Graphical example of BH formation by PAllen

Gold Member
I liked this graphical example of black hole formation posted by PAllen in another thread and I want to discuss it.
I think this point was made earlier in this thread, but I would like to pose it in a graphic form. This is the point that an eternal black hole as described by SC geometry almost certainly does not exist in our universe. Let's instead look at formation of black hole.

To be able to see the formation better from afar, let's have the far fetched scenario of a trillion stars of some super cluster collapsing with no net angular momentum, no accretion disk forming. I pick the far fetched number of a trillion stars because that allows the black hole to form while the stars are still well separated from each other, and individually resolvable (in principle) up until the last moments. Let's further assume there is a background of galaxies behind this collapsing cluster, but nothing in your line of sight in front of it.

What would you see? As the collapse occurred, you would see the cluster, as a whole, reddening, and more and more extreme Einstein rings from galaxies behind the cluster. Up until the last moments, you would see highly red shifted light from stars throughout the cluster - especially, you could see stars in the center of the cluster. Then, in a relatively brief period of time, the cluster would further redden/darken until it was blacker than even CMB radiation. Against the background galaxies, it would look, quite literally, like a black hole in the sky surrounded by an Einstein ring of light from galaxies behind it.

How would you want to interpret this? It is a mathematical fact that this is what you would see. Would you say that a trillion stars have actually vanished? Would you say that the stars in center magically are compressed invisibly against the not quite yet formed horizon (having jumped billions of miles from the center of the cluster to the edge of this black ball)? You could say there is an invisible ball of a trillion frozen stars, a millimeter larger than the theoretical event horizon. Then, if matter falls in, it soon vanishes and the black region grows slightly (after all settles down). Again, you could say the black ball is still just larger than the theoretical event horizon, with frozen stars throughout, and new matter somewhere at the outer edge.

If you prefer this interpretation, it is, indeed, not determinable from outside observations that further collapse has occurred inside the black region. However, I would than ask:

If look like a duck, ... . Isn't black hole a good description of the this scenario? Then if you ask, what would happen according to a ship orbiting one of those interior stars? GR has only one answer - further collapse (to a singularity), in very finite time for the ship.

It is not unusual that arguments defending existence of black hole go like that:
1. Assume that BH exists.
2. Then observer falling into BH ...
I think that point 1. is begging the question fallacy. So any argument defending BH should be about formation of BH.
Therefore I think that this example of PAllen is excellent basis for discussions around black holes and so it is worth a closer look.

Now about example. Let's say that we are looking at star at the far edge of collapsing cluster. Let's assume further that we have idealized situation where light from that star is going exactly through the cluster's center of mass. That particular star will be gravitationally lensed (amplified) and it is a bit unclear if it will be redshifted or blushifted as it is moving toward us not away from us, right?

Another point is where exactly formation of event horizon starts? It does not appear at once but is expanding from some point by engulfing mass. So we have to have some seed black hole that is produced in collision of two stars near cluster's center of mass, right?

Mike Holland
Yes, its an interesting scenario. If the distribution of stars is uniform, then r = 2m will first occur at the centre. But in a uniform sphere the gravitational force is strongest at the perimeter and weakest in the centre, so one might expect to first see the density increasing away from the centre.
The normal scenario for black hole development is a collapsing massive star, after it has exploded a lot of matter away. This version has been simulated on computers and is well known. But is the collapsing star cluster really any different? Essentially it is a lumpy cloud of gas, but because of the lumpiness it has no pressure, and so is very similar to the Oppenheimer-Snyder calculation. Why would stars give a different result to particles of gas in a cloud of similar size?

Mike

Staff Emeritus
I think that point 1. is begging the question fallacy. So any argument defending BH should be about formation of BH.

In my opinion, the primary purpose of PF isn't to "defend" modern physics, but to explain it to those who are interested.

The idea that we "should" explain in detail how a black hole forms may sound appealing , but it's rather unrealistic, requiring advanced knowledge. For pedagogical purposes, it's a disaster.

