Oppenheimer-Snyder model of star collapse

[Mike Holland]

Excuse me butting in here, but I have a big concern about the O-S calculation. You all keep agreeing that it applies to a spherical symmetric collapsing mass. But when you think about how fast a pulsar spins, in general a near-BH mass would spin very much faster, and at best we would have an oblate spheroid. In fact, what I have in mind would look something like an LP record! I realize that the end result BH would be spherical, but I don't see how it can be spherical beore reaching that stage except in very idealised theory.

If this picture is correct, then mass in the polar direction would have very little distance to fall, but there would be very little of it, while the angular momentum of the equatorial regions would delay the collapse significantly. Is this taken into account in any of the calculations that have been done?

A further note - such a flat spinning object would qualify as an axi-symmetric collapse as described by Saul Teukolsky, in which one could at some stage have a naked singularity before the Black Hole forms completely.

I would like to hear your comments.

Mike

 

[PeterDonis]

You all keep agreeing that it applies to a spherical symmetric collapsing mass.

Because that is an assumption of the O-S model.

But when you think about how fast a pulsar spins, in general a near-BH mass would spin very much faster, and at best we would have an oblate spheroid.

If a BH is formed by a process similar to how a spinning pulsar is formed, yes, one would expect it to be spinning fast. The O-S model is not meant to apply to this case.

I realize that the end result BH would be spherical

No, it wouldn't. A spinning BH is not spherical; it's an oblate spheroid. More precisely, its horizon is shaped like an oblate spheroid.

If this picture is correct, then mass in the polar direction would have very little distance to fall, but there would be very little of it, while the angular momentum of the equatorial regions would delay the collapse significantly. Is this taken into account in any of the calculations that have been done?

I'm sure there are numerical models of a collapse with significant angular momentum, which is basically what you're describing. I don't think there is any analytical solution for this case. As I said above, the O-S model was not intended to apply to this case; it was a drastically idealized model that was constructed in order to be able to find an analytical solution whose qualitative properties could be investigated. It was not meant to model a "realistic" stellar collapse.

A further note - such a flat spinning object would qualify as an axi-symmetric collapse as described by Saul Teukolsky, in which one could at some stage have a naked singularity before the Black Hole forms completely.

If there is enough angular momentum compared to the total mass, yes, I believe there could be a naked singularity. In this case, however, I don't think a BH ever forms (meaning I don't think a horizon ever forms). I believe this is the case where the angular momentum per unit mass is large enough to be what the Wikipedia page calls an "over-extreme Kerr solution":

http://en.wikipedia.org/wiki/Kerr_metric#Overextreme_Kerr_solutions

These solutions have no horizon, just a naked singularity.

 

[PAllen]

Excuse me butting in here, but I have a big concern about the O-S calculation. You all keep agreeing that it applies to a spherical symmetric collapsing mass. But when you think about how fast a pulsar spins, in general a near-BH mass would spin very much faster, and at best we would have an oblate spheroid. In fact, what I have in mind would look something like an LP record! I realize that the end result BH would be spherical, but I don't see how it can be spherical beore reaching that stage except in very idealised theory.

If this picture is correct, then mass in the polar direction would have very little distance to fall, but there would be very little of it, while the angular momentum of the equatorial regions would delay the collapse significantly. Is this taken into account in any of the calculations that have been done?

A further note - such a flat spinning object would qualify as an axi-symmetric collapse as described by Saul Teukolsky, in which one could at some stage have a naked singularity before the Black Hole forms completely.

I would like to hear your comments.

Mike

The claim is only that it is more realistic model of BH than SC eternal WH-BH. There is no dispute that it is still extremely unrealistic for the reason you mention - no rotation. Once there is rotation, there is no spherical symmetry.

Teukolsky's model is theoretically very interesting, but also unrealistic - any deviation, however slight, from perfect axial symmetry removes the naked singularity (this is why Penrose's revised bet is still unclaimed - a naked singularity from initial conditions that are perturbatively stable). The best bet for realism are numerical simulations. Whatever the details of collapse, the final, stable form long after last matter infall, is simply the Kerr-Newman metric, at least on the outside. The inside is another matter, that has been discussed above by Pervect.

Please note: this whole line of siscussion would be distraction for this thread, which was opened specifically to discuss Oppenheimer-Snyder.

