Oppenheimer-Snyder model of star collapse

[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!

 

[PeterDonis]

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".

In this paragraph of the paper, O-S are using "time" to mean what we now call "Schwarzschild coordinate time". Slightly later they call it "clock time at r = infinity", and they also use the coordinate t to refer to it. The precise way to express what they are saying here, in modern language, is that the density and pressure of the collapsing star can never become singular in the region of spacetime where Schwarzschild coordinate time is finite (and timelike--even more precisely, we would say "exterior Schwarzschild coordinate time" to make it clear that we are talking about the region outside the horizon, and outside the collapsing matter as long as it is above the horizon).

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.

The correct way to say "infinite time doesn't happen" is "Schwarzschild coordinate time never becomes infinite anywhere along the worldline of the distant observer". That is *not* the same as saying "the region of spacetime in which the worldline of the distant observer lies, and in which Schwarzschild coordinate time is finite, is the entire spacetime". The latter statement is false. The "future realization of a singularity" happens in a different region of spacetime, one which the coordinate chart in which Schwarzschild coordinate time (strictly speaking, *exterior* Schwarzschild coordinate time) is finite does not cover.

 

[harrylin]

In this paragraph of the paper, O-S are using "time" to mean what we now call "Schwarzschild coordinate time". Slightly later they call it "clock time at r = infinity", and they also use the coordinate t to refer to it. The precise way to express what they are saying here, in modern language, is that the density and pressure of the collapsing star can never become singular in the region of spacetime where Schwarzschild coordinate time is finite (and timelike--even more precisely, we would say "exterior Schwarzschild coordinate time" to make it clear that we are talking about the region outside the horizon, and outside the collapsing matter as long as it is above the horizon).

Yes, they consider Schwarzschild coordinate time to be far away clock time - which is approximately the time on our clocks. And that time is according to GR valid for making physical predictions, just as they did and I cited. I understand the language of O-S better than your "modern language". As far as I understand your "modern" paraphrase of what they wrote, it agrees with mine; you just put it in "modern" wrapping. But if not, please correct my paraphrase in standard English.

The correct way to say "infinite time doesn't happen" is "Schwarzschild coordinate time never becomes infinite anywhere along the worldline of the distant observer".

Why would you think that "infinite time doesn't happen" could be incorrect? It simply means that no clock (at least, no clock that is tuned in accordance with theory) will ever indicate ∞.

That is *not* the same as saying "the region of spacetime in which the worldline of the distant observer lies, and in which Schwarzschild coordinate time is finite, is the entire spacetime". The latter statement is false. [..]

Obviously.

The "future realization of a singularity" happens in a different region of spacetime, one which the coordinate chart in which Schwarzschild coordinate time (strictly speaking, *exterior* Schwarzschild coordinate time) is finite does not cover.

In my language your claim is incompatible with the claim of O-S; apparently we don't speak the same language. So be it. I hope to have clarified that Trickydicky's "logical causal future realization" is according to O-S "impossible to develop in a finite time" - with which of course not the proper clock time of an infalling observer is meant, but approximately our clock time.

 

[PAllen]

Yes, they consider Schwarzschild coordinate time to be far away clock time - which is approximately the time on our clocks. And that time is according to GR valid for making physical predictions, just as they did and I cited. I understand the language of O-S better than your "modern language". As far as I understand your "modern" paraphrase of what they wrote, it agrees with mine; you just put it in "modern" wrapping. But if not, please correct my paraphrase in standard English.

Why would you think that "infinite time doesn't happen" could be incorrect? It simply means that no clock (at least, no clock that is tuned in accordance with theory) will ever indicate ∞.

Obviously.

In my language your claim is incompatible with the claim of O-S; apparently we don't speak the same language. So be it. I hope to have clarified that Trickydicky's "logical causal future realization" is according to O-S "impossible to develop in a finite time" - with which of course not the proper clock time of an infalling observer is meant, but approximately our clock time.

A key problem is your insistence that only one type of clock matters, as opposed examining the universe using all clocks. A clock just outside the dust ball, free falling with it, encounters the singularity if finite time. There is no justification for subtracting this clock from reality.

Another key problem is considering any coordinate time physical. All coordinate times can be used to make physical predictions, but no coordinate time, by itself, constitutes a physical prediction or observable quantity - especially even Minkowski coordinate time for inertial observers in SR. The predictions are what is measured by various observers.

The type of one-way causal relations between distant, static, observers and free fall observers is a common characteristic of pseudo-riemannian metrics, and can occur purely in SR, as has been explained a zillion times. The relations between a distant clock and free fall clock in SC geometry has virtually identical features as the relation between a uniformly accelerating observer and an inertial observer in SR, with the distant observer playing the role of the uniformly accelerating observer.

[Edit: Let me try to clarify further between a coordinate statement (of no physical significance whatsoever) versus a statement about observables.

1) Coordinate statement: For distant observer's time, an event horizon never forms. Why is this not a physical statement? Both SR and GR completely reject the concept of global time as a physical concept. Global time is construct of convention or convenience, in all cases. This misleading statement has buried within it a concept there is physical meaning to distant simultaneity : which events in universe correspond to which times on a distant clock. As I have explained numerous times, there are perfectly plausible alternative (to SC coordinate time based simultaneity) simultaneity conventions which relate events inside the horizon to events on distant clocks.

2) Indisputable physical statements: No physical process at or inside the horizon can causally influence a distant observer, even if that distant observer's world line is continued to infinite proper time (while remaining 'distant'). Conversely, distant observers can causally influence infallers up to the moment of their reaching the singularity. In a Kerr black hole (assuming its exact interior actually existed in our universe - it is definitely a predicted possibility of GR), where there are stable interior orbits, an external observer can send messages to such an interior orbiting observer forever; they just can't get a reply.

The difference between (1) and (2) may be subtle, but it is crucial. Note that, as required diffeomorphism invariance, Kruskal coordinates predict (2) just as much as SC coordinates. Further, a simultaneity surface relating interior and exterior event can be defined in SC coordinates; you just need to use limiting processes at the horizon due to the coordinate singularity there.

]

 

[DrGreg]

I haven't read the O-S paper and am not prepared to spend $25 to do so, but it seems to me most of the discussion in this thread isn't about the fine details of O-S collapse, but about more fundamental issues.

If we assume we have a non-rotating uncharged spherically symmetric distribution of matter surrounded by vacuum, then Birkhoff's Theorem tells us that the Schwarzschild solution must apply throughout the vacuum region, which means the vacuum region must look something like this:

The pink grid shows the exterior Schwarzschild coordinates. The blue grid shows the interior Schwarzschild coordinates (that is, inside the event horizon, but still outside the collapsing matter). The purple dotted line is the event horizon. The thick blue line is the singularity. All of these have been plotted accurately using MATLAB and are mandated by Birkhoff's theorem, regardless of the details of the collapse.