For those who might be interested, some aspects of this hard problem are discussed in http://arxiv.org/abs/1010.1269. http://relativity.livingreviews.org/open?pubNo=lrr-2008-1&amp;page=articlesu11.html [Broken] is also intersting.

This is well, well, well beyond naieve and ultimately ill-founded arguments by non-experts who don't understand why black holes are thought to exist.

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TrickyDicky
In my opinion, the primary purpose of PF isn't to "defend" modern physics, but to explain it to those who are interested.
I agree. Though I'm not s sure everyone agrees with the part about "defending" not being the primary pupose of PF. See i.e. the rest of your post
The idea that we "should" explain in detail how a black hole forms may sound appealing , but it's rather unrealistic, requiring advanced knowledge. For pedagogical purposes, it's a disaster.
This seems to contradict your first paragraph, I'm not sure if you're for or against explaining, or just prefer explaining but only those subjects you think are easy to grasp for the average non-expert. But there are a whole lot of threads in PF that deal with just as hard or even harder (this is very subjective) issues like a brief look at this subforum or the Quantum physics, or HEP or condensed matter one... shows, and I'm sure there is always some group of people that benefits from them depending on their background and intelligence. So why is that a pedagogical disaster?

This is well, well, well beyond naieve and ultimately ill-founded arguments by non-experts who don't understand why black holes are thought to exist.
Right, that is why a forum like PF can improve IMHO even if it's just a little, those non-experts understanding of why BH's are thought to exist.

For pedagogical purposes, I think the Oppenheimer-Snyder collapse is reasonable for this highly idealized scenario.

It is, in fact, known that GR predicts the event horizon to form at the center and grow. It is not true that some specific collision at the center produces it - the whole accumulation of matter produces it, without need for collisions (though, of course, they will happen). However, by definition, a distant observer never sees light from (or inside) the horizon at any stage of its existence. What you do see is the light from just outside the growing horizon, barely staying ahead of the horizon. Think about any period well before BH formation. You see newest light from the near edge, older light from the center, oldest from the far edge. For now ignore the issue of light from the far side of the cluster. At each moment you see light from the nearer stars down through the center, at some small time before the horizon has reached that point. A moment later, you see light from closer to when the horizon reached each distance. Then closer yet. Thus, the image is one of all the stars seeming to slow down (e.g. if you had a few binary systems thrown in, they would slow down their period) and redden collectively, until they become invisible by virtue of the highest energy gamma rays being red shifted to below CMB frequency.

As for the stars on the far side of the cluster, I am not sure. Some mixture of blocked light and refracted light would seem likely. This part of the scenario would require careful simulation that would be a major project to perform (and I haven't bumped into any published simulations covering close to this situation - that of the optics of the stars of the far side of a collapsing cluster with no net angular momentum).

The final result would look something like this:

http://commons.wikimedia.org/wiki/File:BH_LMC.png

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Mike Holland
I have just been in touch with Saul Teukolsky, who has been doing simulations involving naked singularities, and his comment was

"A naked singularity would be one not screened by an event horizon. There is no example yet of a naked singularity being generated by the collapse of non-exotic matter in a generic way. But it is also true that there is no theorem that rules it out. This has been the situation for over 30 years, so it's a hard problem. Because all the attempts to make a naked singularity have failed, most experts think you can't do it, but that of course is a psychological statement.

Our work showed strong numerical evidence that you could make a naked singularity by collapsing matter that only interacted gravitationally. However, it was not generic, because the situation we looked at was perfectly axisymmetric. In any event, if you collapse an ordinary star and GR is correct, you definitely get a black hole with an event horizon."

Mike

Gold Member
In my opinion, the primary purpose of PF isn't to "defend" modern physics, but to explain it to those who are interested.
I don't understand. As I see it "defending" is part of "explaining". To explain means to show how to come to certain conclusion so you want to show why other alternatives are untenable, right?

I will get to the rest later after I will look at the links.

