Graphical example of BH formation by PAllen

[Mike Holland]

I assume you are referring to this statement by PAllen?
"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.

Ok, but PAllen said that the Event Horizon can form before all the matter is within its Schwarzschild Radius, and as I pointed out, an external event could prevent the collapse from completing, so the EH has to "know" about what external events are "going to" happen when it decides whether to come into existence or not!

The future null infinity has not been reached when the EH forms, and in the situation I described it is indeterminate until all the mass is within the SR, so the existence of the EH is similarly indeterminate.

Mike

 

[PeterDonis]

Ok, but PAllen said that the Event Horizon can form before all the matter is within its Schwarzschild Radius, and as I pointed out, an external event could prevent the collapse from completing, so the EH has to "know" about what external events are "going to" happen when it decides whether to come into existence or not!

Strictly speaking, the EH does not "decide to come into existence". It is not a "thing". It's just a boundary between two regions of the spacetime. Saying the EH "forms" is not, strictly speaking, correct. The strictly correct way to state it would be to say "looking at the spacetime as a whole, as a 4-dimensional geometric object, this particular null surface is an event horizon". And all the other statements about an EH "forming before all matter is within the Schwarzschild radius", strictly speaking, should be re-stated in a similar manner: for example, "looking at the spacetime as a whole, as a 4-dimensional geometric object, there are spacelike hypersurfaces that intersect this particular null surface, which is an event horizon, but still contain matter outside the Schwarzschild radius".

The reason people so often state things the way PAllen did is that the strictly correct way of stating them is cumbersome. But if you look at the actual literature, such as the book by Hawking and Ellis, you will see that the actual, mathematical definitions of the EH, when translated into English, say only what I said above; they do *not* say that the EH is a "thing" that has to "know" about the future history of the spacetime in order to "come into existence".

 

[PAllen]

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.

Thermodynamics has to be added via additional definitions and assumptions to strictly deterministic theories like GR or classical mechanics, and is altogether not relevant to this discussion.

There is actually nothing so mysterious going on here. Suppose, in Newtonian gravity, we define an 'asteroid escape set' - membership in this set is determined by whether an asteroid ever, to inifinite future, escapes the solar system. Membership in this set now depends on future events. Alternatively, one could define an 'apparent asteroid escape set' of asteroids that are in the process of escaping the solar system now. This last set is clearly a subset of the former set, and any given asteroid joins this set much later than it joins the first set (in this somewhat silly analog, an asteroid is or isn't a member of the true escape set at birth).

Fundamentally, there is nothing more mysterious going on in apparent versus true horizons in GR.

So FRW universes with positive spatial curvature can't have Black holes?

The technical definition of black hole event horizons cannot be satisfied in a closed universe. There is no infinity to escape to.

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.

No, it avoids the future dependency problem precisely because it forms later and is generally inside the true event horizon. As with the asteroid analog, by virtue of forming later and being smaller, it responds to events which are quasi-locally committed, and not to things like a star interacting with a black hole in the future.

 

[PAllen]

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

See my asteroid answer in my prior post to see that there is no causality issue or mystery at all. An event horizon's definition is not causal. It is a feature of a complete spacetime manifold, which is the complete history of some hypothetical universe.

You are the one who want to propose that 'what you see from one side of an event horizon' defines reality. I am describing the many problems with that point of view - one of the minor of which is that an event horizon is acausal.

Of course, much more fundamental is my simple example of a rocket ship in the middle mega-collapsing cluseter of stars. You are communicating with it while it is well within what will soon be the black hole. After you can no longer get signals from it, it still receives signals from you[in the following precise sense: the rocket's last signal you ever get is at rocket time e.g. t=5; however the rocket continues to receive signals from you until rocket t=7 when it is destroyed by tidal forces - which are not strong until near the singularity for such a supermassive cluster. GR makes clear predictions about the fate of this rocket. You propose to censor the predictions you don't like, while still claiming you accept GR.

 

[PAllen]

Strictly speaking, the EH does not "decide to come into existence". It is not a "thing". It's just a boundary between two regions of the spacetime. Saying the EH "forms" is not, strictly speaking, correct. The strictly correct way to state it would be to say "looking at the spacetime as a whole, as a 4-dimensional geometric object, this particular null surface is an event horizon". And all the other statements about an EH "forming before all matter is within the Schwarzschild radius", strictly speaking, should be re-stated in a similar manner: for example, "looking at the spacetime as a whole, as a 4-dimensional geometric object, there are spacelike hypersurfaces that intersect this particular null surface, which is an event horizon, but still contain matter outside the Schwarzschild radius".

The reason people so often state things the way PAllen did is that the strictly correct way of stating them is cumbersome. But if you look at the actual literature, such as the book by Hawking and Ellis, you will see that the actual, mathematical definitions of the EH, when translated into English, say only what I said above; they do *not* say that the EH is a "thing" that has to "know" about the future history of the spacetime in order to "come into existence".

