Graphical example of BH formation by PAllen

[PeterDonis]

This paper suggests it should be perfectly possible to have spherically symmetric collapse with outgoing null radiation (which can represent incoherent light):

http://arxiv.org/pdf/gr-qc/0504045v1.pdf

This particular construction specifies ingoing radiation (incoming Vaidya metric), but it seems very likely to me that you could match outgoing Vaidya to collapsing dust using similar methods. This would be a perfectly spherically symmetric solution.

Hm, good point, the Vaidya null dust is spherically symmetric (I think both ingoing and outgoing are). But the Vaidya null dust does not directly model any "source" for the radiation; you can match it to collapsing matter, as this paper does, but that doesn't really explain how the matter radiates. In particular, I don't believe the Vaidya null dust is derived by solving the combined Einstein-Maxwell equations, so it doesn't necessarily represent a physically reasonable source for EM radiation. But you're right, it is a spherically symmetric metric with radiation present.

 

[PeterDonis]

Do these two cases lead to different horizons with any different characteristics??

The horizon of a charged or rotating BH (both of which have timelike singularities in the idealized case of exact symmetry) does have some different characteristics from that of an uncharged, nonrotating BH (which has a spacelike singularity in the idealized case). However, they're not that much different, certainly not as different as the singularities are. AFAIK the speculation about the timelike singularities not being stable under perturbations does not apply to their corresponding horizons; I believe the horizons themselves are thought to be physically possible, it's just what's hidden deeper inside them that may be very different from the idealized case.

I think the definition I have seen is consistent with 'outward-pointing light rays are actually converging (moving inwards)..'...Do you guys agree??

I do, yes.

Seems like other horizons maybe Rindler, might not meet this 'closed' definition?? Is that correct??

I think so; I don't think it's possible to find a closed 2-surface that is contained in the Rindler horizon, because the motion of the family of observers that define the horizon is not spherically symmetric. In any case, the Rindler horizon is not a trapped null surface (it's null, but it's not trapped).

 

[PAllen]

Hm, good point, the Vaidya null dust is spherically symmetric (I think both ingoing and outgoing are). But the Vaidya null dust does not directly model any "source" for the radiation; you can match it to collapsing matter, as this paper does, but that doesn't really explain how the matter radiates. In particular, I don't believe the Vaidya null dust is derived by solving the combined Einstein-Maxwell equations, so it doesn't necessarily represent a physically reasonable source for EM radiation. But you're right, it is a spherically symmetric metric with radiation present.

It can't possibly represent an exact EM solution for the very reason that even in SR there are no point sources of radiation, only dipole or higher. At a distance, for all practical purposes, you can treat spherical wave front, but not if we are discussing exact spherical symmetry.

However, the Vaidya null dust outgoing radiation could model e.g. massless neutrinos or the like. However, your point about source still remains. You would have to treat it as a causeless source of information about the boundary between matter and 'radiation'.

In any case, the main point is that real world difficulties with exact spherical symmetry does not impede making reasonable conclusions from artificial exact cases. It's one thing to note that the internal region approaching the singularity is likely very inaccurate in the same sense as suggesting that a ring of sharpshooter firing together is a useful way to manufacture canonballs (as opposed to collective suicide). However, in both cases, away from the very center, spherical symmetry is a useful approximation, and there is no reason I know of (or proposed) to doubt general conclusions about horizon formation (here I am talking to Zonde - I know you agree).

 

[zonde]

PAllen,
But do you agree that putting in restriction that there is (asymptotically) no EM radiation is statement about matter configuration?

 

[PAllen]

PAllen,
But do you agree that putting in restriction that there is (asymptotically) no EM radiation is statement about matter configuration?

Not really. Putting in realistic amounts of light emission with infinitesimal deviations from spherical symmetry would greatly complicate the math but not change any of the main conclusions we're drawing in this thread. Note, that you have freely argued from SC coordinates even though they are just one coordinate system on the most perfectly simple geometry, whenever it suits your purpose - which is fine, as long you don't attach significance to the specialized features which would not generalize to realistic situations.