I've thought about the question of stars from the far side of the cluster more. I think I see the answer. Each wave front from the cluster you see from afar will have images of the far side of the cluster closer and closer to the last point when light from the far side will cross the center. Over time, the last light you will see from the far side approaches light reaching the center at the moment the event horizon forms and starts growing. The net result is that there is no unusual change in the appearance of the cluster from afar - you always see the whole cluster as of different points in time. It reddens and disappears, but you never see anything that explicitly confirms an event horizon (as opposed to deducing it).

Gold Member
The idea that we "should" explain in detail how a black hole forms may sound appealing , but it's rather unrealistic, requiring advanced knowledge. For pedagogical purposes, it's a disaster.
The more advanced are the arguments the more error prone and harder to check they are. So less believable is the claim.

It is, in fact, known that GR predicts the event horizon to form at the center and grow. It is not true that some specific collision at the center produces it - the whole accumulation of matter produces it, without need for collisions (though, of course, they will happen).
Okay, but it has to start at some point. And that point can't be just empty space. Next in order for it to grow it must accumulate more matter i.e. collide with it.

I've thought about the question of stars from the far side of the cluster more. I think I see the answer. Each wave front from the cluster you see from afar will have images of the far side of the cluster closer and closer to the last point when light from the far side will cross the center. Over time, the last light you will see from the far side approaches light reaching the center at the moment the event horizon forms and starts growing. The net result is that there is no unusual change in the appearance of the cluster from afar - you always see the whole cluster as of different points in time. It reddens and disappears, but you never see anything that explicitly confirms an event horizon (as opposed to deducing it).
We should see the light from the far side star that bends around mass center. So it seems that this star should turn into Einstein's ring before disappearing.

Gold Member
Yes, its an interesting scenario. If the distribution of stars is uniform, then r = 2m will first occur at the centre. But in a uniform sphere the gravitational force is strongest at the perimeter and weakest in the centre, so one might expect to first see the density increasing away from the centre.
I suppose that we can draw some analogy with gravitation potential of Newton's gravity and gravitation potential is lowest at mass center even as gradient of potential approaches zero. And black hole should be born where gravitation potential is lowest. Hmm, unless some star at it's center has lower potential then center of mass.

The normal scenario for black hole development is a collapsing massive star, after it has exploded a lot of matter away. This version has been simulated on computers and is well known. But is the collapsing star cluster really any different? Essentially it is a lumpy cloud of gas, but because of the lumpiness it has no pressure, and so is very similar to the Oppenheimer-Snyder calculation. Why would stars give a different result to particles of gas in a cloud of similar size?
In case of stars we take degeneracy pressure into consideration. Not sure how to extrapolate it to cluster of stars.

Mike Holland
In case of stars we take degeneracy pressure into consideration. Not sure how to extrapolate it to cluster of stars.

Yes, that's what I meant. In the cluster of stars there is normal pressure in each star, but due to the size of the cluster they are collapsing into a black hole without themselves collapsing into white dwarfs/neutron stars/whatever first. And there is no pressure between the stars as the black hole forms. So degeneracy just doesn't come into the pictrure.

When they are all within the Schwarzschild radius, they are still well separated from each other (assuming uniform distribution).

Mike

Okay, but it has to start at some point. And that point can't be just empty space. Next in order for it to grow it must accumulate more matter i.e. collide with it.

No, this is not necessary. The event horizon is just a surface of last light. There need never be any matter there. There are exact solution for a collapsing shell of matter that produce a black hole. In these, the event horizon starts as one point in the center. There is no matter at or near it when it forms.
We should see the light from the far side star that bends around mass center. So it seems that this star should turn into Einstein's ring before disappearing.

It would take simulation to answer this, so I can't be sure. However, my intuition is that this would happen for galaxies behind the collapsing cluster, but not much, if at all, for stars on the far side of the cluster participating in the collapse. I think the light from these stars emitted after the event horizon forms and starts growing simply never makes it out of the cluster.