The grey line from the bottom to the left is the border of the vacuum region, i.e. the outermost layer of the collapsing matter. It is not possible to continue the diagram to the left of this line because Schwarzschild coordinates do not apply there. I have only sketched an approximate location for this line. The precise shape of this line will depend on the details of the collapse. The O-S model and other models would produce different shapes for the grey curve. But the pink and blue grid to the right of the line is the same for all models that are compatible with my initial assumptions.

If the collapse were resisted by sufficient pressure to prevent the event horizon forming, the grey curve would remain entirely within the pink region and would curve towards the top right of diagram. Otherwise, the curve must enter the blue region and eventually hit the darker blue singularity.

I have drawn this as a Kruskal–Szekeres diagram. There is an invisible, uniformly square, horizontal and vertical grid of Kruskal–Szekeres coordinates not shown. The advantage of a Kruskal–Szekeres diagram is that it shares many features with a Minkowski diagram in flat spacetime: timelike directions are within 45° of vertical, spacelike directions are within 45° of horizontal, and light travels at exactly 45°. The difference is scale. Minkowski maps have a uniform scale: the ratio of 1 cm vertically on the map to 1 second in the Universe is constant, and the ratio of 1 cm horizontally on the map to 1 light-second in the Universe is constant. On a Kruskal–Szekeres map, the map scale is variable (although at every event, the horizontal and vertical scales are the same as each other).

The pink curves are the worldlines of observers hovering at a constant distance from the centre of the collapsing matter, and the radial pink lines are lines of simultaneity for such observers as determined by the convention of Schwarzschild coordinates. For these observers, none of the events in the blue region occur "simultaneously" with an event on the observer's worldline, so you could say the event "never occurs" (within finite time) relative to that observer. But that is just an artefact of the coordinate system chosen. It's unreasonable to say those events "don't exist".

 

[PeterDonis]

Yes, they consider Schwarzschild coordinate time to be far away clock time - which is approximately the time on our clocks. And that time is according to GR valid for making physical predictions, just as they did and I cited.

It's valid for making physical predictions about the region of spacetime in which that time coordinate is finite. It is *not* valid for making physical predictions about any other region of spacetime.

Why would you think that "infinite time doesn't happen" could be incorrect? It simply means that no clock (at least, no clock that is tuned in accordance with theory) will ever indicate ∞.

Yes, no actual clock will ever give an infinite reading. But that does not mean that any clock which gives a finite reading has to be confined to the region of spacetime where the Schwarzschild time coordinate is finite. Clocks don't read coordinate time, they read proper time along their worldlines. Only clocks "at infinity" actually read that time coordinate, because only for clocks "at infinity" is Schwarzschild coordinate time equal to proper time along their worldlines.

In my language your claim is incompatible with the claim of O-S; apparently we don't speak the same language. So be it. I hope to have clarified that Trickydicky's "logical causal future realization" is according to O-S "impossible to develop in a finite time" - with which of course not the proper clock time of an infalling observer is meant, but approximately our clock time.

I'm confused. Are we talking about physics or terminology? I don't care what terminology you use; I'm just trying to figure out whether we disagree about the physics or not.

When O-S (and you) use the words "impossible to develop in a finite time", that can have two meanings:

#1: It doesn't happen anywhere in the region of spacetime covered by the Schwarzschild exterior time coordinate, but it does happen in some *other* region of spacetime.

#2: It doesn't happen *anywhere* in the spacetime, period.

If all you mean is #1, then we agree on the physics; we just disagree on the terminology we're using to describe it. If you mean #2, we disagree on the physics.

It's possible that O-S themselves did not take a position in their paper on this question; in other words, it's possible that all O-S meant by "impossible to happen in a finite time" was this:

#0: It doesn't happen anywhere in the region of spacetime covered by the Schwarzschild exterior time coordinate; we take no position on whether it happens in some other region of spacetime, because our model only covers the region covered by finite values of Schwarzschild coordinate time, and we haven't studied the question of whether or not the spacetime contains other regions besides that one.

But even if it was the case that O-S meant #1, I don't care; I have already said I agree with PAllen that that's a question about history, not physics. Our best current model of gravitational collapse says #1 is true and #2 is false; that's what I mean by "the physics".

 

[harrylin]

A key problem is your insistence that only one type of clock matters, as opposed examining the universe using all clocks.[..]

As I clarified earlier, not at all; you continue to misrepresent what I say. Instead I stressed that in GR all reference systems are valid for the predictions of physics. However, it suddenly becomes really interesting:

Let me try to clarify further between a coordinate statement (of no physical significance whatsoever) versus a statement about observables.

1) Coordinate statement: For distant observer's time, an event horizon never forms. Why is this not a physical statement? Both SR and GR completely reject the concept of global time as a physical concept. Global time is construct of convention or convenience, in all cases. This misleading statement has buried within it a concept there is physical meaning to distant simultaneity : which events in universe correspond to which times on a distant clock.
2) Indisputable physical statements [..]

Here we have a subtle but fundamental disagreement, even concerning SR. You pretend that global coordinate time cannot be used as a physical concept for making predictions about physical events, not even in SR. However, I know with 100% certainty that according to SR the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good (this was restricted to inertial frames). Those laws are for making predictions about observable events. And I think that the same is true according to GR, including accelerating frames. If you like, we can start that as a topic; it is too far off topic to discuss here.

 

[harrylin]

It's valid for making physical predictions about the region of spacetime in which that time coordinate is finite. It is *not* valid for making physical predictions about any other region of spacetime. [..]

Yes of course - that is the only thing that is needed.

Yes, no actual clock will ever give an infinite reading. But that does not mean that any clock which gives a finite reading has to be confined to the region of spacetime where the Schwarzschild time coordinate is finite. Clocks don't read coordinate time, they read proper time along their worldlines. Only clocks "at infinity" actually read that time coordinate

In fact, that is our free choice. In 1911, while developing GR, Einstein proposed that "we must use clocks of unlike constitution, for measuring time at places with differeing gravitational potential". That is in practice exactly what is done based on GR, and there is nothing in theory (both in GR and in theory) to prevent us from doing so for the whole accessible universe.

I'm confused. Are we talking about physics or terminology?

I thought that we were talking about physics; however I found myself being told that what I say is "wrong", without however identifying anything substantial apart of the fact that I do not use modern jargon.

When O-S (and you) use the words "impossible to develop in a finite time", that can have two meanings:

#1: It doesn't happen anywhere in the region of spacetime covered by the Schwarzschild exterior time coordinate, but it does happen in some *other* region of spacetime.

#2: It doesn't happen *anywhere* in the spacetime, period.

Here we discuss not our opinions but that of O-S, and about what follows from their model.
As I mentioned earlier, there appear to be some inconsistencies in formulation in their paper, which made me doubt that O-S had contemplated that question when they wrote it. For that reason I did not make a statement that goes beyond what they explained; my point was that Trickydicky made a claim about O-S that I find hard to rime with what O-S claimed themselves.