 

[my_wan]

I think that the utility of examples with free falling observers dries up at the moment when you try to construct global coordinate system where some background stays more or less static, isotropic and homogenous.

I certainly was not trying to imply some specific "utility" of "free falling observers". I was pointing out that the definition of an event horizon suffered the same limited utility issues that free falling observers do.

It appears to me that you are implying that "free falling observers" lack a certain utility while "event horizons" retain said utility, even though the definition of the "event horizon" itself is an observer dependent construct. Perhaps you intended something more nuanced but, so far as I can see, this is what you implied.

Tidal forces are not exclusively associated with event horizon. Tidal forces are present in any field of gravity.

Really! I thought it was quantum fluctuations.. Just kidding, of course tidal forces are common to all gravitational bodies.

This assumption is problematic if you are trying to construct an argument about possible formation of black hole.
Look up Begging the question fallacy.

Which question is it begging here? It's a matter of historical fact that black holes where theoretical entities long before Einstein. If you thought I "assumed" photons have mass you are wrong. This was merely an assumption that existed before Einstein and QM, on which pre-Einstein black holes were theoretically predicated on. The only feature required to qualify as a black hole is that light can't escape. I was stating a historical fact, not making any claim, or assumption, we know today to be invalid.

 

[zonde]

Not really. Putting in realistic amounts of light emission with infinitesimal deviations from spherical symmetry would greatly complicate the math but not change any of the main conclusions we're drawing in this thread.

Basically you think (believe) that there are no factors that can oppose runaway gravitational collapse given big enough mass, right?

Maybe we can end our discussion there? Your replays where very good but we have to stop somewhere.

 

[PAllen]

Basically you think (believe) that there are no factors that can oppose runaway gravitational collapse given big enough mass, right?

Maybe we can end our discussion there? Your replays where very good but we have to stop somewhere.

Fine, but one slight qualification: I think that is what GR predicts. I do not believe singularities actually form, and I have doubts about the exact nature of event horizons. I distinguish understanding what GR predicts, as a classical theory, from what is likely true in our universe - that GR breaks down in certain regimes, just as Maxwell's equations do.

 

[pervect]

Basically you think (believe) that there are no factors that can oppose runaway gravitational collapse given big enough mass, right?

Maybe we can end our discussion there? Your replays where very good but we have to stop somewhere.

As far as I can tell, (I have only been skimming the thread, because from what I've read it hasn't been going anywhere) the discussion isn't actually about this issue, but it's about something simpler, which is whether there are any factors that can prevent the formation of an event horizon.

And it's pretty clear that the answer to that (in the literature) is no.

 

[zonde]

It appears to me that you are implying that "free falling observers" lack a certain utility while "event horizons" retain said utility, even though the definition of the "event horizon" itself is an observer dependent construct. Perhaps you intended something more nuanced but, so far as I can see, this is what you implied.

If we speak about event horizon as closed surface then we want some global coordinate system. And it seems to me (but you can dispute this) that in any viable global coordinate system event horizon keeps it's place.

Which question is it begging here? It's a matter of historical fact that black holes where theoretical entities long before Einstein. If you thought I "assumed" photons have mass you are wrong. This was merely an assumption that existed before Einstein and QM, on which pre-Einstein black holes were theoretically predicated on. The only feature required to qualify as a black hole is that light can't escape. I was stating a historical fact, not making any claim, or assumption, we know today to be invalid.

I didn't mean assumption that we can model such gravity field that light can't escape. I rather meant assumption that there exists (can form) gravitating object with such gravity field.

 

[zonde]

My intention about this thread was to check out if formation of black hole does not require pre-existing micro black hole. And it seems I got an answer. Apparent event horizon can form at once and as I consider it physically meaningful contrary to absolute horizon it is the answer to my question - pre-existing micro black hole is not required.

 

[my_wan]

If we speak about event horizon as closed surface then we want some global coordinate system. And it seems to me (but you can dispute this) that in any viable global coordinate system event horizon keeps it's place.

You have effectively just defined all possible coordinate systems as on-viable.