 

[PAllen]

If there is enough angular momentum compared to the total mass, yes, I believe there could be a naked singularity. In this case, however, I don't think a BH ever forms (meaning I don't think a horizon ever forms). I believe this is the case where the angular momentum per unit mass is large enough to be what the Wikipedia page calls an "over-extreme Kerr solution":

http://en.wikipedia.org/wiki/Kerr_metric#Overextreme_Kerr_solutions

These solutions have no horizon, just a naked singularity.

I could be wrong, but I recall reading papers showing that an over-extreme Kerr-Newman cannot form. That is, it exists mathematically like a WH solution, but cannot arise from matter satisfying any of the energy conditions, no matter what initial conditions.

 

[PeterDonis]

I recall reading papers showing that an over-extreme Kerr-Newman cannot form.

I vaguely recall reading speculations along these lines (not actual papers). It seems reasonable to me; I also seem to remember that there is a theorem along the lines of: you can't make a Kerr BH into an over-extreme Kerr geometry by adding angular momentum to it; you always end up adding enough mass along with the angular momentum to keep the angular momentum per unit mass below the limit.

 

[Mike Holland]

Please note: this whole line of discussion would be distraction for this thread, which was opened specifically to discuss Oppenheimer-Snyder.

That is a joke after what happened to my "Black Holes - two points of view" thread - a long argument about Killing Vectors! I just abandoned that thread!

I thought my post was about O-S black hole formation. But anyway, thanks for your replies. I shall read up more on Kerr's ideas and their follow-ons. I am particularly interested in the effects of perturbations in the various models, and in numerical simulations of BH formation, but I don't have the maths to follow the calculations.

Mike

 

[PAllen]

I vaguely recall reading speculations along these lines (not actual papers). It seems reasonable to me; I also seem to remember that there is a theorem along the lines of: you can't make a Kerr BH into an over-extreme Kerr geometry by adding angular momentum to it; you always end up adding enough mass along with the angular momentum to keep the angular momentum per unit mass below the limit.

These issues are why Penrose does not consider these solutions a violation of cosmic censorship - unless someone finds a loophole that allows them to form from acceptable initial conditions.

 

[PAllen]

I thought my post was about O-S black hole formation. But anyway, thanks for your replies. I shall read up more on Kerr's ideas and their follow-ons. I am particularly interested in the effects of perturbations in the various models, and in numerical simulations of BH formation, but I don't have the maths to follow the calculations.

Mike

If there is any rotation at all, it is not covered by O-S (for that matter, if there is any pressure it is not governed by O-S). Even vaguely realistic collapse is only described via numeric approximations (which have become very robust and accurate in the last 10 years or so). O-S is an idealized model in the same way as SC geometry, except that it doesn't require 'eternal BH'. Instead, it provides an origin for a perfect spherically symmetric BH; as a consequence, the WH is removed, as is the wormhole to another universe.

 

[harrylin]

[..]
BTW here's the citation history of the OS paper, curious to say the least. It was ignored until 1964, but it wasn't really until the last 5 years that it took off.
http://libra.msra.cn/Publication/19921235/on-continued-gravitational-contraction

That's amazing, yes in fact it started to get more attention since 20 years ago, but even more since 6 years ago.

I had not heard of it before as I saw it cited in a recent paper as I mentioned last month*; and that paper cited it because it agrees in essence with recently expressed opinions and computer simulations.

Thanks for the graph!

*https://www.physicsforums.com/showthread.php?p=4134963&highlight=Oppenheimer#post4134963

 

[PAllen]

That's amazing, yes in fact it started to get more attention since 20 years ago, but even more since 6 years ago.

I had not heard of it before as I saw it cited in a recent paper as I mentioned last month*; and that paper cited it because it agrees in essence with recently expressed opinions and computer simulations.

Thanks for the graph!

*https://www.physicsforums.com/showthread.php?p=4134963&highlight=Oppenheimer#post4134963

If you mean Yi, his position is a crank position, not shared by any serious scholar of GR (coordinate quantities are physical; Einstein would vomit); if you mean Krauss, you misinterpret his paper seriously (applying quantum conclusions to the classical case).