Right, to actually talk about differences between apparent and true horizons for different scenarios, we would need circumlocutions like:

Imagine two solutions (complete spacetime manifolds) to the EFE for initial conditions identical except for the history of one baseball. We find these solutions are each an open manifold. Event horizons in each are determined by the manifold geometry as a whole. There is one particular event horizon we compare between the two manifolds. Despite great technical difficulties in general (for matching arbitrary GR solutions in some physically meaningful way), the similarity of these manifolds allows meaningful comparisons between them. We impose a similar family of spacelike slices in the manifolds, parameterize them with a similar time parameter. We have some physically significant reference event in each that we can call t=0 in both (say a slice on which we specified the initial conditions). Then we ask what is the earliest t from which there is an event from which no null paths escape to infinity? This will be slightly earlier in one manifold than the other (earlier in the one where the baseball falls in). Further, in every spatial slice that intersects the event horizon 3-surface, the spatial 2-surface in one manifold will be larger (again, we rely on the similarity of the two solutions to give meaning to this otherwise problematic statement).

Then we can ask about apparent horizons. Here (depending on the exact definition), the earliest time of a spatial slice containing a any point of the apparent horizons will be much closer between the two manifolds than was true of the true horizon. The apparent horizons will only begin to differ when (in the base ball fall in manifold) the base ball is captured by the black hole, and can't escape.

 

[TrickyDicky]

Saying the EH "forms" is not, strictly speaking, correct. The strictly correct way to state it would be to say "looking at the spacetime as a whole, as a 4-dimensional geometric object, this particular null surface is an event horizon".

An event horizon's definition is not causal. It is a feature of a complete spacetime manifold, which is the complete history of some hypothetical universe.

You both seem to take for granted a kind of GR's "block universe" interpretation that requires an 5th dimensional ambient space for an observer capable of seeing the whole spacetime as a frozen , acausal geometric object in order to explain the event horizon.

But no such observer is thought to exist.

 

[PAllen]

You both seem to take for granted a kind of GR's "block universe" interpretation that requires an 5th dimensional ambient space for an observer capable of seeing the whole spacetime as a frozen , acausal geometric object in order to explain the event horizon.

But no such observer is thought to exist.

No 5th dimension is required. No embedding is required to discuss a spacetime geometry.

GR is a deterministic, classical, theory for which a solution is complete history of a 'universe'. How do you have a non-deterministic interpretation of GR?

 

[TrickyDicky]

No 5th dimension is required. No embedding is required to discuss a spacetime geometry. GR is a deterministic, classical, theory for which a solution is complete history of a 'universe'.

Sure,I have not said anything contradicting these facts. I'm referring to something very specific, that you are postulating the need of a certain kind of observer to explain Event horizons, and I'm just pointing out these observers are not physical.

How do you have a non-deterministic interpretation of GR?

I'd be delighted to have a philosophic discussion, but this is not the place, there's a nice guy locking threads that get into that.

 

[PAllen]

There is actually nothing so mysterious going on here. Suppose, in Newtonian gravity, we define an 'asteroid escape set' - membership in this set is determined by whether an asteroid ever, to inifinite future, escapes the solar system. Membership in this set now depends on future events. Alternatively, one could define an 'apparent asteroid escape set' of asteroids that are in the process of escaping the solar system now. This last set is clearly a subset of the former set, and any given asteroid joins this set much later than it joins the first set (in this somewhat silly analog, an asteroid is or isn't a member of the true escape set at birth).

Fundamentally, there is nothing more mysterious going on in apparent versus true horizons in GR.

Expanding on this a little, the 'cause' of membership in the 'asteroid escape set' is initial conditions, and the same is true of event horizons in GR (at least in the case of an initial conditions formulation). Thus, for both my analog and event horizons, there is no causality issue within the framework of a classical deterministic theory.

What is slightly puzzling, but shouldn't be, is the correlation between two consequences of initial conditions: 'beginning of event horizon' and 'a given body merges with a black hole', which happens later. But this is just a correlation of consequences, not a causal relation.

 

[zonde]

I think this discussion somewhere just went astray.
If we would have to wait eternity to define black hole we could safely discard the concept as non-scientific crap.

So let's look at the problem of event horizon definition. We have observer A and object B. Light is continuously traveling from B to A. If at some moment somewhere along the way light gets stuck (in a global coordinate system of A) we say that event horizon has formed between A and B. As I see this is very close to real life cases where we speak about black holes.

Question could be if inside observer can decide about event horizon. I think he can't directly. But if he lives some extended time without anything horrible happening to him that he could interpret as hitting singularity then he is not inside black hole.

 

[PAllen]

I think this discussion somewhere just went astray.
If we would have to wait eternity to define black hole we could safely discard the concept as non-scientific crap.

You don't have to wait for eternity. You just need to be able to calculate ultimate escape.

The definition you call 'crap' is the only one accepted by all professional experts on GR: Hawking, Ellis, Penrose, Geroch, Poisson, anyone you can name.

So let's look at the problem of event horizon definition. We have observer A and object B. Light is continuously traveling from B to A. If at some moment somewhere along the way light gets stuck (in a global coordinate system of A) we say that event horizon has formed between A and B. As I see this is very close to real life cases where we speak about black holes.

What is the definition of stuck? Taking one year to escape? Ten years ?

There is a way to get at what I think you are looking for. It is the apparent horizon versus the true horizon.