I didn't mean assumption that we can model such gravity field that light can't escape. I rather meant assumption that there exists (can form) gravitating object with such gravity field.

So I get from this you don't believe black holes exist. Nothing wrong with questioning their legitimacy, in whole or in part, but to simply deny their existence is just as wrong as an insistence they must exist a priori. Given our observational data at present denying the possibility of such an assumption requires some major contortions of logic.

 

[zonde]

So I get from this you don't believe black holes exist. Nothing wrong with questioning their legitimacy, in whole or in part, but to simply deny their existence is just as wrong as an insistence they must exist a priori. Given our observational data at present denying the possibility of such an assumption requires some major contortions of logic.

We have theoretical concept called "black hole" and we have observed objects that we call "black holes". Both things got the same name ... logically it is the same thing, right?
Is this how you think?

 

[my_wan]

We have theoretical concept called "black hole" and we have observed objects that we call "black holes". Both things got the same name ... logically it is the same thing, right?
Is this how you think?

I'm quiet willing to entertain the notion that the things we observe and label "black holes" may not strictly be the things we describe them to be. However, only a single property is required to keep the label "black hole", that being that light cannot escape its interior.

In spite of this willingness to entertain alternative descriptions of what we are observing, it's going to require something far more specific than a rejection of the standard description to be of interest.

 

[zonde]

I'm quiet willing to entertain the notion that the things we observe and label "black holes" may not strictly be the things we describe them to be. However, only a single property is required to keep the label "black hole", that being that light cannot escape its interior.

There shouldn't be anything that can escape it's interior to call it "black hole".

In spite of this willingness to entertain alternative descriptions of what we are observing, it's going to require something far more specific than a rejection of the standard description to be of interest.

So you do not take answer "we don't know" as acceptable, right?

 

[my_wan]

So you do not take answer "we don't know" as acceptable, right?

If we did factually know I wouldn't be willing to entertain alternative models of our observations. Hence your presumption of what I find acceptable is most definitely in error. I also spend some time arguing how we can't be as certain about many things as we tend to like to believe, on a wide variety of issues.

If you want to reject BH physics as we know it fine. I entertain all kinds of wild ideas for creative reasons. If you want to convince anybody else you need a far more specific argument than "we don't know". Among those issues that needs to be addressed, which I think PAllen's approach was an admirable attempt at doing, is how you can think a global coordinate system can be selected that is somehow more meaningful than what can be provided by the observations of a free falling observer. A coordinate system is, by definition, an observer construct. Even what constitutes a "closed surface" is an observer dependent construct. You can't cling to one while rejecting the other, at least not without making some fundamental arguments that go well beyond just BH physics.

 

[zonde]

If we did factually know I wouldn't be willing to entertain alternative models of our observations. Hence your presumption of what I find acceptable is most definitely in error. I also spend some time arguing how we can't be as certain about many things as we tend to like to believe, on a wide variety of issues.

If you want to reject BH physics as we know it fine. I entertain all kinds of wild ideas for creative reasons. If you want to convince anybody else you need a far more specific argument than "we don't know".

Hmm, maybe you have just misunderstood me. I was not trying to argue against BH with this "Begging the question" argument. I was just saying that some arguments defending BH are better than others.

If you want arguments against BH then state that question so that I know about what we are talking.

 

[my_wan]

Hmm, maybe you have just misunderstood me. I was not trying to argue against BH with this "Begging the question" argument. I was just saying that some arguments defending BH are better than others.

If you want arguments against BH then state that question so that I know about what we are talking.

And all I was pointing out, when you responded with the 'begging the question' response, was that even in the absents of GR theoretical grounds remain for the existence of lack holes. Hence any argument against them must be more expansive than the issues GR alone dictates. This was in turn predicated on what you said you wanted to discuss, which said: "1. Assume that BH exists."

Let's assume the opposite, such that they don't exist. This means, irrespective of GR, there no regions of spacetime which light can't escape. This entails an absolute limit in the potential change in depth of a gravitational field.