[Edit: Note, you have been called out by several people here on citing only half of the O-S abstract. On the Krauss et. al. paper note the following:

"However, we find
that Schwarzschild coordinates are sufficient to answer
the very specific set of questions we ask from the asymptotic
observer’s viewpoint. Namely, does the asymptotic
observer see objects disappear into a black hole in the
time that he sees the collapsing body evaporate? And, is
the spectrum of the radiation received ever truly thermal
(even in the semiclassical approximation)?
In Sec. III we verify the standard result that the formation
of an event horizon takes an infinite (Schwarzschild)
time if we consider classical collapse. This is not
surprising and is often viewed as a limitation of the
Schwarzschild coordinate system. To see if this result
changes when quantum effects are taken into account, we
address the problem of quantum collapse using a minisuperspace
version of the functional Schrödinger equation
[2] in Sec. IV."

All of the revisions to the classical GR results, as explained by many here, are qualified (in Krauss et.al.) by quantum corrections, and also qualified by 'see'. In this sense, this paper is scholarly, unlike the Yi paper which is crank.]

 

[harrylin]

[..] One of this qualifications I tried to explain was that even if not in the OS model the logical causal future of the collapsing model is a BH with a singularity, and for the non-charged, non-rotating case the only mathematical model we have of that is an exact solution of the EFE is the extended Schwarzschild spacetime.

Can you put that in other words? I don't understand the "even if not" ...

 

[TrickyDicky]

Can you put that in other words? I don't understand the "even if not" ...

I was just highlighting the difference between the collapse OS model, and the Schwarzschild spacetime.

 

[harrylin]

I was just highlighting the difference between the collapse OS model, and the Schwarzschild spacetime.

Yes. And I like to know what you tried to say, as I cannot parse your sentence.

 

[TrickyDicky]

That if one imagines the future of the OS collapsing star the logic of causality, and the condition of being a solution of the EFE leads naturally to the Sch. black hole.

 

[harrylin]

That if one imagines the future of the OS collapsing star the logic of causality, and the condition of being a solution of the EFE leads naturally to the Sch. black hole.

Thus you meant: even in the OS model the logical causal future of the collapsing model is a Schwartzschild black hole with a singularity. Yes?

I think that that is wrong, or at least in contradiction with Oppenheimer-Snyder; I will check it later!

Also:

If you mean [..] crank position [..] Einstein would vomit[..]

while wrong in every aspect that post is unworthy of this forum, so I won't feed it. Insofar as it is relevant for this thread, misconceptions will become clearer in the discussion.

 

[PeterDonis]

I think that that is wrong, or at least in contradiction with Oppenheimer-Snyder; I will check it later!

My understanding is the same as TrickyDicky's: the spacetime of the O-S model contains an event horizon, black hole interior region, and future singularity. Certainly that's the way MTW describes the model; see the sections I've already referenced.

 

[TrickyDicky]

Thus you meant: even in the OS model the logical causal future of the collapsing model is a Schwartzschild black hole with a singularity. Yes?

No, the logical causal future realization is not in the OS model, but in the Sch. BH model, thus my "even if not...".

 

[PeterDonis]

No, the logical causal future realization is not in the OS model

Maybe I spoke too soon when I said our understanding was the same. If by "logical causal future realization" you mean something like you were saying before, where we somehow have to substitute the full maximally extended Schwarzschild spacetime for the O-S collapsing spacetime, that is *not* correct. I thought you just meant that the O-S model's spacetime contains a future horizon, a BH interior vacuum region, and a future singularity, in addition to the region occupied by the collapsing matter. And I should add that my understanding is that those regions I just listed are the *entire* spacetime; there is nothing else, so there is no past horizon and no white hole region. The maximally extended Schwarzschild spacetime contains a past horizon and white hole, but the O-S spacetime does *not*.

 

[PeterDonis]

The maximally extended Schwarzschild spacetime contains a past horizon and white hole, but the O-S spacetime does *not*.

I can't remember if this particular page on Hamilton's site has been linked to in this thread, but it shows a Penrose diagram of the O-S model:

http://casa.colorado.edu/~ajsh/collapse.html#penrose

Compare with the Penrose diagram of the maximally extended Schwarzschild geometry here:

http://online.kitp.ucsb.edu/online/colloq/hamilton1/oh/penrose_Schwpar.html

 

[harrylin]

Thus you meant: even in the OS model the logical causal future of the collapsing model is a Schwartzschild black hole with a singularity. Yes?