The differences between it and the true horizon are not great for the cases discussed in this thread. For example, for a collapsing shell, the true horizon starts forming while the shell is still a little beyond its SC radius, and it starts at a point. The apparent horizon forms a little later, when the shell is at the point of no return, and it can jump into existence at a finite radius. It is still true that there is no matter at the center and no singularity when the apparent horizon has formed.

Question could be if inside observer can decide about event horizon. I think he can't directly. But if he lives some extended time without anything horrible happening to him that he could interpret as hitting singularity then he is not inside black hole.

The event horizon doesn't exist for a free falling observer. This is the same as a Rindler horizon - it only exists for accelerating observers, not for inertial observers. Inertial observers receive signals from the accelerating observer forever. Free falling observers receive signals from the outside until they hit the singularity. The black hole event horizon only exists for external observers (which have nonzero proper acceleration).

The apparent horizon is the best that can be done if you want a quasilocal definition rather than a global definition.

 

[TrickyDicky]

You don't have to wait for eternity. You just need to be able to calculate ultimate escape.

One can calculate the most fantastic things, that by itself doesn't make them physically plausible. What I mean is that not everything that can be calculated is physical, when those calculations are purely coordinate-dependent the result of the calculations is not physical according to our current understanding.
Only physical objects can be properly said to "form" in a causal way.
You keep saying the event horizon is acausal and yet in your examples you give a completely causal narrative about how and when it forms, that is because all physical observers are causal.

The event horizon doesn't exist for a free falling observer. This is the same as a Rindler horizon - it only exists for accelerating observers, not for inertial observers. Inertial observers receive signals from the accelerating observer forever. Free falling observers receive signals from the outside until they hit the singularity. The black hole event horizon only exists for external observers (which have nonzero proper acceleration).

Only if we agree that event horizons are not physical objects at all, but simply coordinate-dependent mathematical boundaries, can we agree that they are acausal.
Do you agree that Rindler horizons are purely coordinate artifacts? And that coordinate effects are not necessarily physical (they are physical precisely when they are coordinate-invariant)? The classical example in two dimensions is the coordinate singularity at the poles of the sphere, it is removed by changing the coordinate chart, in the same way one can remove the Rindler horizon or the apparent singularity at the Schzwarzschild radius by simply changing the coordinates.
If you agree sofar you must agree that the event horizon is a purely coordinate-dependent mathematical entity, and it cannot be endowed with any physical property. If that is the case you cannot use it to derive any physical consequences for any observer (either external, infalling, static...), you just can mathematically calculate certain outcomes when certain coordinate charts are used.

So we need to have the real singularity at r=0 (the one that cannot be removed by changing charts), not just the event horizon, to derive physical consequences for particles that are at a certain distance r from the true singularity, depending on wheteher they are at one side or the other of said distance.
But then you cannot really say that you can have a black hole and/or an event horizon without a true singularity as you were suggesting. All the physical effects of event horizons on any observer are due to the singularity and its infinite curvature and don't exist if there is no true singularity at the center. All observable horizons(be it the sea horizon or the cosmological one) are due to curvature by the way, and don't imply that anything strange is going on at the physical region where we see the horizon.
Any EH needs to have a singularity inside, by definition.

 

[PeterDonis]

Do you agree that Rindler horizons are purely coordinate artifacts?

I can't speak for PAllen, but I don't. Some statements regarding Rindler horizons are coordinate-independent, invariant statements. Here's one: "No light signal can travel from any event in the region 'inside' the Rindler horizon, to any event in the region 'outside' the Rindler horizon."

Of course, which null surface counts as the "Rindler horizon" depends on where you put the "origin" of the light cone that defines it--or, equivalently, which particular set of accelerating Rindler observers you pick to define the horizon. The event horizon of a black hole differs from the Rindler horizon in that respect, because Schwarzschild spacetime has only one timelike Killing vector field, but Minkowski spacetime has an infinite number of them--in fact, two infinite families of them (one family for all possible inertial frames, and one family for all possible Rindler frames). So Minkowski spacetime has an infinite number of possible Rindler horizons, but Schwarzschild spacetime has only one event horizon.

However, you can still make a similar statement to the one I made above, about the event horizon of a black hole. The fact that the exact location of the horizon (i.e., exactly *which* null surface it is that defines the boundary between the two regions) can't be known without knowing the entire future of the spacetime, doesn't make the statement any less invariant.

 

[PAllen]

I can't speak for PAllen, but I don't. Some statements regarding Rindler horizons are coordinate-independent, invariant statements. Here's one: "No light signal can travel from any event in the region 'inside' the Rindler horizon, to any event in the region 'outside' the Rindler horizon."

I will respond more later, but a way I would phrase it is that a horizon is observer dependent, but this attribute off a particular observer or class of observers is not coordinate dependent.

To make this explicit for the Rindler horizon, note that there is no need to use Rindler coordinates. It is, in fact, easier to derive the Rindler horizon in Minkowski coordinates, though I have seen few authors approach it this way: draw the world line of an accelerating observer; compute the envelope of all its backward light cones; this envelope remains completely below the 45° asymptote of the world line. Thus, the spacetime above this asymptote is causally disconnected from the accelerating observer.