The only way I know to attempt this is to assume the relative mass (not necessarily proper mass) decreases as the gravitational depth increases, such that a collapsing body can only asymptotically approach the creation of an event horizon. Much the same way an accelerated mass can only asymptotically approach the speed of light. In such a case, light would still escape, but even x-rays would escape at such long wavelengths (low energy) as to effectively be radio waves to the external observer.

If you assume the Nordtvedt effect is valid, and the Strong Equivalence Principle is violated, then that would cut short such an argument. However, no evidence of such a violation exist. You can then try to impose such a limit, but that still requires that the proper mass of the material making up the black hole has no direct observational meaning for an external observer. Much like the apparent mass increase in GR, as a mass decreases its gravitational depth, or its binding energy is reduced. This would also imply that the 'proper' mass has no more absolute meaning than any other arbitrarily chosen relativistic measure.

Is that the kind of argument you wanted to discuss?

 

[zonde]

And all I was pointing out, when you responded with the 'begging the question' response, was that even in the absents of GR theoretical grounds remain for the existence of lack holes. Hence any argument against them must be more expansive than the issues GR alone dictates. This was in turn predicated on what you said you wanted to discuss, which said: "1. Assume that BH exists."

I wanted to discuss PAllens example with collapsing cluster of stars. And I tried to explain why I consider it better than other examples (with observers in free fall). And the difference is that in PAllens example we do not assume anything about existence/non-existence of BH. We just play the situation forward according to our understanding of physical laws.

Let's assume the opposite, such that they don't exist. This means, irrespective of GR, there no regions of spacetime which light can't escape. This entails an absolute limit in the potential change in depth of a gravitational field.

The only way I know to attempt this is to assume the relative mass (not necessarily proper mass) decreases as the gravitational depth increases, such that a collapsing body can only asymptotically approach the creation of an event horizon. Much the same way an accelerated mass can only asymptotically approach the speed of light. In such a case, light would still escape, but even x-rays would escape at such long wavelengths (low energy) as to effectively be radio waves to the external observer.

If you assume the Nordtvedt effect is valid, and the Strong Equivalence Principle is violated, then that would cut short such an argument. However, no evidence of such a violation exist. You can then try to impose such a limit, but that still requires that the proper mass of the material making up the black hole has no direct observational meaning for an external observer. Much like the apparent mass increase in GR, as a mass decreases its gravitational depth, or its binding energy is reduced. This would also imply that the 'proper' mass has no more absolute meaning than any other arbitrarily chosen relativistic measure.

Hmm, but why would you associate this with Nordtvedt effect. Strong Equivalence Principle can hold just the same. I can say that inertial mass=active gravitating mass=passive gravitating mass is reduced.

And I see another possibility what can prevent BH formation. It is degeneracy of matter.

 

[my_wan]

I mentioned the Nordtvedt effect because if it held, which is pretty unlikely, such that the gravitational self-energy contributed to its total gravitational mass, then you can't get a relativistic reduction of gravitational mass for an external observer, since even if the inertial mass is reduced its gravitational mass would remain. That's why it would violate the strong equivalence principle. I don't take this effect seriously, but it would render the scenario I described as moot.

I'm not sure how degenerate matter can be exploited to prevent BH formation. In PAllen's scenario the matter density never even observably got especially dense in any local sense. Even in the event it did, you still have to presume the Fermi-pressure would grow indefinitely as the total gravitational pressure increased. Possible I suppose, but fails completely in PAllen's scenario.

 

[zonde]

I mentioned the Nordtvedt effect because if it held, which is pretty unlikely, such that the gravitational self-energy contributed to its total gravitational mass, then you can't get a relativistic reduction of gravitational mass for an external observer, since even if the inertial mass is reduced its gravitational mass would remain. That's why it would violate the strong equivalence principle. I don't take this effect seriously, but it would render the scenario I described as moot.

So are you saying that I misunderstood you? You was presenting kind of possible (not very strong) argumentation against mass reduction by binding energy?