In case of stars we take degeneracy pressure into consideration. Not sure how to extrapolate it to cluster of stars.

This would relate to formation of the singularity, not the event horizon. The event horizon is a global phenomenon - it actually depends on the whole spacetime, including the indefinite future. For example, if you have a stable black hole and a baseball is going to fall into it next year, there is some photon so slowly 'escaping' that this future infall means it will never escape. Thus, this future event affects the location of the event horizon now.

There actually need not even be light trapped at the event horizon. It is a theoretical surface defined by the escape to infinity, in infinite time, of null paths.

One further consequence of the definition of event horizons is that for a closed universe, you cannot actually define an event horizon because there is no future infinity.

There is a related concept of apparent horizon, which has a local definition, and is (almost always) inside the true horizon.

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Yes, that's what I meant. In the cluster of stars there is normal pressure in each star, but due to the size of the cluster they are collapsing into a black hole without themselves collapsing into white dwarfs/neutron stars/whatever first. And there is no pressure between the stars as the black hole forms. So degeneracy just doesn't come into the pictrure.

When they are all within the Schwarzschild radius, they are still well separated from each other (assuming uniform distribution).

Mike

Exactly! That was the whole point of my construction. There is no limit to how low the density of a black hole can be, if you have enough total mass.

TrickyDicky
The "trillion stars collapsing" scenario giving rise to a "blacker than the CMB" region, would be at least ideally and according to theory what we would observe if we had the chance to witness the formation of such a supermassive black hole. Now this hasn't happen yet and we probably won't ever see it as this supermassive black holes usually form in galactic bulges and represent a tiny fraction of the bulge or the AGN, so there is no way to discern from afar.
So basically the only way we have to infer their presence is the tremendous quantities of radiation of different wavelengths they radiate from infalling matter accreted around the black hole.
If a physically reasonable mechanism of production of that massive radiation was found, there would be real motives to seriously question the black hole paradigm IMO, this hasn't happen yet, at least that I know.

The "trillion stars collapsing" scenario giving rise to a "blacker than the CMB" region, would be at least ideally and according to theory what we would observe if we had the chance to witness the formation of such a supermassive black hole. Now this hasn't happen yet and we probably won't ever see it as this supermassive black holes usually form in galactic bulges and represent a tiny fraction of the bulge or the AGN, so there is no way to discern from afar.
So basically the only way we have to infer their presence is the tremendous quantities of radiation of different wavelengths they radiate from infalling matter accreted around the black hole.
If a physically reasonable mechanism of production of that massive radiation was found, there would be real motives to seriously question the black hole paradigm IMO, this hasn't happen yet, at least that I know.

Yes, these are all valid observations. However, I thought the intent of the thread was to get at the frequently recurring questions of interpretation what GR says about black holes (separately from any questions about how strong the evidence for them is).

The point of my admittedly absurd scenario of a trillion stars collapsing with no net angular momentum in a completely smooth way was to confront a common resistance to admitting the reality of the 'inside' of a collapsed object that comes from excessive focus on matter falling into a pre-existing perfect SC black hole. I think my scenario succeeded admirably in this.

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Mike Holland
There are exact solution for a collapsing shell of matter that produce a black hole. In these, the event horizon starts as one point in the center. There is no matter at or near it when it forms.

PAllen, I find this fascinating because it is so counter-intuitive (to me, at least!). Can you supply any references to articles discussing this?

From a Newtonian perspective, the whole interior of a uniform shell should have the same gravitational potential and zero gravitational force.

Mike

Edit: I have read in many places that a black hole (and event horizon) only forms when all the matter is within its Schwarzschild radius!

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PAllen, I find this fascinating because it is so counter-intuitive (to me, at least!). Can you supply any references to articles discussing this?

From a Newtonian perspective, the whole interior of a uniform shell should have the same gravitational potential and zero gravitational force.

Mike

Edit: I have read in many places that a black hole (and event horizon) only forms when all the matter is within its Schwarzschild radius!