It's possible that O-S themselves did not take a position in their paper on this question; in other words, it's possible that all O-S meant by "impossible to happen in a finite time" was this:

#0: It doesn't happen anywhere in the region of spacetime covered by the Schwarzschild exterior time coordinate; we take no position on whether it happens in some other region of spacetime, because our model only covers the region covered by finite values of Schwarzschild coordinate time, and we haven't studied the question of whether or not the spacetime contains other regions besides that one.

Yes, that almost matches my opinion, except that perhaps they had not yet developed that thought - I suppose that they would have made such a statement if they had.

But even if it was the case that O-S meant #1, I don't care; I have already said I agree with PAllen that that's a question about history, not physics. Our best current model of gravitational collapse says #1 is true and #2 is false; that's what I mean by "the physics".

Concerning modern models, I have not yet been won over to consider option #1 as possibly physical; that makes option #2 more plausible for me, at least for from the outside infalling matter. And as you know that option was promoted 5 years ago by Vachaspati et al in phys. review D. However, I cannot judge the quality of their model, or how well their non-QM simulation matches the O-S model. That's just my 2cts.
Once more, the blog of their university: http://blog.case.edu/case-news/2007/06/20/blackholes

 

[PAllen]

Here we have a subtle but fundamental disagreement, even concerning SR. You pretend that global coordinate time cannot be used as a physical concept for making predictions about physical events, not even in SR. However, I know with 100% certainty that according to SR the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good (this was restricted to inertial frames). Those laws are for making predictions about observable events. And I think that the same is true according to GR, including accelerating frames. If you like, we can start that as a topic; it is too far off topic to discuss here.

Here is how subtle things are: I absolutely agree and have explicitly stated numerous times that any coordinate time is valid for making physical predictions. But since this is true for any coordinate time, to me (and, I absolutely believe, Einstein, but not necessarily Lorentz), the implications is none can be physically preferred, and none have physical meaning beyond convention (thus Einstein's careful use in e.g. his 1905 paper: we stipulate; we define; the pure conventions are separated from physical predictions).

So, to you: useful for making physical predictions = physical reality. To me, this follows only if the thing under discussion is, itself, observable. The statement 'my time at a distant place' is not a physically verifiable statement at all. The statement: if I assign time to distant events in one of many ways, I can readily compute physical predictions: this is indisputable. Since nothing more can be given verifiable meaning, I believe nothing more than this.

 

[PAllen]

And as you know that option was promoted 5 years ago by Vachaspati et al in phys. review D. However, I cannot judge the quality of their model, or how well their non-QM simulation matches the O-S model. That's just my 2cts.
Once more, the blog of their university: http://blog.case.edu/case-news/2007/06/20/blackholes

What part of this is non-quantum? As I read it, none is non quantum because it all based on radiation computed using functional schrodinger formalism. We seem to read English differently.

 

[PeterDonis]

In fact, that is our free choice. In 1911, while developing GR, Einstein proposed that "we must use clocks of unlike constitution, for measuring time at places with differeing gravitational potential". That is in practice exactly what is done based on GR, and there is nothing in theory (both in GR and in theory) to prevent us from doing so for the whole accessible universe.

Yes, it's our free choice what clocks to use and what worldlines they follow. But it is *not* our free choice, once the clocks and worldlines are given, to decide what those clocks will read. That's determined by physics. It's also not our free choice what regions of spacetime are present, and what kinds of relationships are possible between the readings on clocks following worldlines that become spatially separated; that's also determined by physics.

Concerning modern models, I have not yet been won over to consider option #1 as possibly physical; that makes option #2 more plausible for me, at least for from the outside infalling matter.

Huh? It's a simple question: is there an event horizon and black hole region anywhere in the spacetime, or not? "Modern models" give an unequivocal answer for the case of classical GR (no quantum corrections): yes. Any paper, whether it's "modern" or not, that claims otherwise is not a reputable paper (or else you're misunderstanding the paper to be talking about the classical case when it's actually talking about the quantum case--see below).

There is not an unequivocal answer when quantum corrections are included; but the O-S paper was not about the quantum case, it was about the classical case, so for this thread, I was assuming that any "modern models" we wanted to talk about would also be about the classical case, not the quantum case. If we want to talk about the quantum case we should probably start a separate thread.

And as you know that option was promoted 5 years ago by Vachaspati et al in phys. review D. However, I cannot judge the quality of their model, or how well their non-QM simulation matches the O-S model. That's just my 2cts.
Once more, the blog of their university: http://blog.case.edu/case-news/2007/06/20/blackholes

This is talking about the quantum case, not the classical case. What "non-QM simulation" are you referring to? I don't see any such thing here.

 

[harrylin]

Here is how subtle things are: I absolutely agree and have explicitly stated numerous times that any coordinate time is valid for making physical predictions. But since this is true for any coordinate time, to me (and, I absolutely believe, Einstein, but not necessarily Lorentz), the implications is none can be physically preferred, and none have physical meaning beyond convention (thus Einstein's careful use in e.g. his 1905 paper: we stipulate; we define; the pure conventions are separated from physical predictions).

So, to you: useful for making physical predictions = physical reality. To me, this follows only if the thing under discussion is, itself, observable. The statement 'my time at a distant place' is not a physically verifiable statement at all. The statement: if I assign time to distant events in one of many ways, I can readily compute physical predictions: this is indisputable. Since nothing more can be given verifiable meaning, I believe nothing more than this.

Apart of an untreatable mutual misunderstanding, we absolutely agree on this. Distant clock time is only physical reality in the sense that a distant clock must indicate a time, which in principle allows for verification of predictions.
There is in principle nothing that prevents us from putting clocks in orbit around a black hole, approximately tuned to the ECI coordinate system.

 

[harrylin]

[..] Huh? It's a simple question: is there an event horizon and black hole region anywhere in the spacetime, or not? "Modern models" give an unequivocal answer for the case of classical GR (no quantum corrections): yes. Any paper, whether it's "modern" or not, that claims otherwise is not a reputable paper (or else you're misunderstanding the paper to be talking about the classical case when it's actually talking about the quantum case--see below).

I'm not sure what you mean with "is there a black hole region in the spacetime"; that seems to be a technical term. Probably you will conclude that in their model there isn't one, if you use the same definitions as them in their press statement. You can decide for yourself:

[..] What "non-QM simulation" are you referring to? I don't see any such thing here.

As already mentioned in the latest thread, section III of http://arxiv.org/abs/gr-qc/0609024

 

[PAllen]

I don't know what you mean with "region in the spacetime"; that seems to be a technical term. But that doesn't matter, as you can decide for yourself:

As already mentioned in the latest thread, section III of http://arxiv.org/abs/gr-qc/0609024

Section three of that paper contains nothing new, and its authors don't claim anything new in this section (they footnote these results to an ancient paper by Townsend). They use this formalism to then establish new results using quantum methods.