Similarly, for the class of BH external observers, spacetime inside the horizon is causally disconnected from these observers. However, a free fall observer's past light cones include events on both sides of the horizon (as well as events from possible world lines of earlier infallers) right up until the singularity. Thus, the feature of the horizon being a causal boundary is, as with Rindler, observer dependent but not coordinate dependent (everything above is phrased in terms of observer light cones, not coordinates).

 

[PeterDonis]

Similarly, for the class of BH external observers, spacetime inside the horizon is causally disconnected from these observers. However, a free fall observer's past light cones include events on both sides of the horizon (as well as events from possible world lines of earlier infallers) right up until the singularity.

A small point, but I think it should be clarified: the infalling observer's past light cones only include events inside the horizon once that observer himself has fallen inside the horizon. While he is still outside the horizon, even though he is freely falling inward, his past light cones only contain events outside the horizon.

Because of this, I would *not* say that a BH's horizon is observer-dependent; all observers agree (in the idealized case where everyone knows the entire future of the spacetime) on which null surface in the spacetime is the horizon. The only difference between the observers is whether their worldlines enter the region behind the horizon or not.

Note, again, that this is *different* from the case of a Rindler horizon, which I *would* say is observer-dependent. This goes back to what I said before, that Minkowski spacetime has an infinite number of timelike Killing vector fields, while Schwarzschild spacetime has only one. So in Minkowski spacetime, different observers can have different horizons, depending on which timelike Killing vector field they pick out as defining what the "horizon" is. There is no such freedom of choice in Schwarzschild spacetime; there is only one timelike Killing vector field, and all observers agree on which null surface is the boundary of the region of spacetime in which that Killing vector field is timelike.

 

[zonde]

The definition you call 'crap' is the only one accepted by all professional experts on GR: Hawking, Ellis, Penrose, Geroch, Poisson, anyone you can name.

From wikipedia article about Absolute horizon:
"The definition of an absolute horizon is sometimes referred to as teleological, meaning that it cannot be known where the absolute horizon is without knowing the entire evolution of the universe, including the future."
I take "theological" as approximately equivalent to non-scientific [crap]. (have to admit however that there is no reference for that statement in wikipedia)

What is the definition of stuck? Taking one year to escape? Ten years ?

There is a way to get at what I think you are looking for. It is the apparent horizon versus the true horizon.

I think that definition of apparent horizon is fine.

The differences between it and the true horizon are not great for the cases discussed in this thread. For example, for a collapsing shell, the true horizon starts forming while the shell is still a little beyond its SC radius, and it starts at a point. The apparent horizon forms a little later, when the shell is at the point of no return, and it can jump into existence at a finite radius. It is still true that there is no matter at the center and no singularity when the apparent horizon has formed.

Apparent horizon is observer dependent. Then if we pick distant observer that is not gravitationally bound to collapsing object there should be no difference for apparent horizon and absolute horizon as long as we don't look into too distant future.
With that on mind I do not understand how you can claim that "apparent horizon forms a little later".

 

[PAllen]

A small point, but I think it should be clarified: the infalling observer's past light cones only include events inside the horizon once that observer himself has fallen inside the horizon. While he is still outside the horizon, even though he is freely falling inward, his past light cones only contain events outside the horizon.

Your wording " the infalling observer's past light cones only include events inside the horizon once that observer himself has fallen inside the horizon" could lead to an incorrect impression. Once past the horizon, the free faller's past light cone includes events both inside and outside the horizon. If you look at the causal past of the free faller's world line as a whole, the horizon is not a causal boundary.

Because of this, I would *not* say that a BH's horizon is observer-dependent; all observers agree (in the idealized case where everyone knows the entire future of the spacetime) on which null surface in the spacetime is the horizon. The only difference between the observers is whether their worldlines enter the region behind the horizon or not.

The horizon is a surface - a geometric object - and is thus invariant. However, it represents a causal boundary only for some observers (a very large class - those that remain forever outside). Radial infallers do not perceive it as a causal boundary.

Note again, an analogy with the Rindler horizon is possible. Define such a horizon by a particular observer's accelerating world line. Then no observer has events 'beyond the horizon' in its causal past until such observer has 'crossed' the horizon.

 

[PAllen]

Apparent horizon is observer dependent. Then if we pick distant observer that is not gravitationally bound to collapsing object there should be no difference for apparent horizon and absolute horizon as long as we don't look into too distant future.
With that on mind I do not understand how you can claim that "apparent horizon forms a little later".

1) For an eternally external observer, no events on or inside the true horizon ever reach them. This is also true for the apparent horizon (which is almost always either inside or the same as the absolue horizon). For this observer, it is meaningless to talk about the difference because both represent events causally disconnected from the external observer.

2) For a radial free falling observer, you can make the statement that they cross the true horizon before an apparent horizon (when they are different). Thus, for every world line that crosses both, the the apparent horizon crossing occurs later. Taking this as a collective observation of infalling observers, I loosely speak of the apparent horizon forming later. Specifically, looking at world lines of observers remaining near the center of a collapsing cluster, they encounter the true horizon before the apparent horizon. This is because there is brief time period where their outgoing light moves outward (yet is eventually trapped - so is inside the true horizon).

 

[PeterDonis]

If you look at the causal past of the free faller's world line as a whole, the horizon is not a causal boundary.