No, it happens also when any subset of matter is within a critical radius (which, in realistic cases - the whole universe is not spherically symmetric around one collapsing mass, that has no rotation), which is not exactly the same as the SC radius. But that is all irrelevant - in the shell example: when the whole shell is within the SC radius there is still no matter at the center. You don't need anything to derive the properties of this scenario (for the spherically symmetric non-rotating case) beyond Birkhoff's theorem.

Newtonian analog is not relevant - collapsing objects are the case when GR has maximum disagreement with Newtonian gravity. Even so, potential and force have nothing to do with event horizon. The event horizon (over time) is just 3 surface (2 space x 1 time) in the spacetime manifold from which no forward going null paths escape to infinity. So, as an object collapses (pretend it is transparent), there is a point where null paths from the center reach the outside only when the whole object is within the SC radius. Since getting from center to outside takes time (during which time the object is collapsing), rays from the center are trapped earlier than rays from the outside. This serves to qualitatively establish that the event horizon must grow from the inside out for typical collapse scenarios.

Mentor
The event horizon (over time) is just 3 surface (2 space x 1 time) in the spacetime manifold from which no forward going null paths escape to infinity.

Actually, the EH is a null surface--more precisely, it has two spacelike and one null dimension. The third dimension has to be null for the EH to have the property you describe (which it does).

Actually, the EH is a null surface--more precisely, it has two spacelike and one null dimension. The third dimension has to be null for the EH to have the property you describe (which it does).

Oops, you're right.

Gold Member
No, it happens also when any subset of matter is within a critical radius
Sphere with certain radius is not a spacetime object so to define it you need to define some kind of simultaneity.
Another thing is that you can define your example as spherically symmetric but we expect from physical laws that they are more universal i.e. they should work for not so symmetrical case the same way. So I say that matter needs to "communicate" it's distribution to the point of event so that different possible scenarios can happen.

And yet another thing. Event horizon of a static black hole moves at speed of light, right? Then how does it expand for growing black hole? It can't move FTL. So you have to provide reasonable model how does it happens.

Gold Member
Actually, the EH is a null surface--more precisely, it has two spacelike and one null dimension. The third dimension has to be null for the EH to have the property you describe (which it does).

Oops, you're right.

PeterDonis, PAllen: what the hell is null dimension? (reference?)

What dimensions it has?

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Gold Member
Yes, that's what I meant. In the cluster of stars there is normal pressure in each star, but due to the size of the cluster they are collapsing into a black hole without themselves collapsing into white dwarfs/neutron stars/whatever first.
I do not agree. At least it is far from obvious.
If we have mass within SC radius then some inner mass in the gravity field of outer mass is too within critical radius, right? So outer mass lowers critical barrier for inner mass.

And there is no pressure between the stars as the black hole forms. So degeneracy just doesn't come into the pictrure.
There is some interesting thing about degeneracy. It is usually called "degeneracy pressure" while it is very unlike pressure.
And I think that degeneracy comes into picture.

Sphere with certain radius is not a spacetime object so to define it you need to define some kind of simultaneity.
Another thing is that you can define your example as spherically symmetric but we expect from physical laws that they are more universal i.e. they should work for not so symmetrical case the same way. So I say that matter needs to "communicate" it's distribution to the point of event so that different possible scenarios can happen.

And yet another thing. Event horizon of a static black hole moves at speed of light, right? Then how does it expand for growing black hole? It can't move FTL. So you have to provide reasonable model how does it happens.

I don't understand what you are asking about. The snippet you quoted is that in a general region of matter in spacetime, there can be multiple regions such that the matter is within a critical volume that (per GR) it will collapse to s singularity, and be surrounded by an event horizon. In the general case, the definition of such critical volumes is very complex and is not the SC radius. Is there something about this you dispute, as a prediction of GR (separate issue is whether you think it is true of our universe)?