 

[PAllen]

Apart of an untreatable mutual misunderstanding, we absolutely agree on this. Distant clock time is only physical reality in the sense that a distant clock must indicate a time, which in principle allows for verification of predictions.
There is in principle nothing that prevents us from putting clocks in orbit around a black hole, approximately tuned to the ECI coordinate system.

Of course we can put clocks all over in different states of motion, and modify their 'natural readings' as desired. However (and maybe you don't disagree) it remains purely a matter of convention or definition which reading on one clock is considered 'the same time' as which reading on another clock.

Note, we can readily do this between an infalling clock and a distant clock such that 3 pm on both clocks corresponds to the infalling clock a microsecond before hitting the singularity. Each clock would read its own proper time, and the relation between their world lines would be based on GP time coordinate instead of SC time coordinate (the time coordinate just being used to establish simultaneity relations).

 

[harrylin]

Section three of that paper contains nothing new, and its authors don't claim anything new in this section (they footnote these results to an ancient paper by Townsend). [..]

Indeed, as we already discussed in this thread, their opinion, already from their "classical" analysis, that "the horizon does not form in a finite time" is nothing new; and as I already mentioned in the new thread, Kraus pretends that it is "controversial" - which is obviously true, as can be seen by the reactions to their publication by some on this forum. I will not elaborate on it in this thread, as that would distract from the O-S model.

 

[PAllen]

Indeed, as we already discussed in this thread, their opinion, already from their "classical" analysis, that "the horizon does not form in a finite time" is nothing new; and as I already mentioned in the new thread, Kraus pretends that it is "controversial" - which is obviously true, as can be seen by the reactions to their publication by some on this forum. I will not elaborate on it in this thread, as that would distract from the O-S model.

If you look carefully, the word 'controversial' in that part of the quote is the journalist's word not Kraus's word. The part actually quoted to Kraus is non-controversial. Again, nothing said in the paper or any of the commentary you link to is inconsistent with:

- starting from established classical results, wondering if one way causality and behavior of SC coordinate time may provide a hint at quantum treatment,

- we then treat the the collapse quantum mechanically (using SC coordinates) and find that evaporation beats collapse. Therefore the information paradox never arises. And backfitting this result, we may choose to ignore anything classical GR says about the horizon and interior.

And counter arguing papers are all on the second bullet above: you don't escape the information paradox that easily. Evaporation does not beat collapse. A deeper solution to the information paradox is needed.

 

[PeterDonis]

I'm not sure what you mean with "is there a black hole region in the spacetime"; that seems to be a technical term.

I suppose it could be called technical, but it's not very technical. Look at the diagram that DrGreg posted of a spherically symmetric gravitational collapse. The blue region in that diagram is the "black hole region", and it is part of the spacetime because it appears on the diagram. That's all "there is a black hole region in the spacetime" means.

But since the blue region is above the horizon line (the 45 degree line going up and to the right), light signals from the black hole region can never get out to the gray region, which is the region covered by the distant observer's time coordinate. That's why the black hole region is not "visible" to the distant observer; he can never see light signals from it. But the region is there.

Probably you will conclude that in their model there isn't one, if you use the same definitions as them in their press statement.

As already mentioned in the latest thread, section III of http://arxiv.org/abs/gr-qc/0609024

This section does give a "classical model", but in that model, there *is* an event horizon and a black hole region; it's just not visible to the asymptotic observer (because no light signals from the EH or the BH region can get back out to the asymptotic observer). In other words, it's qualitatively the same as what I have been calling the best current classical GR model of gravitational collapse. If you drew a spacetime diagram of it in the appropriate coordinates, it would look similar to DrGreg's diagram, including the blue region.

 

[PeterDonis]

There is in principle nothing that prevents us from putting clocks in orbit around a black hole, approximately tuned to the ECI coordinate system.

But you won't be able to extend the ECI coordinates inside the horizon; they will become singular at the horizon just like standard Schwarzschild coordinates do. So ECI coordinates won't cover the black hole region.

 

[PAllen]

Indeed, as we already discussed in this thread, their opinion, already from their "classical" analysis, that "the horizon does not form in a finite time" is nothing new; and as I already mentioned in the new thread, Kraus pretends that it is "controversial" - which is obviously true, as can be seen by the reactions to their publication by some on this forum. I will not elaborate on it in this thread, as that would distract from the O-S model.

Forget journalism and press releases (though it is clear to me you misinterpret the press release). Here is the brief description of the results of section III by the author's intended for a scientific audience:

"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.."

Let me emphasize:

- verify standard results

- infinite Schwarzschild time

No where are they claiming a new classical result; no where do they dispute (nor mention) the classical result that the in other coordinates the EH happens in finite coordinate time, and that the dust cloud crosses the EH in finite time for a clock following just above its surface. These are not concerns of the paper. The paper is clearly concerned with quantum corrections, wherein (if they are right) these other features go away. They believe in coordinate invariance, so the implication is that if quantum analysis says the collapsed object evaporates before the EH is formed in SC coordinates, then this means, in any coordinates, and for any observer, there is no EH at all. This is the new and fairly radical claim - all based on quantum corrections. If piece of matter transforms to radiation before a horizon is formed in coordinate system, the fact must be true in all. This is the controversial aspect of their work.

[Edit: in reference to Dr. Greg's beautiful illustration in #64, the key point of the Krauss,et.al. paper is to argue that [due to quantum behavior - evaporation], the grey line curves up asymptotically to the top 45 degree line of the pink region, never entering the blue region. This means the blue region is not part of the solution at all. This is all coordinate independent geometry. The claim is not about interpreting something like classical O-S spacetime; it is that, when quantum effects are considered, classical O-S spacetime does not occur. What does occur looks very much like it, for a distant observer, for a very long time, but eventually, it can be distinguished - via the radiation - that the actual spacetime was never similar to an O-S spacetime, in that the blue region never existed - at all, for any observer.

If we translate the Krauss et. all. proposal to the experience of an observer on the collapsing shell, we get, instead of:

- reaching a horizon, then a singularity, in finite clock time (for that observer)

we get:

- being converted to not quite thermal radiation, in finite clock time, without ever reaching the critical radius.

If their result holds, and also applies to dust ball collapse, as they hope it does, then an interior observer of such a collapse would experience:

- in finite time, evaporating to non quite thermal radiation before reaching a minimum radius.
]

 

[PeterDonis]

in reference to Dr. Greg's beautiful illustration in #64, the key point of the Krauss,et.al. paper is to argue that [due to quantum behavior - evaporation], the grey line curves up asymptotically to the top 45 degree line of the pink region, never entering the blue region. This means the blue region is not part of the solution at all.

PAllen, great summary. The only point I would add is that, in reference to DrGreg's diagram, it's not enough for just the grey line to curve up and to the right as you describe; the entire interior of the collapsing matter has to do so. DrGreg did not show that region in his diagram; the grey line is just the outer surface of the collapsing matter.