I like this wording better, because it makes clear that you are looking at the union of the past light cones of *all* events on the observer's worldline, not just the past light cone of a single event.

The horizon is a surface - a geometric object - and is thus invariant. However, it represents a causal boundary only for some observers (a very large class - those that remain forever outside). Radial infallers do not perceive it as a causal boundary.

With the definition of "causal boundary" using the wording above, yes, I agree. But IMO it's also important to guard against the opposite misunderstanding: saying "the horizon is not a causal boundary" without the above wording could give the impression that free-fallers can somehow get signals from events inside the horizon while they're still outside it, even though static observers can't.

Note again, an analogy with the Rindler horizon is possible. Define such a horizon by a particular observer's accelerating world line. Then no observer has events 'beyond the horizon' in its causal past until such observer has 'crossed' the horizon.

Yes, I agree. But in the case of the Rindler horizon, you have to add that qualification "define such a horizon by a particular observer's accelerating worldline", which defines which timelike Killing vector field's horizon you are talking about. In Schwarzschild spacetime, no such qualification is necessary.

 

[zonde]

I am not entirely happy with the idea of horizon forming at the center of collapsing shell so I propose for consideration slightly different scenario.
Let's say that we have shell-like distribution of many smaller black holes that are collectively collapsing. When they get inside SC radius of summary mass of all the smaller black holes their event horizons are simply joined together, right? It is not exactly meaningful to say that black hole is falling into black hole, right?

 

[TrickyDicky]

I can't speak for PAllen, but I don't. Some statements regarding Rindler horizons are coordinate-independent, invariant statements. Here's one: "No light signal can travel from any event in the region 'inside' the Rindler horizon, to any event in the region 'outside' the Rindler horizon."

Maybe I should have qualified better my question, but I thought it was obvious what I was referring to.
The statement you call coordinate-invariant is not about horizons, but about physical properties of light and the Rindler observers motion, both of which are indeed coordinate-invariant.
In the Rindler case the horizon is just the boundary of the Rindler cordinates, and given the fact that light has finite speed it is only natural that observers at constant position in that universe that must have constant proper acceleration perceive a horizon at a certain distance where they can no longer receive light, but that is not a property of the physical location of the horizon they perceive, it is a property of light motion and their own motion (constant proper acceleration).
Of course the Rindler coordinates are a chart of an idealized spacetime, the Minkowski one, that is not ours. Rindler observers are in a certain frame and can't change their state of motion as long as they want to keep being Rindler observers.


So in this limited sense sure the horizon is "real" for the external observer, meaning it is perceived as such by him, but that has nothing to do with the horizon in itself, it is not something physical.

That is common to all horizons when you think you're going to reach them they keep receding (this includes also the perception of an BH's infaller observer until it hits the putative singularity).
This is reminiscent of the old time legends about sailors that thought the Earth was flat and that in the horizon line something like a cliff existed that sucked ships down, however no matter how fast they chased the horizon the precipice was nowhere to be found.
So I'll ask again, do you think a Rindler observer can ever "reach" the Rindler horizon?

 

[PeterDonis]

The statement you call coordinate-invariant is not about horizons, but about physical properties of light and the Rindler observers motion, both of which are indeed coordinate-invariant.

This looks to me like an issue with choice of words, not physics. I don't object to that per se, since I was also raising an issue with choice of words (how PAllen was describing the horizon of a black hole).

However, if we're going to talk about choice of words, it would seem more relevant to talk about the words we're using to describe the event horizon of a black hole, since that's the actual scenario under discussion in this thread. Rindler horizons were only brought up by analogy, and one of the points I've been making is about ways in which the analogy does *not* hold. In particular, you don't appear to be taking into account the key difference between Rindler horizons in Minkowski spacetime and the event horizon of a black hole: there are an infinite number of the former, but only one of the latter.

That is common to all horizons when you think you're going to reach them they keep receding (this includes also the perception of an BH's infaller observer until it hits the putative singularity).

This is a statement about the light images that the infalling observer sees: yes, an infaller sees images that make it "appear" that the horizon is receding from him, even after he has crossed the horizon. This is yet another reason why choice of words can be important: we've had at least one thread recently where people were claiming that no one can ever cross the horizon, because the horizon always appears to recede from them.

 

[TrickyDicky]

I will respond more later, but a way I would phrase it is that a horizon is observer dependent, but this attribute off a particular observer or class of observers is not coordinate dependent.

Sure, (see my previous post), that attribute is simply light invariant speed, that is what determines what any observer can or can't "observe" , that perceptual part related to light properties is of course coordinate-independent too.
But the the horizon in itself as an object in a certain position is purely coordinate-dependent, a coordinate singularity, that is always calculated using a certain chart and depends on that specific chart.
Light finite speed is an invariant for any observer, it is observer-independent, horizons are not observer dependent, all physical observers perceive the horizon ahead of them.

To make this explicit for the Rindler horizon, note that there is no need to use Rindler coordinates. It is, in fact, easier to derive the Rindler horizon in Minkowski coordinates, though I have seen few authors approach it this way: draw the world line of an accelerating observer; compute the envelope of all its backward light cones; this envelope remains completely below the 45° asymptote of the world line. Thus, the spacetime above this asymptote is causally disconnected from the accelerating observer.

