiatio was based on the existence of a horizon. Thus Hawking radiation without a semiclassical horizon is nonsense. The consensus here is that you have something that macroscopically behaves like a horizon but microscopically does not.
As an example, consider a spacetime containing a smooth spherically symmetric time varying matter density such that at early times there are no horizons, at intermediate times a collapse occurs such that a trapped region forms, and at late times (for whatever reason) the collapse reverses and the trapped region disappears. What happens to

Without major violation of GR, this scenario is impossible. It is impossible with any of QG corrections of GR that I am familiar with. That is, the reversal of collapse after a macroscopic horizon forms is impossible.
It must either exit to the untrapped region (in which case the trapping surface isn't an event horizon) or it must be destroyed. If Hawking radiation is completely thermal then that suggests that information from those events is destroyed, but that view seems to be falling out of favor. If so, if the information persists in some way, then I do not see how it is possible to call the surface bounding the trapped region an event horizon.

My original point was that quantum effects seem to imply that event horizons are not impermeable; thus, I feel that apparent horizons are more physically relevant and interesting. I stated this poorly (and incorrectly) above, and you were right to call me on it; my apologies.

I mostly agree with this last paragraph with some caveats. Hawking radiation for a stellar black hole (let along a supermassive black hole) is at a lower temperature than CMB radiation. Thus all black holes in the current universe are growing, not shrinking (even if there is no matter at all nearby). The time frame in which black holes decay is well after the heat death of all stars.

Note: you have several times now used language like:

"they will see an infalling object remain just above their own apparent horizon while the black hole evaporates, and the object will appear to cross the horizon at the exact moment the horizon disappears"

This is what is refuted in this thread. For a collapsing supercluster observed from afar, matter is seen in the center of the forming black region, until the whole region goes black. The idea that the matter is seen only outside what appears to be the horizon is false for a collapse. It is true only for an eternal black hole, which is a pretty absurd concept.

Austin0
Huh? Of course there would. To an outside observer the object appears to freeze just outside of the horizon while the light from that object redshifts away. Sure, it'll very quickly become colder than the CMB, but it'll never become completely black.

OK we have an infalling object which approaches and passes through the horizon.
At the surface we imagine a certain number of photons which are permanently trapped.
Another quantity of photons which would not be trapped but whose coordinate speed is slowed down to the point of extremely delayed emergence to reach outside observers.

But from the outside, the whole passage occurred in an extremely short coordinate time interval. SO the number of these time release photons would actually be exceedingly small compared to the many billions of years of BH lifespan before possible evaporation.

Do you think a small number of photons making their way out per year would be in some way detectable or capable of discrimination from the background of radiation from infalling matter and Hawking radiation?
Do you think "seeing" is an apt descriptive term to apply to an image based on the assumption of these undetectable photons?

My assumption would be that in the real universe , infalling matter would disperse and/or redirect inward all of the retarded photons so that in a relatively short period of time there would be no trace image, even in abstract principle.
Just my opinion.

Gold Member
And who arranges the rocks? Why not just admit:

1) You think what GR predicts violates your sense of plausibility.
GR is not so monolithic as you are trying to imply by your phrase "GR predicts".

But yes, what you claim violates my sense of plausibility. But there might be different reasons for that. For example, we understand the same term differently and that leads to different conclusions.

2) As a result you think the GR is incorrect in this scenario.
I can't make consistent picture out of the things that you claim.

That would be honorable rather than claiming that the top experts in GR (not me, others who I study) are misinterpreting its equations.
Scepticism is important thing in science. That is basically what makes it differ from religion.

Further, except for the details of where it breaks down, you would then be in good company - many serious physicists think GR breaks down in the vicinity of horizons, and many more near the singularity. But that is different from disputing what GR predicts.

(Personally, I think, at least macroscopically, GR only breaks down near the singularity, and the horizon behavior you don't like actually occurs. I think, microscopically, a horizon may not be a true horizon, but macroscopically it behaves as GR predicts. Horizon behavior is likely to be subject to observational test within the next decades. )
I am interested what is out there. And I am interested how much GR can help in finding that out.