As I read it, the model in the Krauss paper is somewhat different from the "O-S" model (by which I mean the modern version, not necessarily the version in the O-S paper). The Krauss paper models a collapsing "domain wall", which means a very thin spherical shell of stress-energy. In this model, the grey line in DrGreg's diagram *would* indeed be the entire "collapsing matter", since that matter is supposed to be very thin. Obviously this is much less realistic, physically, than the collapse of spherically symmetric dust as in the standard O-S type model (which itself is highly idealized, of course, with zero pressure and perfect spherical symmetry). They appear to be willing to make the educated guess that the qualitative conclusions from their model would still hold in a more realistic model; but they don't actually show that.

However, that leaves a very big open question in my mind: what is *inside* the domain wall? The classical GR conclusion would be that it is a flat Minkowski spacetime region, which would shrink as the domain wall collapses. However, I don't see such a region included in the Krauss paper's model at all. I haven't read any of the papers making counter-arguments, so I don't know if this issue has been raised.

Just off the top of my head, including the flat region interior to the domain wall, if the conclusion of Krauss et al. is true that quantum effects stop the collapse by converting the domain wall's stress-energy into outgoing radiation before it forms a horizon, would change the whole spacetime diagram; it would no longer look like DrGreg's. (Actually, if Krauss et al. are correct and a horizon doesn't form when quantum effects are included, that would change the diagram in any case; the 45 degree line up and to the right is the horizon, and if there is no horizon that changes the whole causal structure.) This is probably getting pretty far off topic for this thread, though.

 

[PAllen]

Just off the top of my head, including the flat region interior to the domain wall, if the conclusion of Krauss et al. is true that quantum effects stop the collapse by converting the domain wall's stress-energy into outgoing radiation before it forms a horizon, would change the whole spacetime diagram; it would no longer look like DrGreg's. (Actually, if Krauss et al. are correct and a horizon doesn't form when quantum effects are included, that would change the diagram in any case; the 45 degree line up and to the right is the horizon, and if there is no horizon that changes the whole causal structure.) This is probably getting pretty far off topic for this thread, though.

The main refutation seems to be the long Padnanabhan paper I linked. I have only skimmed it and much of it is too far beyond my expertise to read. However, they do raise, as one of several errors, that, if Krauss et.al. are right about the evaporation process, then they are wrong about using exterior SC geometry, even if spherical symmetry is assumed (due to the radiation).

Without radiation, and without a horizon, you could still the geometry as a large part of Dr. Greg's pink region. The grey line would bend up below the 'horizon that isn't there'. Anything outside (left of) the grey line would not be SC geometry, and we could cover it with a different chart. However, the remaining pink part could still represent the exact SC geometry outside the non-collapsing shell.

 

[PeterDonis]

The main refutation seems to be the long Padnanabhan paper I linked. I have only skimmed it and much of it is too far beyond my expertise to read. However, they do raise, as one of several errors, that, if Krauss et.al. are right about the evaporation process, then they are wrong about using exterior SC geometry, even if spherical symmetry is assumed (due to the radiation).

Ok, that means I didn't guess too badly.

Without radiation, and without a horizon,

Does this make sense? Isn't the Krauss argument that the horizon doesn't form because the stress-energy in the collapsing domain wall gets converted into radiation? If there is no radiation, what stops the horizon from forming?

The grey line would bend up below the 'horizon that isn't there'. Anything outside (left of) the grey line would not be SC geometry, and we could cover it with a different chart. However, the remaining pink part could still represent the exact SC geometry outside the non-collapsing shell.

I see the general point, but I'm not sure about it, because the "shape" of the pink region depends on their being a horizon; if the upward 45 degree line isn't there, because the horizon isn't there, there is no reason for the grey line to "bend up below the horizon that isn't there". There is no singularity "above" the horizon line if quantum effects mean the horizon doesn't form, so with no horizon timelike lines could extend "upwards" indefinitely and still be able to send light signals to infinity.

 

[PAllen]

Does this make sense? Isn't the Krauss argument that the horizon doesn't form because the stress-energy in the collapsing domain wall gets converted into radiation? If there is no radiation, what stops the horizon from forming?

mass of shell too small

I see the general point, but I'm not sure about it, because the "shape" of the pink region depends on their being a horizon; if the upward 45 degree line isn't there, because the horizon isn't there, there is no reason for the grey line to "bend up below the horizon that isn't there". There is no singularity "above" the horizon line if quantum effects mean the horizon doesn't form, so with no horizon timelike lines could extend "upwards" indefinitely and still be able to send light signals to infinity.

I have SC geometry for r > r0 for some r0 > SC radius (where Birkhoff applies). Within this region I use SC coordinates. Now, I apply the transform to Kruskal for this region of spacetime. I get section of Dr. Greg's pink region to the right of the r0 curve.

 

[PeterDonis]

mass of shell too small

But the shell is collapsing; if radiation doesn't continually carry away its mass, eventually it will collapse far enough to form a horizon. If there's no radiation, there's no method of carrying away any of its mass, so it will *have* to eventually form a horizon, regardless of how small its mass is; that's the point of the classical analysis in section III of the paper.

I have SC geometry for r > r0 for some r0 > SC radius (where Birkhoff applies). Within this region I use SC coordinates. Now, I apply the transform to Kruskal for this region of spacetime. I get section of Dr. Greg's pink region to the right of the r0 curve.

Yes, I understand that; I'm just trying to understand what the rest of the spacetime would look like (the part occupied by the non-collapsing wall and the interior Minkowski region) in these coordinates. Probably I need to first think about a simpler case, a static spherically symmetric star surrounded by vacuum, and how that would look when transformed to Kruskal-like coordinates.

 

[PAllen]

But the shell is collapsing; if radiation doesn't continually carry away its mass, eventually it will collapse far enough to form a horizon. If there's no radiation, there's no method of carrying away any of its mass, so it will *have* to eventually form a horizon, regardless of how small its mass is; that's the point of the classical analysis in section III of the paper.

I haven't looked at whether they exclude pressure from the Lagrangian. However, for any realistic equation of state for matter, there is a shell mass below which collapse will simply stop at some point. Dr. Greg referred to this possibility. It is also discussed at some length in the Padmanabhan paper, where they show some claims of the Krauss et.al. paper lead to rather silly conclusions for this case.

Yes, I understand that; I'm just trying to understand what the rest of the spacetime would look like (the part occupied by the non-collapsing wall and the interior Minkowski region) in these coordinates. Probably I need to first think about a simpler case, a static spherically symmetric star surrounded by vacuum, and how that would look when transformed to Kruskal-like coordinates.

I was positing a simpler way of handling it. Use the section of Kruskal I described for a vacuum. Use a completely different chart for the non-vacuum. For the non-vacuum, you must satisfy junction conditions. However, Birkhoff allows you to ignore that for the vacuum part.