Exactly that is because as I said horizons as something perceived by any observer is an invariant related to the speed of light.

Similarly, for the class of BH external observers, spacetime inside the horizon is causally disconnected from these observers. However, a free fall observer's past light cones include events on both sides of the horizon (as well as events from possible world lines of earlier infallers) right up until the singularity. Thus, the feature of the horizon being a causal boundary is, as with Rindler, observer dependent but not coordinate dependent (everything above is phrased in terms of observer light cones, not coordinates).

Again, see above. All observations any observer can perform related with light is observer-independent due to light's invariant properties. I'm of course only referring to observers that are considered physical, that is those that cannot accelerate to light speed.

 

[TrickyDicky]

This looks to me like an issue with choice of words, not physics. I don't object to that per se, since I was also raising an issue with choice of words (how PAllen was describing the horizon of a black hole).

However, if we're going to talk about choice of words, it would seem more relevant to talk about the words we're using to describe the event horizon of a black hole, since that's the actual scenario under discussion in this thread. Rindler horizons were only brought up by analogy, and one of the points I've been making is about ways in which the analogy does *not* hold. In particular, you don't appear to be taking into account the key difference between Rindler horizons in Minkowski spacetime and the event horizon of a black hole: there are an infinite number of the former, but only one of the latter.

Yes, let's go back to BHs, I acknowledge the difference, and the difference is that in the BH case there is a true singularity in the center, unlike the Rindler case.
That is why all this was to point out that if you eliminate the singularity as PAllen suggested, it basically makes no sense to tlk about event horizons.

This is a statement about the light images that the infalling observer sees: yes, an infaller sees images that make it "appear" that the horizon is receding from him, even after he has crossed the horizon. This is yet another reason why choice of words can be important: we've had at least one thread recently where people were claiming that no one can ever cross the horizon, because the horizon always appears to recede from them.

Well, that might be related to the fact that an event horizon has never been observed.
But certainly I agree it is predicted by GR.

 

[PAllen]

That is why all this was to point out that if you eliminate the singularity as PAllen suggested, it basically makes no sense to tlk about event horizons.

This is not true, per GR. For one thing, Birkhoff's theorem alone establishes that as soon as spherically symmetric mass distribution is inside its SC radius, you have an event horizon. The singularity comes later. Further, it is non-trivial consequence, requiring some extra assumptions, to conclude that the mass inside an event horizon must ultimately form a singularity. Birkhoff's theorem requires no energy conditions assumptions - it is strictly a consequence of EFE. The singularity theorems require additional assumptions. There are a number of known GR solutions with slightly exotic matter, that constitute stable, non-singular black holes (i.e. spherical event horizon with stable, nonsingular, exotic matter distribution inside).

[Edit: I should also emphasize that for all scenarios discussed in this thread, there is a singularity formed. The point of discussion was that the event horizon forms before the singularity in any reasonable collapse, and may form in a vacuum region - that will later be occupied by matter, and then a singularity.]

 

[PeterDonis]

But the the horizon in itself as an object in a certain position is purely coordinate-dependent, a coordinate singularity, that is always calculated using a certain chart and depends on that specific chart.

No, this is not correct, at least not for the event horizon of a black hole. (It's not really correct for Rindler horizons either, but I'm not sure it's worth going into that, though it can be seen using the same idea I'm about to use for the BH case.)

The event horizon is a particular null surface in the spacetime, and can be defined in a coordinate-free manner, without reference to any chart. I hinted at the definition in earlier posts, but here it is explicitly: the event horizon is the boundary of the region in which the Killing vector field of the "time translation" isometry of Schwarzschild spacetime is timelike--i.e., the EH is the Killing horizon associated with the "time translation" isometry. There is a proof--I believe it's in Hawking & Ellis--that the event horizon of any stationary BH must be a Killing horizon, so this idea doesn't just apply to Schwarzschild BH's, it applies to the whole family of generalized Kerr-Newman BH's.

The fact that there is also a coordinate singularity at the EH in a particular chart *is*, of course, dependent on that specific chart; but you don't need that fact to define the EH itself and its properties.

 

[PAllen]

I am not entirely happy with the idea of horizon forming at the center of collapsing shell so I propose for consideration slightly different scenario.
Let's say that we have shell-like distribution of many smaller black holes that are collectively collapsing. When they get inside SC radius of summary mass of all the smaller black holes their event horizons are simply joined together, right? It is not exactly meaningful to say that black hole is falling into black hole, right?

This is a fairly complicated scenario. I can make some educated guesses, but cannot be sure about the following observations:

1) When the shell of small BHs is of 'large' radius, there are separate event horizons for each BH. (uncontroversial).

2) Shortly before the shell reaches the collective SC radius, the total event horizon consists of a horizon around each BH plus a growing sphere in the empty center. The reason for this remains as I gave before: light on this inner spherical surface will not arrive at the shell until the shell has reached (or passed) the SC radius, so this light gets trapped. Meanwhile, light emitted from inside the shell but further from the center, and between any shell BHs can still escape. I believe this argument justifies my proposed shape for the horizon at this point.

3) Once the shell of BHs has reached SC radius, there is one collective event horizon.