 

[my_wan]

I was asking if that was the kind of argument you had in mind back in the opening post, where you also characterized "Assume that BH exists" as begging the question. Limiting the creation of black holes through mass reduction by binding energy would be ruled out by the Nordtvedt effect. I only mentioned it to be inclusive of possibilities that contradicted the mechanism I described. Since I don't take the Nordtvedt effect very seriously it actually strengthens the argument. Apparently the answer is no, given your responses.

Consider the apparent relative mass increase of a planet like Mercury, as defined by GR, at its aphelion compared to its perihelion. Conversely a mass minimum at its perihelion. Hence the total relative mass apparently decreases as the mass density increases. For a far removed observer, wouldn't this then indicate that the total relative mass of the system varies inversely with density? This then implies that for a given constant volume of space, as defined by some external observer, the addition of masses to this volume would then add up in a manner similar to the relativistic addition of velocities, as seen by the external observer.

This wouldn't necessarily invalidate an event horizon, for the same reason that an apparent horizon can be present in a particle's accelerating reference, beyond which events are unobservable. This actually makes it possible to accelerate fast enough to prevent a photon from ever catching you.

Anyway, I started thinking about this in response to your apparent objection to assuming black holes exist. Because if your going to object to that assumption some mechanism for avoiding them is required. "We don't know", however valid in general, is not sufficient when specific mechanism are required to avoid black holes.

 

[zonde]

I'm not sure how degenerate matter can be exploited to prevent BH formation. In PAllen's scenario the matter density never even observably got especially dense in any local sense. Even in the event it did, you still have to presume the Fermi-pressure would grow indefinitely as the total gravitational pressure increased. Possible I suppose, but fails completely in PAllen's scenario.

This is rather complicated topic and I would like to discuss it only if we can dedicate some time for that topic alone.

Consider the apparent relative mass increase of a planet like Mercury, as defined by GR, at its aphelion compared to its perihelion. Conversely a mass minimum at its perihelion. Hence the total relative mass apparently decreases as the mass density increases. For a far removed observer, wouldn't this then indicate that the total relative mass of the system varies inversely with density? This then implies that for a given constant volume of space, as defined by some external observer, the addition of masses to this volume would then add up in a manner similar to the relativistic addition of velocities, as seen by the external observer.

I can't consider this scenario. I don't know how to model it.
And I am not sure about the term "relative mass". I imagined it as something like proper mass minus binding energy, is this in the right direction? But then I don't know how it can be represented in GR as I don't know how (or if) binding energy is represented in GR.

Anyways I know we can speak about binding energy as we compare one equilibrium state with another equilibrium state. But I'm not sure how to model dynamics between equilibrium states in respect of binding energy. And certainly aphelion and perihelion of Mercury are not equilibriums states for the whole system.

 

[zonde]

I'm not sure how degenerate matter can be exploited to prevent BH formation. In PAllen's scenario the matter density never even observably got especially dense in any local sense. Even in the event it did, you still have to presume the Fermi-pressure would grow indefinitely as the total gravitational pressure increased. Possible I suppose, but fails completely in PAllen's scenario.

Okay one question is what happens when matter is degenerate but you try to contain it within some volume. I think that degenerate matter can not be contained by other particles i.e. it does not participate in elastic collisions. I am not sure if I can propose solid arguments why it should be so from perspective of QM. The problem is with interpretation of "quantum state" in case of free particles. Anyways we can speculate that this is the case with neutrinos - they are very degenerate and after encounter with other particles they fall back on the same trajectory (the same momentum/position state) as before collision with very high probability.

Speaking about degeneracy and density dependence. To claim that the two are varying proportionally we have to assume that there is some cut-off distance for quantum level occupancy, meaning that particles don't compete for quantum state given sufficient distance. However we can assume that this "quantum level occupancy" effect drops as inverse square law. And in this case density factor does not exactly determine degeneracy level and it is more related to number of particles and distance to them.
And assuming this PAllen's scenario is still subject to questions about degeneracy levels as number of particles is much higher even so the distances are bigger as well.