 

[PeterDonis]

I haven't looked at whether they exclude pressure from the Lagrangian. However, for any realistic equation of state for matter, there is a shell mass below which collapse will simply stop at some point. Dr. Greg referred to this possibility. It is also discussed at some length in the Padmanabhan paper, where they show some claims of the Krauss et.al. paper lead to rather silly conclusions for this case.

Hm, yes, I wasn't considering pressure. I'll have to look at the paper again to see exactly how they model the domain wall; I had thought it was simply a shell of dust, but I may be wrong.

I was positing a simpler way of handling it. Use the section of Kruskal I described for a vacuum. Use a completely different chart for the non-vacuum.

There's nothing requiring the use of a specific chart, true. The standard Kruskal chart only works for vacuum regions anyway. But in order to show the causal structure of the spacetime, I would want to find a chart for the non-vacuum region that still shows radial null curves as 45 degree lines; I don't know if such a chart has ever been used. [Edit: Actually a Penrose chart does this, and those do exist for FRW spacetimes, so one can certainly draw one for the standard O-S type model where an FRW interior is matched to a Schwarzschild exterior; I've seen that done. I haven't seen one for a "domain wall" type of model.]

 

[harrylin]

[..] you don't escape the information paradox that easily. Evaporation does not beat collapse. A deeper solution to the information paradox is needed.

I had not seen this. Contrary to you, I can find no paradox at all, except with your interpretation.
But probably we will discuss that in your new thread, https://www.physicsforums.com/showthread.php?t=652839

 

[PAllen]

I had not seen this. Contrary to you, I can find no paradox at all, except with your interpretation.
But probably we will discuss that in your new thread, https://www.physicsforums.com/showthread.php?t=652839

The 'information paradox' is a general concern of quantum mechanics + gravity. It is universally accepted that there must be some solution (well, except for Penrose, who believes information is truly lost in a BH, and QM must be superseded). A great many possible solutions have been proposed. As I read the Krauss et.al. paper and other paper citing it, it is proposal in this general field: the information paradox is resolved because it never occurs, because the collapsed object evaporates before EH is formed. Most other solutions involve quantizing the EH (and interior) in some way, with various models of how the information paradox gets solved in the particular model.

But again, as seem so common, I am not sure I understand what your are getting at. Probability of this seems 99% bidirectional between us.

 

[harrylin]

[..] But again, as seem so common, I am not sure I understand what your are getting at. Probability of this seems 99% bidirectional between us.

Yes, that is too often a problem. But not this time: I made sure to not clarify it here, because I want to discuss it there - and knowing you, if I clarify it here then you will start to discuss it here.

 

[harrylin]

[..] probably we will discuss that in your new thread, https://www.physicsforums.com/showthread.php?t=652839

The discussion there was for me very surprising. The discussion quickly zoomed in on O-S model predictions - and that brings me back to this thread:

they consider Schwarzschild coordinate time to be far away clock time - which is approximately the time on our clocks. And that time is according to GR valid for making physical predictions, just as they did and I cited.

[SC coordinate map] is valid for making physical predictions about the region of spacetime in which that time coordinate is finite. It is *not* valid for making physical predictions about any other region of spacetime.

The only sense in which the maps "disagree about events" is that one map (SC coordinates) can't assign coordinates to some events (those on or inside the horizon), while another map (e.g., Painleve coordinates) can.

Actually they don't disagree about events. With one convention, assign remote times ranging to infinity for all the events I will ever see. I still compute that physical law says there are other events I will never actually see.

Time codes emitted from Earth are received by Voyager just fine at τ=42, and indeed all the way up to τ=48.

Inspired by that last comment, I will here expand on that simple example.

Voyager 35 is sent to a newly discovered black hole only about 20 light years away and which for simplicity we assume to be eternal static, and in rest wrt the solar system. The Voyager is indestructible and always in operation.

A time code is emitted from Earth that can be received by Voyager. Voyager emits its proper time code s1 that is sent back to Earth together with the then received time stamp t1 from Earth (we'll ignore the technical difficulties).

An observer on Earth with the name Kraus calculates the expected (s1,t1) signal from Voyager as function of expected UTC, for the approximation or assumption that the black hole is completely formed. He stresses that he could choose other coordinates, but that the "SC" of Oppenheimer-Snyder-1939 are fine and valid for making predictions about what can be observed on Earth, making small corrections for Earth's gravitational field and orbit. He finds something like the following (I pull this out of my hat, just for the gist of it):

UTC , (s1 , t1)
--------------
100 , 40.3, 200
1E3 , 41.2, 1.5E3
1E4 , 41.5, 1E5
1E5 , 41.7, 1E7
1E6 , 41.9, 1E10
1E100 42.0, 1E1000

My question: Please give an illustration of time codes t1 from Earth that reach Voyager at τ=43, as it has gone through the horizon.

 

[Mike Holland]

I have trouble imagining the Krauss quantum phenomena in the case of PAllen's trillion star contractring cluster. Surely in this case an event horizon would form long before any quantum radiation is emittted. The stars are still well separated when the black hole forms!

Mike

 

[PeterDonis]

A time code is emitted from Earth that can be received by Voyager. Voyager emits its proper time code s1 that is sent back to Earth together with the then received time stamp t1 from Earth (we'll ignore the technical difficulties).

An observer on Earth with the name Kraus calculates the expected (s1,t1) signal from Voyager as function of expected UTC, for the approximation or assumption that the black hole is completely formed.

Ok, just to make sure I understand:

- Earth emits a signal time stamped with the time t1 of emission according to Earth clocks.

- Voyager receives the signal, and emits a return signal time stamped with the time s1 of emission according to Voyager's clock, plus the Earth emission timestamp t1 of the Earth signal just received.

- Earth wants to predict the (s1, t1) pairs that it will receive in Voyager's return signal, as a function of the time UTC that it receives the return signal.

UTC , (s1 , t1)
--------------
100 , 40.3, 200
1E3 , 41.2, 1.5E3
1E4 , 41.5, 1E5
1E5 , 41.7, 1E7
1E6 , 41.9, 1E10
1E100 42.0, 1E1000

Assuming my understanding above is correct, the first and last columns are wrong as given. The last column is reasonable as a set of "UTC" values; the first column isn't usable at all as given.

A correct set of numbers would look something like this (I haven't calculated these numbers exactly, I've just tried to give a fair approximation of the qualitative behavior):

t1, s1, UTC
-------------
40, 40.3, 200
40.5, 41.2, 1.5E3
40.7, , 41.5, 1E5
40.8. , 41.7, 1E7
40.9, 41.9, 1E10
40.99, 41.99, 1E1000
(...)
41, 42, (Earth never receives any return signal from here on)
41.3, 43
41.6, 44
41.8, 45
42, 46
42.2, 47
42.3, 48
42.300001, (Voyager never receives any Earth signal from here on, it is destroyed in the singularity at tau = 48)

 

[PeterDonis]

To help make sense of the numbers in my last post, attached is a Kruskal-type plot of the scenario. (I made it using fooplot.com, which seems like a neat if simple online tool for generating plots.)