I can't make consistent picture out of the things that you claim.

Does it boil down to: you cannot accept that a 2-sphere horizon forms around empty space with no matter inside (in the case of a collapsing shell of matter, or shell like distribution of black holes)?

If so, I note that while I give a logical argument for this (not a 'reference to authority' - though I could ask you to study the Vaidya dust analysis in Poisson's "A Relativists's Toolkit", where growth of horizon from inside out is discussed in great detail), your response appears to me to be absolutely nothing except "I don't like it". Who is being closed minded rather than logical here?

Gold Member
Does it boil down to: you cannot accept that a 2-sphere horizon forms around empty space with no matter inside (in the case of a collapsing shell of matter, or shell like distribution of black holes)?
Yes. Look you are using word "forms" and that rather clearly says: "I am thinking about this in a time ordered manner." But when I read definition of absolute horizon it's clear that this definition is very inconvenient for time ordered approach (you have to extrapolate things from present to infinite future and then back). So this definition makes more sense for blockworld and even then it is quite inconvenient (to me it seems non-local).

Another problem with time ordered approach is that you need definition of simultaneity. But relativity uses round trip of light for that and so that definition breaks down for black holes. We of course can extrapolate simultaneity from regions where it is defined. And in that case it seems that we should get Schwarzschild coordinates. (Or maybe we can make testable predictions for different simultaneity? In SR this obviously is not possible but in GR?)

Mentor
Yes. Look you are using word "forms" and that rather clearly says: "I am thinking about this in a time ordered manner." But when I read definition of absolute horizon it's clear that this definition is very inconvenient for time ordered approach (you have to extrapolate things from present to infinite future and then back). So this definition makes more sense for blockworld and even then it is quite inconvenient (to me it seems non-local).

This has already been addressed upthread. See post #37. PAllen is thinking about the EH "forming" in a time-ordered manner, but that doesn't mean the EH is *defined* that way. Its definition does not require any assumptions about time ordering or simultaneity. It is "non-local" in a sense, since its definition requires knowledge of the entire spacetime as a 4-D manifold; but that doesn't imply that information about the future has to "propagate" back into the past.

This has already been addressed upthread. See post #37. PAllen is thinking about the EH "forming" in a time-ordered manner, but that doesn't mean the EH is *defined* that way. Its definition does not require any assumptions about time ordering or simultaneity. It is "non-local" in a sense, since its definition requires knowledge of the entire spacetime as a 4-D manifold; but that doesn't imply that information about the future has to "propagate" back into the past.

See also #53 and #66 where I give physically motivated definitions for this time ordering.

Gold Member
This has already been addressed upthread. See post #37. PAllen is thinking about the EH "forming" in a time-ordered manner, but that doesn't mean the EH is *defined* that way. Its definition does not require any assumptions about time ordering or simultaneity. It is "non-local" in a sense, since its definition requires knowledge of the entire spacetime as a 4-D manifold; but that doesn't imply that information about the future has to "propagate" back into the past.
Absolute horizon is defined in retrospective manner and that is incompatible with time ordered approach.

We can look at things from perspective of blockworld but then you have to take an extra step in sorting out which statements are scientific and which are not.

Gold Member
See also #53 and #66 where I give physically motivated definitions for this time ordering.
You can give definitions for time ordering but they relay on locally defined things (events). They don't work for retrospectively defined thing.
Retrospectively defined thing don't "form" and you can't "encounter" retrospectively defined thing.

Mentor
Absolute horizon is defined in retrospective manner and that is incompatible with time ordered approach.

No, it isn't. Any statement about the EH "existing" at a certain "time", or "forming" by a certain "time", given a specific simultaneity convention (and PAllen specified that), can be translated into a statement about a particular spacelike hypersurface intersecting a particular null surface in the 4-D spacetime. There's no problem at all.

Gold Member
Absolute horizon is defined in retrospective manner and that is incompatible with time ordered approach.
No, it isn't. Any statement about the EH "existing" at a certain "time", or "forming" by a certain "time", given a specific simultaneity convention (and PAllen specified that), can be translated into a statement about a particular spacelike hypersurface intersecting a particular null surface in the 4-D spacetime. There's no problem at all.
You are talking past the statement you quoted.