I think that that is wrong, or at least in contradiction with Oppenheimer-Snyder; I will check it later! [..]

My understanding is the same as TrickyDicky's: the spacetime of the O-S model contains an event horizon, black hole interior region, and future singularity. Certainly that's the way MTW describes the model; see the sections I've already referenced.

I don't have MTW but I do have O-S, which is what matters; see next.

No, the logical causal future realization is not in the OS model, but in the Sch. BH model, thus my "even if not...".

OK. I now checked it more carefully, and I maintain the paper essentially agrees with Einstein's paper of that same year; however I had not noticed that it is in fact a bit inconsistent. Still, the paper denies for forming black holes the future realisation of a singularity; thus "collapse" in the summary apparently refers to the shrinking to its gravitational radius. Here are the IMHO pertinent passages (I hope that I cite little enough not to infringe copyright):

Near the surface of the star,
where the pressure must in any case be low, we
should expect to have a local observer see matter
falling inward with a velocity very close to that
of light; to a distant observer this motion will be
slowed up by a factor (1-r_o/r_b). All energy
emitted outward from the surface of the star
will be reduced very much in escaping, by the
Doppler effect from the receding source, by the
large gravitational red-shift, (1-r_o/r_b)^½, and by
the gravitational deflection of light which will
prevent the escape of radiation except through a
cone about the outward normal of progressively
shrinking aperture as the star contracts. The star
thus tends to close itself off from any communi-
cation with a distant observer; only its gravi-
tational field persists.
[..]
Further, a star
in its early stage of development would not
possess a singular density or pressure; it is
impossible for a singularity to develop in a finite
time.
[..]
we see that for a fixed
value of R as t tends toward infinity, τ tends to a
finite limit, which increases with R. After this
time τ_o an observer comoving with the matter
would not be able to send a light signal from the
star; the cone within which a signal can escape
has closed entirely.

 

[PeterDonis]

I don't have MTW but I do have O-S, which is what matters

It does if we are trying to establish what O-S said in their original paper, yes. But there is also a separate question, which is, what is the best currently accepted "O-S" model, i.e., the best currently accepted spacetime that models the collapse of a massive object like a star? We may be talking past each other if you are trying to answer the first question while I am trying to answer the second, and the answers are different (see below).

OK. I now checked it more carefully, and I maintain the paper essentially agrees with Einstein's paper of that same year; however I had not noticed that it is in fact a bit inconsistent. Still, the paper denies for forming black holes the future realisation of a singularity; thus "collapse" in the summary apparently refers to the shrinking to its gravitational radius.

Hm, yes, I see what you mean; they don't seem fully consistent in what they say, and this language doesn't seem fully consistent with the abstract. So it may indeed be that the answers to the two questions above are different. I can't say for sure without seeing the whole paper. If the answers are different, then we have indeed been talking past each other, since I have been talking about question #2, the best current model, in the belief that (as presented in MTW) that was also the model O-S had derived. For example, the Penrose diagram I posted a link to in my last post was for the best current model.

(I hope that I cite little enough not to infringe copyright)

A side note, off-topic but this is a pet peeve of mine: the fact that you even have to worry about this is outrageous. If only they had had arxiv.org in 1939...

 

[PAllen]

I don't have MTW but I do have O-S, which is what matters; see next.

Unfortunately, it remains a bit difficult to interpret these snippets without the fuller context, which you obviously can't provide (and which most of us here cannot access). However, it seems quite possible to interpret these snippets as given in a way perfectly consistent with the modern understanding of O-S collapse:

OK. I now checked it more carefully, and I maintain the paper essentially agrees with Einstein's paper of that same year; however I had not noticed that it is in fact a bit inconsistent. Still, the paper denies for forming black holes the future realisation of a singularity; thus "collapse" in the summary apparently refers to the shrinking to its gravitational radius. Here are the IMHO pertinent passages (I hope that I cite little enough not to infringe copyright):

Near the surface of the star,
where the pressure must in any case be low, we
should expect to have a local observer see matter
falling inward with a velocity very close to that
of light; to a distant observer this motion will be
slowed up by a factor (1-r_o/r_b). All energy
emitted outward from the surface of the star
will be reduced very much in escaping, by the
Doppler effect from the receding source, by the
large gravitational red-shift, (1-r_o/r_b)^½, and by
the gravitational deflection of light which will
prevent the escape of radiation except through a
cone about the outward normal of progressively
shrinking aperture as the star contracts. The star
thus tends to close itself off from any communi-
cation with a distant observer; only its gravi-
tational field persists.