Quick description of the plot:

- The horizontal and vertical axes are the Kruskal U and V coordinate axes.

- The black hyperbola at the top is the singularity at r = 0.

- The crossing 45 degree gray lines are the horizon (up and to the right) and the antihorizon (up and to the left). In a more realistic model where the black hole was formed by the collapse of a massive object, the antihorizon would not be there; instead, there would be the surface of the collapsing object on the left as in the diagram DrGreg posted some time ago.

- The blue hyperbola on the right is the Earth's worldline.

- The dark red curve that leaves Earth at U = 0 (i.e., just as Earth crosses the horizontal axis--this is also t = 0 on Earth's clock) is Voyager's worldline; Voyager leaves Earth and falls into the hole.

- The three progressively darker green lines, running from Earth up and to the left towards Voyager, are three of the light signals emitted from Earth, at Earth times (according to the numbers in my previous post) 40 (more or less--the qualitative behavior is the key here, not the exact numbers), 41, and 42.3. Note what happens to them:

Signal #1 reaches Voyager before it crosses the horizon; Voyager then emits a return signal (the 45 degree line going up and to the right from where #1 reaches Voyager), which reaches Earth further up its worldline, at t = 200 (more or less). You can see that signals emitted in between #1 and #2 from Earth will be received by Voyager closer and closer to the horizon, so Voyager's return signals will reach Earth further and further up its worldline, i.e., at later and later times, increasing without bound.

Signal #2 reaches Voyager just as it crosses the horizon. Voyager's return signal therefore stays at the horizon; it never reaches Earth. Signals emitted from Earth between #2 and #3 will reach Voyager between the horizon and the singularity, so its return signals will stay below the horizon and also never reach Earth (eventually each of these return signals will hit the singularity).

Signal #3 reaches Voyager just as it hits the singularity. Any signal emitted from Earth after #3 will never reach Voyager, because it is destroyed in the singularity; these signals will hit the singularity instead.

 

[harrylin]

Ok, just to make sure I understand:

- Earth emits a signal time stamped with the time t1 of emission according to Earth clocks.

- Voyager receives the signal, and emits a return signal time stamped with the time s1 of emission according to Voyager's clock, plus the Earth emission timestamp t1 of the Earth signal just received.

- Earth wants to predict the (s1, t1) pairs that it will receive in Voyager's return signal, as a function of the time UTC that it receives the return signal.

Assuming my understanding above is correct, the first and last columns are wrong as given.

Oops yes, sorry for the glitch - indeed I swapped the two Earth times in the table.

The last column is reasonable as a set of "UTC" values; the first column isn't usable at all as given.

A correct set of numbers would look something like this (I haven't calculated these numbers exactly, I've just tried to give a fair approximation of the qualitative behavior):

t1, s1, UTC
-------------
40, 40.3, 200
40.5, 41.2, 1.5E3
40.7, , 41.5, 1E5
40.8. , 41.7, 1E7
40.9, 41.9, 1E10
40.99, 41.99, 1E1000
(...)
41, 42, (Earth never receives any return signal from here on) [..]

I suppose that with "from here on" you mean after UTC > 1E10000000000000000000000000000000000000000.
Correct?

The t1 numbers in the beginning are surprising to me; you seem not to account for the ca. 20 light years in "distant" units in your estimated prediction. And/or you assume that the different time dilation factors largely compensate each other.

[Addendum]: in fact I assumed the Voyager to circle for some years in orbit, thus ticking slower; and I suddenly realize that I added instead of subtracted the 20 years - I was in a hurry! What could be relevant for this discussion (although likely also not) is your (t1,s1) = (40.99, 41.99). I don't know how you get that 1 year difference, is that just a coincidence?

Now I'll study the rest; the issue is really (t1,s1)= (41.3, 43).
I do think that Earth must get a signal back (41.3, 41.9999999999) according to O-S-1939.

 

[PeterDonis]

Oops yes, sorry for the glitch - indeed I swapped the two Earth times in the table.

Ok, good.

I suppose that with "from here on" you mean after UTC > 1E100000000000000 - correct?

No, I mean that signals emitted by Voyager at or after s1 = 42 are never received by Earth (because they remain at or inside the horizon). There is no invariant way to relate that to a "time" on Earth's worldline; it depends on which simultaneity convention you choose. Some conventions (like that of standard SC coordinates) don't allow you to assign a "t" coordinate to events on Voyager's worldline with s1 >= 42 at all; no surface of simultaneity in that convention passes through any event on or inside the horizon. Other conventions (like that of Painleve coordinates or Eddington-Finkelstein coordinates) allow you to assign a finite "time" coordinate in those charts to events on or inside the horizon.

The s1 numbers in the beginning don't make sense to me. I accounted (very roughly) for about a factor 2 time dilation due to the high speed of Voyager on its way towards the black hole, aas measured in Schwartzschild time t. I find that time dilation lacking in your estimation. However, that is perhaps not important for this discussion.

I don't think the exact numbers are important (I wasn't trying to get them exact anyway), but the qualitative behavior is. Your t1 numbers were *larger* than your s1 numbers, and your t1 numbers increased very fast (though not as fast as your UTC numbers) as your s1 numbers approached 42. That's wrong. The t1 numbers should be *less* than the s1 numbers, and the t1 numbers should, if anything, grow more slowly than the s1 numbers as the s1 numbers approach 42, because the t1 timestamps are made before the Earth light signals travel inward towards Voyager; that light-speed travel time delay should more than cancel out the time dilation factor due to Voyager's inward motion (though I'm not quite as sure about that last; I'll have to do the calculation when I get a chance). Looking at the diagram I posted may be helpful.

 

[harrylin]

Oops I was still editing my post, trying to reconstruct what went wrong in not -so-important details.

[..] No, I mean that signals emitted by Voyager at or after s1 = 42 are never received by Earth (because they remain at or inside the horizon). There is no invariant way to relate that to a "time" on Earth's worldline; it depends on which simultaneity convention you choose.

I specified that the black hole and solar system are in rest wrt to each other, and that that time convention is used for t. t>∞ is in number simulation indicated as t>1E100000000000000. As a reminder, the O-S model:
"we see that for a fixed value of R as t tends toward infinity, τ tends to a finite limit".
That is also what online simulators find (in fact I now found a nice one in Java. )

your t1 numbers increased very fast (though not as fast as your UTC numbers) as your s1 numbers approached 42. That's wrong. The t1 numbers should be *less* than the s1 numbers, and the t1 numbers should, if anything, grow more slowly than the s1 numbers as the s1 numbers approach 42, because the t1 timestamps are made before the Earth light signals travel inward towards Voyager; that light-speed travel time delay should more than cancel out the time dilation factor due to Voyager's inward motion (though I'm not quite as sure about that last; I'll have to do the calculation when I get a chance). Looking at the diagram I posted may be helpful.