Definition of horizon is question about picking out particular null surface from other null surfaces as special.

Gold Member
PAllen,
How vital is concept of absolute horizon for this discussion (formation of black hole)?
Isn't it possible to define black hole using apparent horizon? At least in some specific cases if not all?

Mentor
You are talking past the statement you quoted.

Definition of horizon is question about picking out particular null surface from other null surfaces as special.

Yes. So what? It's still a null surface, and you can still translate any statement about the horizon "existing" at a particular "time" into a statement about the particular null surface that is the horizon intersecting a particular spacelike surface.

PAllen,
How vital is concept of absolute horizon for this discussion (formation of black hole)?
Isn't it possible to define black hole using apparent horizon? At least in some specific cases if not all?

Sure you can choose not to worry about the true horizon. Once a black hole is stable, with no more matter falling in for for quite a while, they coincide to any limit of measurement. Apparent horizons are more complex to derive for the scenarios under discussion. However, we can say the following:

1) For the collapsing spherical shell, by the time the the shell is inside its SC radius, the apparent horizon is at the SC radius. At this time (as observed by interior observers), there is not yet any singularity, nor (necessarily) any high density of matter (if the shell is enormous enough). Note, it is guaranteed that a singularity will form as the shell cannot stop collapsing at this point. (per GR of course).

2) For the collapsing star cluster, a similar observation is true. As soon as the cluster is within its SC radius, we know the apparent horizon is at the SC radius. There is no requirement that any stars have collided, nor any singularity exist yet (for interior observers). Again, per GR, it is guaranteed that a singularity will form.

The only thing I can't fill in (with my available time and resources) is the early history of the apparent horizon in these two scenarios. The true horizon is easier to derive general features of using general principles.

Gold Member
1) For the collapsing spherical shell, by the time the the shell is inside its SC radius, the apparent horizon is at the SC radius.
Let's take a closer look at SC solution and how much does it applies to collapsing body.
SC solution describes gravity around static (existing in equilibrium state) body. Now we take series of SC solutions with the same mass and progressively smaller radius. As mass is the same and radius shrinks it seems like we can claim that this series of SC solutions describes collapsing body.
But each solution for certain radius describes static body. And in order for the same body to go from larger radius to smaller radius and then reach equilibrium state at smaller radius it should release binding energy (reducing it's mass by appropriate amount). And that makes quite different series of SC solutions.

So in order to claim that this series of SC solutions with the same mass and progressively smaller radius describe collapsing body we have to assume equivalence between
compressed smaller body (less particles) that has not yet released binding energy
and
bigger body (more particles) at the same radius that has already released binding energy.

Does it make sense so far?

Let's take a closer look at SC solution and how much does it applies to collapsing body.
SC solution describes gravity around static (existing in equilibrium state) body. Now we take series of SC solutions with the same mass and progressively smaller radius. As mass is the same and radius shrinks it seems like we can claim that this series of SC solutions describes collapsing body.
But each solution for certain radius describes static body. And in order for the same body to go from larger radius to smaller radius and then reach equilibrium state at smaller radius it should release binding energy (reducing it's mass by appropriate amount). And that makes quite different series of SC solutions.

So in order to claim that this series of SC solutions with the same mass and progressively smaller radius describe collapsing body we have to assume equivalence between
compressed smaller body (less particles) that has not yet released binding energy
and
bigger body (more particles) at the same radius that has already released binding energy.

Does it make sense so far?

There is no need for such complexity unless you reject pure math: Birkhoff's theorem. Assuming spherical symmetry, and any shell of matter just inside its SC radius, it is guaranteed that the true horizon is at the SC radius and the apparent horizon is inside it by some infinitesimal amount. If you don't want to accept this, you have no choice but to admit that you reject GR, because this is pure mathematical proof. Unlike the singularity theorems, Birkhoff's theorem makes no assumptions about 'reasonable matter states'. Nothing is assumed except the Einstein Field equations.