4) Once the BH's are inside the collective SC radius, they no longer have individual true event horizons. However (until they get close enough to each other) I think they still have individual apparent horizons for observers inside the collapsing shell. This is because light can still proceed locally outward if it originates far enough away from each BH. This last discussion is in the spirit of a sufficiently large shell that the BH's can be well separated well within the SC radius.

5) At some time, well before the BH singularities have coalesced in some way, when they get 'too close', you would have one merged apparent horizon.

 

[TrickyDicky]

This is not true, per GR. For one thing, Birkhoff's theorem alone establishes that as soon as spherically symmetric mass distribution is inside its SC radius, you have an event horizon. The singularity comes later. Further, it is non-trivial consequence, requiring some extra assumptions, to conclude that the mass inside an event horizon must ultimately form a singularity. Birkhoff's theorem requires no energy conditions assumptions - it is strictly a consequence of EFE. The singularity theorems require additional assumptions. There are a number of known GR solutions with slightly exotic matter, that constitute stable, non-singular black holes (i.e. spherical event horizon with stable, nonsingular, exotic matter distribution inside).

[Edit: I should also emphasize that for all scenarios discussed in this thread, there is a singularity formed. The point of discussion was that the event horizon forms before the singularity in any reasonable collapse, and may form in a vacuum region - that will later be occupied by matter, and then a singularity.]

I disagree. I think you read too much in Birkhoff's theorem, it actually doesn't say what you claim it says. It says that a vacuum, spherically symmetric solution must be static and the Schwarzschild solution. The consequences that you extract from this were originated in an interpretation of the Schwarzschild solution that came many years after the theorem was proved by Birkhoff. Your wording gives the misleading impression that the original theorem included event horizons and its causal relations with singularities in its text.
And your argument sounds weird after your insistence that the spherically symmetric eternal BH is unphysical.
I don't quite understand why you previously insisted on the acausality of the EH and seem now so interested in remarking it "forms" before the singularity.
The relevant fact is that before, after or at the same time, the EH in GR is associated always to a singularity, as Penrose for instance states according to wikipedia.

 

[TrickyDicky]

No, this is not correct, at least not for the event horizon of a black hole. (It's not really correct for Rindler horizons either, but I'm not sure it's worth going into that, though it can be seen using the same idea I'm about to use for the BH case.)

The event horizon is a particular null surface in the spacetime, and can be defined in a coordinate-free manner, without reference to any chart. I hinted at the definition in earlier posts, but here it is explicitly: the event horizon is the boundary of the region in which the Killing vector field of the "time translation" isometry of Schwarzschild spacetime is timelike--i.e., the EH is the Killing horizon associated with the "time translation" isometry. There is a proof--I believe it's in Hawking & Ellis--that the event horizon of any stationary BH must be a Killing horizon, so this idea doesn't just apply to Schwarzschild BH's, it applies to the whole family of generalized Kerr-Newman BH's.

The fact that there is also a coordinate singularity at the EH in a particular chart *is*, of course, dependent on that specific chart; but you don't need that fact to define the EH itself and its properties.

But that definition is coordinate dependent, it is precisely dependent on the fact that removing the coordinate singularity implies a coordinate transformation that swaps timelike and spacelike Killing vector fields. In doing so it transforms invariants, but as long as it is considered a licit transformation...

 

[PeterDonis]

But that definition is coordinate dependent, it is precisely dependent on the fact that removing the coordinate singularity implies a coordinate transformation that swaps timelike and spacelike Killing vector fields.

No, it isn't. You have things backwards. The Killing vector field is not "swapped" by a coordinate transformation; it is a feature of the underlying geometry, independent of the choice of coordinates. So is the timelike, spacelike, or null nature of the Killing vector field at a particular event or within a particular region of the spacetime. Whether or not a particular chart is singular at a particular location is *derived* from the underlying geometry, plus the definition of the chart.

The correct way of describing the behavior of the "time translation" Killing vector field, which makes explicit the fact that its definition is completely coordinate-free, is that it is timelike outside the horizon, null on the horizon, and spacelike inside the horizon. A still more precise statement would involve defining "outside" and "inside" the horizon in terms of the physical areas of 2-spheres: 2-spheres with areas greater than $16 \pi M^2$ lie outside the horizon, 2-spheres with areas less than $16 \pi M^2$ lie inside the horizon, and the 2-sphere with area equal to $16 \pi M^2$, of course, lies on the horizon.

None of the above requires defining any coordinate charts at all, nor does it make any use of the fact that Schwarzschild coordinates are singular at the horizon.

 

[PAllen]

I disagree. I think you read too much in Birkhoff's theorem, it actually doesn't say what you claim it says. It says that a vacuum, spherically symmetric solution must be static and the Schwarzschild solution. The consequences that you extract from this were originated in an interpretation of the Schwarzschild solution that came many years after the theorem was proved by Birkhoff. Your wording gives the misleading impression that the original theorem included event horizons and its causal relations with singularities in its text.

There is no real difference in our interpretation of what Birkhoff's theorem. However, I am applying it in a slightly clever way, which is well known. If a manifold is spherically symmetric, and is vacuum outside some 2-sphere, then the uniqueness of vacuum spherically symmetric metrics requires that the solution be exactly SC metric in the vacuum region outside a sphere containing all the matter. This requires that as soon as a collapsing spherical shell is inside the SC radius, the vacuum solution outside the shell must be exactly SC geometry - including the event horizon.