I'm too tired now, it was a long day and I squeezed this example in-between. But yes, you are certainly right about that point (except that I did not assume Voyager to free-fall straight towards the black hole).
The real issue is the last point in my addendum, which was also the intended point of the illustration. To be discussed tomorrow!

 

[PeterDonis]

I specified that the black hole and solar system are in rest wrt to each other, and that that time convention is used for t.

Which is fine for events outside the horizon; but you can't just declare by fiat that those are the only events that exist. If you want to say that, for purposes of your scenario, those are the only events we can consider, then some of the questions you are trying to ask simply do not have answers at all.

(except that I did not assume Voyager to free-fall straight towards the black hole).

That's the simplest assumption from a mathematical standpoint, so it's the one I used. A more complicated assumption would not change the central conclusions, it would just make the calculations more complicated.

I'll comment on your addendum in a separate post.

 

[PeterDonis]

[Addendum]: in fact I assumed the Voyager to circle for some years in orbit, thus ticking slower

Doing that just adds a long period of time where Voyager can exchange light signals with Earth before it falls in. There are no stable orbits inside r = 6M (three times the horizon radius), and no orbits at all, even unstable ones that have to constantly be maintained by rocket thrust, inside r = 3M (1.5 times the horizon radius). Time dilation at those altitudes is not very great by relativisitic standards, and anyway, as I said, the period of orbiting is irrelevant to the central question we're addressing.

What could be relevant for this discussion (although likely also not) is your (t1,s1) = (40.99, 41.99). I don't know how you get that 1 year difference, is that just a coincidence?

As I said, I wasn't calculating exact numbers, just trying to qualitatively describe the general pattern; so if any numbers happen to match something else, it's just a coincidence. I won't have time to do any detailed calculations until after this weekend.

Now I'll study the rest; the issue is really (t1,s1)= (41.3, 43).
I do think that Earth must get a signal back (41.3, 41.9999999999) according to O-S-1939.

O-S 1939 is consistent with everything I said up to (t1, s1, UTC) -> (41, 42, infinity) (qualitatively speaking--as I said, I haven't done detailed calculations of the exact numbers). After that point O-S 1939 doesn't cover the scenario at all; they don't say it's possible and they don't say it's impossible. They simply leave their analysis incomplete. (Their analysis has been completed since--for example, it's in MTW and other GR textbooks--and the completion of the analysis is what I've used to generate the qualitative behavior I illustrated.)

O-S do say, however, that when the surface of the infalling matter reaches the horizon radius (what they call r_0)--this corresponds to Voyager's clock reaching tau = 42--outgoing light can no longer escape (hence the infinity as the limit of the UTC times above as t1, s1 -> 41, 42). This seems like a pretty clear indication that *if* O-S had continued their analysis and discovered that points on Voyager's worldline with tau > 42 could exist, they would find (as modern analyses have found) that those points would not be able to send light signals back to Earth; since if outgoing light can't escape from the event where tau = 42, at r = r_0, any event with tau > 42 must have r < r_0 (since r > r_0 would require Voyager to move faster than light from the tau = 42 event, and even r = r_0 would require Voyager to move at the speed of light from the tau = 42 event), and would also not be able to send signals back to Earth (since those signals would also have to move faster than light).

If you think otherwise, please give specific references from the paper. I've read it through now and what I've said about the model in that paper and its limitations is based on what I've read.

A final note about the 20 light-year distance: that would just add an irrelevant constant to every s1 value and every UTC value. Instead of triples like (40, 40.3, 200), you would get, for example, (40, 40.3 + 20 years, 200 + 20 years); and instead of triples like (40.99, 41.99, 1E1000), you would get, for example, (40.99, 41.99 + 20 years, 1E1000 + 20 years), which works out to a very good approximation to (40.99, 41.99 + 20 years, 1E1000). So the 20 years quickly becomes negligible compared to the huge increase in UTC values compared to the other two.

Rather than add 20 years to the s1 and UTC values as above, I chose to ignore the 20 light year distance and assume that Earth was much closer to the hole. But I can put back in the 20 light year distance when I do the detailed calculations if you think it's really important (I don't think it is, since it doesn't change the qualitative behavior).

 

[harrylin]

[..] As I said, I wasn't calculating exact numbers, just trying to qualitatively describe the general pattern; so if any numbers happen to match something else, it's just a coincidence. I won't have time to do any detailed calculations until after this weekend.

Surely that won't be needed. For general interest for this kind of discussions, the following simulation program that I found yesterday may be handy:

http://www.compadre.org/osp/items/detail.cfm?ID=7232
Put r=7.414 and τ gets to nearly 42 as in my original illustration.

O-S 1939 is consistent with everything I said up to (t1, s1, UTC) -> (41, 42, infinity) (qualitatively speaking--as I said, I haven't done detailed calculations of the exact numbers). After that point O-S 1939 doesn't cover the scenario at all; they don't say it's possible and they don't say it's impossible. They simply leave their analysis incomplete. [..]

Sure. To me their model looks straightforward enough to discuss qualitatively (for high numerical precision we should write a little program). Their model is based on standard stationary space of Einstein's GR that is also used in Schwartzschild's model, right?

[..] *if* O-S had continued their analysis and discovered that points on Voyager's worldline with tau > 42 could exist, they would find (as modern analyses have found) that those points would not be able to send light signals back to Earth

in fact, I cited them as saying just that - see my post #50.

However there was an essential point that I overlooked: in the model of a fully formed black hole Voyager remains in free-fall towards the centre, so that it may be expected to outrun certain radio waves (thanks for pointing that out Atyy!).

Consequently I will almost certainly agree with your calculation about by us observable events - thank you too.

 

[PeterDonis]

For general interest for this kind of discussions, the following simulation program that I found yesterday may be handy:

http://www.compadre.org/osp/items/detail.cfm?ID=7232

This looks cool, thanks for the link!

Their model is based on standard stationary space of Einstein's GR that is also used in Schwartzschild's model, right?

For the portion of the spacetime that is vacuum (i.e., outside the collapsing matter), yes. For the portion of the spacetime that is not vacuum (i.e., inside the collapsing matter), no: that portion of the spacetime is not vacuum (of course), it's stationary (it's collapsing), and the boundary between it and the vacuum region is not stationary either (it's shrinking).

However there was an essential point that I overlooked: in the model of a fully formed black hole Voyager remains in free-fall towards the centre, so that it may be expected to outrun certain radio waves (thanks for pointing that out Atyy!).

Yes, that's reflected in my numbers: in my numbers, Voyager will "outrun" any radio wave emitted by Earth after t1 = 42.3, in the sense that Voyager will hit the singularity before the radio wave reaches it.