The more specific point I was making is that if the total mass of a spherical shell and its size are such that it is all within the SC radius for that mass, Birkhoff's theorem guarantees that the metric outside the shell is the SC metric, meaning that the horizon already exists, even though the center is empty space. Via singularity theorems, it is also guaranteed that the shell will soon form a singularity. I also describe how, if the above is true, the event horizon (as defined in GR) must have started forming in an empty region inside the shell while the collapsing shell was still outside the SC radius. Again, I am not sure if you dispute that this is what GR says, or do you dispute the truth of GR for our universe. For the purposes of this thread, I am not interested in discussing the breakdwon of GR (though I do have some specific ideas on where that happens).

As to the question of defining spherical, that is easy to do in an invariant way. A really old fashioned, but easy to understand, way is to ask whether a coordinate transform is possible that puts the metric into form in which spherical symmetry is evident. This is phrased in terms of coordinates, but it either is or isn't true of a manifold (or 'sufficiently isolated section' of a manifold).

As to your last point, an event horizon for a strictly static black hole moves at the speed of light for a local inertial frame. However, an event horizon, in general, is a mathematical surface (not the path along which light or matter travel). It is not limited to any speed, and is not a local observable. Its definition is global and requires waiting for eternity to decide its exact location.

PeterDonis, PAllen: what the hell is null dimension? (reference?)

What dimensions it has?

Null dimension is not the best choice of words. Lightlike basis would be better. Or even that we have a 3-surface a pseudo-riemannian 4-manifold that has a light like tangent at every point on it.

I do not agree. At least it is far from obvious.
If we have mass within SC radius then some inner mass in the gravity field of outer mass is too within critical radius, right? So outer mass lowers critical barrier for inner mass.

There is some interesting thing about degeneracy. It is usually called "degeneracy pressure" while it is very unlike pressure.
And I think that degeneracy comes into picture.

I don't know what you are going on about here. There is simply no requirement in GR that a black hole have any particular density or pressure. A collection of ordinary dust particles, of the average density of air, will constitute a black hole if it occupies a spherical region large enough to be within its collective SC radius.

Mike Holland
The more specific point I was making is that if the total mass of a spherical shell and its size are such that it is all within the SC radius for that mass, Birkhoff's theorem guarantees that the metric outside the shell is the SC metric, meaning that the horizon already exists, even though the center is empty space. Via singularity theorems, it is also guaranteed that the shell will soon form a singularity. I also describe how, if the above is true, the event horizon (as defined in GR) must have started forming in an empty region inside the shell while the collapsing shell was still outside the SC radius. Again, I am not sure if you dispute that this is what GR says, or do you dispute the truth of GR for our universe. For the purposes of this thread, I am not interested in discussing the breakdwon of GR (though I do have some specific ideas on where that happens).

OK, so we have your trillion stars colllapsing, but a void in the centre with a candle burning in the center to give as a stream of photons. The stars are not yet within the SR, only the inner few have reached it. So an Event Horizon comes into existence in the centre and chases the last photons outwards. Photons within this EH cannot flow outwards, because space and time get all mixed up inside and they can only flow towards the coming singularity in the centre.

But now a large rogue Black Hole comes shooting past at high speed, and scatters the stars. So our Black Hole is never formed, and the orphaned Event Horizon has nowhere to go. What happens to it, and what happens to the photons emitted just after it formed? This problem arises in any situation where all the mass is not yet within the SR. It can, at least in principle, still escape. So if an Event Horizon has already appeared, that Event Horizon has a big problem!

Mike

OK, so we have your trillion stars colllapsing, but a void in the centre with a candle burning in the center to give as a stream of photons. The stars are not yet within the SR, only the inner few have reached it. So an Event Horizon comes into existence in the centre and chases the last photons outwards. Photons within this EH cannot flow outwards, because space and time get all mixed up inside and they can only flow towards the coming singularity in the centre.

But now a large rogue Black Hole comes shooting past at high speed, and scatters the stars. So our Black Hole is never formed, and the orphaned Event Horizon has nowhere to go. What happens to it, and what happens to the photons emitted just after it formed? This problem arises in any situation where all the mass is not yet within the SR. It can, at least in principle, still escape. So if an Event Horizon has already appeared, that Event Horizon has a big problem!