 

[PAllen]

Okay one question is what happens when matter is degenerate but you try to contain it within some volume. I think that degenerate matter can not be contained by other particles i.e. it does not participate in elastic collisions. I am not sure if I can propose solid arguments why it should be so from perspective of QM. The problem is with interpretation of "quantum state" in case of free particles. Anyways we can speculate that this is the case with neutrinos - they are very degenerate and after encounter with other particles they fall back on the same trajectory (the same momentum/position state) as before collision with very high probability.

Speaking about degeneracy and density dependence. To claim that the two are varying proportionally we have to assume that there is some cut-off distance for quantum level occupancy, meaning that particles don't compete for quantum state given sufficient distance. However we can assume that this "quantum level occupancy" effect drops as inverse square law. And in this case density factor does not exactly determine degeneracy level and it is more related to number of particles and distance to them.
And assuming this PAllen's scenario is still subject to questions about degeneracy levels as number of particles is much higher even so the distances are bigger as well.

So, you propose two stars 10 million miles apart are fine, but add more, further away, there is a problem of quantum occupancay? It would be a wild theory, different from any currently known, to have such an effect. Which all gets back to: you can say BH don't form if and only if you admit you say GR is seriously wrong. Which is fine, but be willing to say it.

 

[zonde]

you can say BH don't form if and only if you admit you say GR is seriously wrong.

It would be nice to be as confident as you are ... but I am not.

Say I have heard that particles without any forces applied to them follow geodesics. But as I look more into details it turns out it is an approximation. You have to assume that particle has zero (negligible) mass.
So maybe you can tell me (if you know) - when we take into account particle's own gravity in what (space-time) direction it deviates from original geodesic (if we speak about particle falling radially toward gravitating mass)?

 

[PAllen]

It would be nice to be as confident as you are ... but I am not.

Say I have heard that particles without any forces applied to them follow geodesics. But as I look more into details it turns out it is an approximation. You have to assume that particle has zero (negligible) mass.
So maybe you can tell me (if you know) - when we take into account particle's own gravity in what (space-time) direction it deviates from original geodesic (if we speak about particle falling radially toward gravitating mass)?

It won't change direction. It will emit some amount of gravitational radiation and slow down (assuming the initial configuration had exactly zero angular momentum).

Let's turn it around: on what basis are your doubts about what GR predicts (as opposed to any beliefs about reality)? Note that we have the following:

- artificially perfect exact solutions showing formation of black holes
- theorems with very weak assumptions showing black hole formation is inevitable under general, realistic conditions
- ever more precise numeric simulations of black hole formation
- no theoretical counter arguments I've seen that don't actually modify GR (e.g. incorporating some model of quantum correction).

Note, even your argument about quantum occupancy is an argument that GR is incorrect, since such cannot be represented in a stress energy tensor, and cannot be described classically. If your actual argument is that there exist approaches to apply quantum arguments to GR that avoid singularities and event horizons, this is a no brainer. I can link dozens of such arguments, some may be close to how the world works, but none are statements about what GR predicts as a classical theory; all are modifications of GR in the same spirit as QED is to Maxwell EM.

 

[zonde]

It won't change direction. It will emit some amount of gravitational radiation and slow down (assuming the initial configuration had exactly zero angular momentum).

Let's turn it around: on what basis are your doubts about what GR predicts (as opposed to any beliefs about reality)?

I have doubts about exactness of GR predictions. It's too open for interpretation.

Note that we have the following:

- artificially perfect exact solutions showing formation of black holes

Are there any exact solution for runaway gravitational collapse? No? Then you can't claim that.
Obviously you need such a solution to claim that massive body undergoing runaway gravitational collapse and not emitting gravitational waves is a valid solution to EFE.

EFE take as arguments continuous 4D tensor fields. I simply do not get why I should believe it's something calculable without radical approximations.

You need coordinate system to express continuous tensor field. And this coordinate system is supposedly defined using this same tensor field. To me it seems like circular definition.

Hyperbolic coordinates is a dirty cheat unless you can provide a very serious arguments why they should be considered physically meaningful. So I do not believe argument about coordinate singularity in SC coordinates is valid (as I see "frozen star" is equivalent to "exterior of black hole").