I don't see anything wrong with this description. I see it as not even addressing the question of what happens to the infalling matter. It simply says, whatever happens is unable to causally influence (in any way) a distant observer. This is indisputable.

[..]
Further, a star
in its early stage of development would not
possess a singular density or pressure; it is
impossible for a singularity to develop in a finite
time.

There are two parts to this. Early : no singularity; obvious, no dispute.
For the second part, the issue is 'whose time, defined or measured how'. Much more context would be needed to resolve this.

[..]
we see that for a fixed
value of R as t tends toward infinity, τ tends to a
finite limit, which increases with R. After this
time τ_o an observer comoving with the matter
would not be able to send a light signal from the
star; the cone within which a signal can escape
has closed entirely.

The last is perfectly consistent with my understanding.

The upshot is these quotes leave me convinced that were I to read the paper, I would find it mostly agreeing with modern understanding.

As for modern understanding, since MTW is not accessible on line, with some effort, I found the following, which gives a really good introduction to dust collapse:http://www.aei.mpg.de/~rezzolla/lnotes/mondragone/collapse.pdf

Chapters 3 and 4 give a good treatment, with both illustrations and explanations that can be somewhat separated from the math, if that is too much for the reader (though the level is easier than MTW, at least in this section). See, especially, the illustration at the top of p. 31.

[Edit: Final note: Even if we could all access the O-S paper, and concluded some statements disagreed with later understandings, this would be purely of historic interest. There are not multiple versions of GR equations; nor multiple definitions of physical observables (right from 1915, Einstein defined these invariant quantities computed from covariant objects). On the other hand, mathematical technique and results are cumulative. It is not shocking to say an early practitioner was mistaken about some consequence of the theory. Einstein, for example, flip flopped 3 times on whether gravitational waves were a real prediction of his theory - ending with the view that they definitely were.]

 

[PeterDonis]

I don't see anything wrong with this description. I see it as not even addressing the question of what happens to the infalling matter. It simply says, whatever happens is unable to causally influence (in any way) a distant observer. This is indisputable.

I agree. I didn't mean to imply that I thought what harrylin quoted was inconsistent with modern understanding; what he quoted clearly states that the proper time experienced by an infalling observer riding on the surface of the collapsing star, from a finite radius R down to the horizon, is finite. That's the modern understanding. I was only saying that I had thought the O-S paper itself addressed what happens *after* the infalling matter reaches the horizon; but it appears it may not.

[Edit: Final note: Even if we could all access the O-S paper, and concluded some statements disagreed with later understandings, this would be purely of historic interest.

I agree; that's why I made a point of drawing a distinction between my question #1--what is the model presented in the original O-S paper?--and question #2--what is the best current model? Everything I have said in this thread is really directed at #2, not #1.

 

[Mike Holland]

http://www.aei.mpg.de/~rezzolla/lnotes/mondragone/collapse.pdf

Thanks for that reference, PAllen. It answers my queries about rapidly rotating disc-like collapsing stars.
I shall continue lurking around your discussion.
Mike

 

[harrylin]

[..] the proper time experienced by an infalling observer riding on the surface of the collapsing star, from a finite radius R down to the horizon, is finite. That's the modern understanding. [..]

I'm surprised to hear that there ever was a different understanding about the proper time of the infalling observer; I did not find any disagreement on that point in the literature. Do you have a reference to such a paper?

 

[PAllen]

I'm surprised to hear that there ever was a different understanding about the proper time of the infalling observer; I did not find any disagreement on that point in the literature. Do you have a reference to such a paper?

I wouldn't say there was ever a disagreement, but the fact wasn't known (as far as I know) before the O-S paper, and not well known for 20 years later. Note, what Peter is referring to is an observer following a collapsing surface - not the test trajectory for an SC geometry. Collapse was not understood to any degree before the O-S paper. Even the behavior of test trajectories to or through an SC geometry EH wasn't at all well known until the late 1930s.