And your argument sounds weird after your insistence that the spherically symmetric eternal BH is unphysical.

This thread has not focused on eternal BH, and this argument is not about eternal BH - it uses Birkhoff to make inferences about spherically symmetric collapse using a standard argument.

I don't quite understand why you previously insisted on the acausality of the EH and seem now so interested in remarking it "forms" before the singularity.
The relevant fact is that before, after or at the same time, the EH in GR is associated always to a singularity, as Penrose for instance states according to wikipedia.

That a BH horizon is associated with a singlularity in all physically plausible scenarios is not in dispute. However, the fact that the EH is a feature of the solution as a whole (and is acausal in that sense) does not make it meaningless to talk about the time ordering of events for a class of observers. As I made clear in a reply to Zonde a few posts back, a way to disambiguate as well as make physical statements like "EH forms before apparent horizon forms before singularity" is to ask about the order of events on various time like world lines. If a timelike world line in which the COM of a collapsing cluster is stationary, and which is at the center of such a cluster when it is large (and remains at the center), encounters first the EH, then the apparent horizon, then the singularity, it is meaningful to loosely speak as I have: EH forms first, then AH, then singularity.

 

[zonde]

2) Shortly before the shell reaches the collective SC radius, the total event horizon consists of a horizon around each BH plus a growing sphere in the empty center. The reason for this remains as I gave before: light on this inner spherical surface will not arrive at the shell until the shell has reached (or passed) the SC radius, so this light gets trapped. Meanwhile, light emitted from inside the shell but further from the center, and between any shell BHs can still escape. I believe this argument justifies my proposed shape for the horizon at this point.

What changes if we take away growing sphere of EH in the empty center?
Nothing.

Basically in scenario of collapsing spherically symmetric shell we can say that event horizon appears at once at a finite radius.

 

[PAllen]

What changes if we take away growing sphere of EH in the empty center?
Nothing.

Basically in scenario of collapsing spherically symmetric shell we can say that event horizon appears at once at a finite radius.

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?

 

[TrickyDicky]

The fact that there is also a coordinate singularity at the EH in a particular chart *is*, of course, dependent on that specific chart; but you don't need that fact to define the EH itself and its properties.

But you do need it, (at least for non-rotating black holes), it is not a problem of that particular chart only.
For instance if you consider that the EH/Sch. radius is part of the vacuum exterior solution of a collapsing object. In Schwarzschild coordinates, the EH is defined as the coordinate singularity using those coordinates. This is independent of the fact that using different coordinates you can extend the solution to the shell interior.
Or if you think considering the EH to belong to the exterior region solution is some kind of "word trick" you can use the Kruskal -Szekeres coordinates if you want to, you have exactly the same situation, the EH lies at the coordinate singularity, meaning the transformation between Schwarzschild coordinates and Kruskal–Szekeres coordinates is defined for r > 0, r ≠ 2GM, and −∞ < t < ∞, so it is not defined at the EH.

From wikipedia:

All this leads to an apparently paradoxical situation because there is a ctually a kind of twisted sense in which the EH is coordinate-independent as you claim: in the sense that no coordinates exist that cover the transition between the exterior and the interior regions, all coordinates have the EH as a limit coordinate singularity, but I gues this is not what you mean by the EH being coordinate-independent, among other things because a coordinate-independent singularity is no longer a coordinate singularity.

The Killing vector field is not "swapped" by a coordinate transformation; it is a feature of the underlying geometry, independent of the choice of coordinates. So is the timelike, spacelike, or null nature of the Killing vector field at a particular event or within a particular region of the spacetime. Whether or not a particular chart is singular at a particular location is *derived* from the underlying geometry, plus the definition of the chart.

Ok, I won't dispute this here. But actually this doesn't contradict my claim about the EH being coordinate-dependent. See above.

None of the above requires defining any coordinate charts at all, nor does it make any use of the fact that Schwarzschild coordinates are singular at the horizon.

See above.

 

[PeterDonis]

For instance if you consider that the EH/Sch. radius is part of the vacuum exterior solution of a collapsing object. In Schwarzschild coordinates, the EH is defined as the coordinate singularity using those coordinates.

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.

Or if you think considering the EH to belong to the exterior region solution

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.

you can use the Kruskal -Szekeres coordinates if you want to, you have exactly the same situation, the EH lies at the coordinate singularity

There is no coordinate singularity in K-S coordinates.

meaning the transformation between Schwarzschild coordinates and Kruskal–Szekeres coordinates is defined for r > 0, r ≠ 2GM, and −∞ < t < ∞, so it is not defined at the EH.

So what? That's a problem with the Schwarzschild coordinates, not with the EH. And it's not the correct definition of a coordinate singularity; see below.

All this leads to an apparently paradoxical situation because there is a ctually a kind of twisted sense in which the EH is coordinate-independent as you claim

It's not a paradox at all to me.

in the sense that no coordinates exist that cover the transition between the exterior and the interior regions

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. 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.

all coordinates have the EH as a limit coordinate singularity

No, they don't. See above.