Mike

This is indeed a subtle point. It gets at the reason why various definitions of apparent horizon were introduced. The true event horizon is determined by the entire future of the universe - considered as a complete spacetime manifold. Remember my example earlier - the exact location of the event horizon 'now' is affected by whether a baseball will fall into a black hole a year from now. So, in your case, the event horizon forms or doesn't form depending on whether the collapse will or will not proceed in the future. Recall also, that the mathematical definition of event horizon and even black hole cannot be satisfied in a closed universe, because the 'escape to infinity' cannot be defined.

Apparent horizon is a quasi-local concept without such strong future dependent behavior. It was introduced as an alternative to the true horizon. In examples such as yours, it will reflect (roughly - there are multiple definitions) the trapping of light irrespective of future; that is, it is determined by collapse past the point of no return.

TrickyDicky
The true event horizon is determined by the entire future of the universe - considered as a complete spacetime manifold. Remember my example earlier - the exact location of the event horizon 'now' is affected by whether a baseball will fall into a black hole a year from now. So, in your case, the event horizon forms or doesn't form depending on whether the collapse will or will not proceed in the future.

Hmmm... how can an event (such as the formation of an EH, we are not even talking about the singularity here) in a causal time asymmetric universe with a working 2nd law of thermodynamics, be determined by future events? Certainly causality here goes out the window, this looks more like "Back to the future" the movie series.

Recall also, that the mathematical definition of event horizon and even black hole cannot be satisfied in a closed universe, because the 'escape to infinity' cannot be defined.
So FRW universes with positive spatial curvature can't have Black holes?
It gets at the reason why various definitions of apparent horizon were introduced...
Apparent horizon is a quasi-local concept without such strong future dependent behavior. It was introduced as an alternative to the true horizon. In examples such as yours, it will reflect (roughly - there are multiple definitions) the trapping of light irrespective of future; that is, it is determined by collapse past the point of no return.
But an apparent horizon, unless we were considering the eternal Black hole of the Kruskal-Szekeres space, which is not the case, always forms after the EH and inside it, so if it forms it will have the same future dependency problem as the EH.

Mike Holland
So causality works in reverse! PAllen, don't you want to go away and think about this for a while? Chuckle, chuckle!

I think I will stick with Eternally Collapsing Objects. Much easier on the brain.

Mike

Gold Member
The more specific point I was making is that if the total mass of a spherical shell and its size are such that it is all within the SC radius for that mass, Birkhoff's theorem guarantees that the metric outside the shell is the SC metric, meaning that the horizon already exists, even though the center is empty space.
Wikipedia says that:
"In general relativity, Birkhoff's theorem states that any spherically symmetric solution of the vacuum field equations must be static and asymptotically flat. This means that the exterior solution must be given by the Schwarzschild metric."

I am interpreting this as a statement that dynamic solution can't be spherically symmetric. Do you agree?

Mentor
Null dimension is not the best choice of words. Lightlike basis would be better. Or even that we have a 3-surface a pseudo-riemannian 4-manifold that has a light like tangent at every point on it.

Yes, the more precise way of stating what I said before is that the event horizon is a 3-surface whose tangent space at each point can be given a basis that has two spacelike basis vectors and one null basis vector.

Mentor
I am interpreting this as a statement that dynamic solution can't be spherically symmetric. Do you agree?

No. The implication only goes one way: spherically symmetric + vacuum => static. A non-static spacetime can still be spherically symmetric.

The FRW spacetimes used in cosmology are examples of solutions that are dynamic and spherically symmetric.

Mentor
So causality works in reverse!

I assume you are referring to this statement by PAllen?

The true event horizon is determined by the entire future of the universe

"Is determined by" in that statement does not mean "is caused by". It merely refers to the fact that the event horizon is *defined* by which null curves can reach future null infinity, and which can't.