Also, note that Einstein's paper of the same year as O-S argued that no real collapse could ever happen as envisaged in the O-S paper. Modern understanding is that Einstein's argument was simply wrong; while the O-S paper is viewed as the simplest case that has the general characteristics of realistic collapse analyzed by numerical GR (and is also consistent with the singularity theorems).

 

[harrylin]

I wouldn't say there was ever a disagreement, but the fact wasn't known (as far as I know) before the O-S paper [..]

OK

Also, note that Einstein's paper of the same year as O-S argued that no real collapse could ever happen as invisaged in the O-S paper. Modern understanding is that Einstein's argument was simply wrong; while the O-S paper is viewed as the simplest case that has the general characteristics of realistic collapse analyzed by numerical GR (and is also consistent with the singularity theorems).

Contrary to you, I noted no essential difference between those two papers; the O-S paper doesn't describe what you would call a "real collapse". Further, I can see no way to counter the arguments in those papers; and I have seen no counter argument in any peer reviewed paper or in the recent discussions here, nor in the discussion about the more recent understanding on the other forum to which I gave the link.

 

[PAllen]

OK

Contrary to you, I noted no essential difference between those two papers; the O-S paper doesn't describe what you would call a "real collapse". Further, I can see no way to counter the arguments in those papers; and I have seen no counter argument in any peer reviewed paper or in the recent discussions here, nor in the discussion about the more recent understanding on the other forum to which I gave the link.

That's because many here interpret those papers differently from you. I see the O-S paper (from its abstract) arguing for a real collapse that is not seen by an outside observer. Consensus source available on line discussing this exact same mathematics agree with this. Einstein argued, on the contrary, that this solution could not be realized in the real world - it was not a matter of how to interpret a solution but an argument that the O-S solution (without being mentioned by name) would not actually occur from a reasonable starting point. Einstein's argument in this particular paper is generally recognized as wrong.

As for the Krauss paper, I have read it twice; I and others here disagree with your interpretation. To me it says:

- Authors review aspects of the classical collapse. Wording is very careful to convey (to me) that nothing new or different from consensus is claimed here. Certain aspects are emphasized. It is emphasized that, by pure choice (not physical significance - classically) they are using SC t=constant slices; these are qualified in the intro section to be just a choice useful to the paper's overall purpose:

- Quantum mechanical arguments are presented to show that the EH never actually forms when quantum phenomena are considered; specifically evaporation rates.

I also disagree with your view of the status of this paper. It has the status of an interesting contribution, not a consensus, even for the quantum features. There are literally hundreds of papers in the last 5 years offering many different takes on how QM modifies the classical GR collapse process. This paper is just one respectable contribution to that.

 

[George Jones]

OK. I now checked it more carefully, and I maintain the paper essentially agrees with Einstein's paper of that same year;

See below.

however I had not noticed that it is in fact a bit inconsistent. Still, the paper denies for forming black holes the future realisation of a singularity;

As others have noted, this is just plain wrong.

Also, note that Einstein's paper of the same year as O-S argued that no real collapse could ever happen as envisaged in the O-S paper. Modern understanding is that Einstein's argument was simply wrong; while the O-S paper is viewed as the simplest case that has the general characteristics of realistic collapse analyzed by numerical GR (and is also consistent with the singularity theorems).

In his 1939 paper, Einstein drew the wrong conclusion from his calculations. What he actually showed (or came close to showing) was that, below the event horizon, Schwarzschild spacetime is not stationary.

 

[harrylin]

[..]
As others have noted, this is just plain wrong.

Hi George, please elaborate: according to O-S, and consistent with their preceding discussion, for a star that does not yet "possess a singular density or pressure" "it is impossible for a singularity to develop in a finite time". But you say that it is "plain wrong" to think that this means that the future realisation of a singularity does not happen for forming black holes. Surely infinite time doesn't happen, so I don't follow your thinking.

In his 1939 paper, Einstein drew the wrong conclusion from his calculations. What he actually showed (or came close to showing) was that, below the event horizon, Schwarzschild spacetime is not stationary.

That paper is a bit off topic so I won't elaborate on that. Just one question, in view of your claim: do you (or anyone else) know a paper that proves that he drew the wrong conclusion?
Thanks in advance!