Not to mention that I still don't know how binding energy can be represented in GR. And I consider it important in order to understand GR.




So make your pick.

 

[PAllen]

I have doubts about exactness of GR predictions. It's too open for interpretation.

You could say this about quantum mechanics, QFT, etc. It is a vacuous statement without specific arguments.

Are there any exact solution for runaway gravitational collapse? No? Then you can't claim that.

Sure there are. It's just that the exact ones are implausibly symmetric. How is this different from many other theories where approximation is required for realistic cases?

Obviously you need such a solution to claim that massive body undergoing runaway gravitational collapse and not emitting gravitational waves is a valid solution to EFE.

.
GW emission is expected for any collapse in the real world. Not sure why you thought otherwise. It is only known (mathematically) not to occur for perfect spherical symmetry, which will never exist in the real world. For realistic scenarios, we have (at least) 4 strong reasons to say GR predicts black holes, and you have still not provided a single reason for believing GR does not:

(1) simple, exact solutions (considered as indicative of general features of more realistic cases)
(2) general singularity theorems
(3) absence of any process with GR + classical matter models + reasonable quantum models that could prevent super massive BH formation (that is, matter coalescing within the horizon radius; any type of horizon you like).
(4) numeric models of ever growing sophistication (these, for example, model the precise GW emission spectrum expected from realistic collapses).

EFE take as arguments continuous 4D tensor fields. I simply do not get why I should believe it's something calculable without radical approximations.

see above

You need coordinate system to express continuous tensor field. And this coordinate system is supposedly defined using this same tensor field. To me it seems like circular definition.

This makes no sense to me. You need coordinate charts to define manifold topology. You do not define a coordinate system from a tensor field. This circularity is your invention or misunderstanding.

Hyperbolic coordinates is a dirty cheat unless you can provide a very serious arguments why they should be considered physically meaningful. So I do not believe argument about coordinate singularity in SC coordinates is valid (as I see "frozen star" is equivalent to "exterior of black hole").

1) So you reject 'general covariance' or diffeomorphism invariance: a definitional principle of GR. This is completely equal to the statement that you reject GR, which for some reason you are unwilling to admit.
2) Are you aware that you can derive the Kruskal metric directly from the EFE without ever introducing the SC coordinates? (I'm guessing that by hyperbolic coordinates you mean Kruskal).
3) Lemaitre coordinates are not hyperbolic and have no horizon singularity, and can also be derived directly from the EFE.

Not to mention that I still don't know how binding energy can be represented in GR. And I consider it important in order to understand GR.

1) To the extent this argument is valid, it is an argument against the validity of GR, which for some reason you remain resistant to admit.

2) In any case, GR says plenty about binding energy, but there are loose ends and open issues. First, in any asymptotically flat spacetime, there is globally conserved energy. Binding energy for non-catastrophic collapse is modeled by emission of ordinary radiation + GW. It is true that without an asymptotic geometry assumption, GR cannot account for total energy conservation, and that none of quasi-local approaches is fully satisfactory. However, for practical purposes, you can take a sufficiently isolated region, and model it as if it were embedded in asymptotically flat spacetime. To the extent this is a cheat (and it is, technically), your issue is with GR itself. Another anomaly of GR itself is that catastrophic collapse is predicted to be irreversible to an extent beyond what can be explained with binding energy (e.g. the Oppenheimer-Snyder collapse emits no radiation at all (GW or regular), yet is irreversible in the sense that you can't continue the forward time solution from after the horizon forms to a re-expansion without violating the EFE. Note, within the Lemaitre-Tolman generalization of Oppenheimer-Snyder, you can have WH->BH solutions but not BH->WH solutions. Time reverse WH->BH and you still have WH->BH.)

 

[TrickyDicky]

1) So you reject 'general covariance' or diffeomorphism invariance: a definitional principle of GR. This is completely equal to the statement that you reject GR, which for some reason you are unwilling to admit.

Then many relativists "reject GR", because there is an unsolved controversy (mostly from the LQG people) about what exactly is "general covariance" for dynamical theories.