Oppenheimer-Snyder model of star collapse

[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.

 

[harrylin]

This looks cool, thanks for the link!

Yes I also think that it's cool, The orbiter can be repositioned and double-clicking on it gives the energy. Seeing such nice programs encourages me to get back to doing some programming . Regretfully I don't know Java.

Now that I finally got an understanding of the "inside region" arguments, I can zoom in on the real issues - which did not go away. But before continuing I want to make sure of one thing:

[..] 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 [not]stationary (it's collapsing), and the boundary between it and the vacuum region is not stationary either (it's shrinking). [..] .

I think that you misunderstood. What I meant is that O-S are developing further Schwartzschild's model, which uses stationary space coordinates. That is consistent with Einstein's 1905 purpose ("the view here to be developed will not require an “absolutely stationary space” provided with special properties, nor assign a velocity-vector to a point of the empty space in which electromagnetic processes take place").
Like me, you seem to relate the motion of matter with respect to such a reference system in which space does not have a velocity vector; and my impression is that the O-S model that they presented is consistent with that.

 

[PeterDonis]

I think that you misunderstood. What I meant is that O-S are developing further Schwartzschild's model, which uses stationary space coordinates.

They use these coordinates in the first part of the paper; but in the second part of the paper they use different coordinates, ones which are comoving with the collapsing matter.

However, I wasn't making a statement about coordinates; I was making a statement about physics. The original Schwarzschild model was of a spacetime that is entirely static--nothing changes with time. The O-S model is of a spacetime that is only partially static; the region containing the collapsing matter is not static, it changes with time, and so does the radius of its boundary with the vacuum region. So if I am at a certain radius that is greater than the radius r_0 (what we would now call the horizon radius), the metric in my vicinity only becomes static once the collapsing matter falls past me to a smaller radius. That's true regardless of what coordinates I use.

That is consistent with Einstein's 1905 purpose ("the view here to be developed will not require an “absolutely stationary space” provided with special properties, nor assign a velocity-vector to a point of the empty space in which electromagnetic processes take place").

I don't have any particular problem with this, but I don't see how it's relevant to what we're discussing here. A coordinate system that is comoving with the collapsing matter doesn't have to "assign a velocity-vector to a point of the empty space", any more than a stationary coordinate system does.

 

[harrylin]

Atyy gave a for me useful reference about a nearly equivalent system with accelerating rockets, http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/RindlerHorizon.html#FREEFALL

The interesting phrase for me is:
"Eve could claim that Adam never reaches the horizon as far as she's concerned. However, not only is it clear that Adam really does cross the horizon".

I agree with that, but it appears for different reasons than some others.

In fact, according to 1916 GR, Eve's point of view is equally valid as that of Adam; according to that, acceleration and gravitation are just as "relative" as velocity, and their coordinate systems are valid GR systems.
However, the interpretation of what "really" happens is very different, even qualitatively; and in modern GR many people reject "induced gravitation" and agree that we can discern the difference between gravitation and acceleration.

We thus distinguish in that example that Eve's acceleration is real, and that her gravitational field is only apparent because the effect is not caused by the nearby presence of matter. For that reason I think that we should prefer Adam's interpretation. Similarly, in case of a real gravitational field that we ascribe to the presence of matter, it is Eve's interpretation that we should prefer.

Now, it is still not clear to me if O-S used what Einstein called a Gaussian coordinate system, or if they fitted two such systems together that correspond to the same interpretation, or with conflicting interpretations. So, I want to make sure that their model is self-consistent. I guess that it is; the only difference between their inner and outer region modelling is the presence of matter - correct?

 

[Dale]

However, the interpretation of what "really" happens is very different, even qualitatively; and in modern GR many people reject "induced gravitation" and agree that we can discern the difference between gravitation and acceleration.

We thus distinguish in that example that Eve's acceleration is real, and that her gravitational field is only apparent because the effect is not caused by the nearby presence of matter. For that reason I think that we should prefer Adam's interpretation. Similarly, in case of a real gravitational field that we ascribe to the presence of matter, it is Eve's interpretation that we should prefer.

The problem with this whole statement is not the different perspectives or interpretations, but the ambiguous term "real" which makes both paragraphs rather handwavy. What definition of "real" are you using, and can you demonstrate that Eve's acceleration indeed qualifies as "real" under that definition?

 

[PeterDonis]

In fact, according to 1916 GR, Eve's point of view is equally valid as that of Adam; according to that, acceleration and gravitation are just as "relative" as velocity, and their coordinate systems are valid GR systems.

For the region of spacetime that both coordinate systems cover, yes, this is true. However, if Adam's coordinate system covers a portion of spacetime that Eve's does not (in the scenario on Egan's web page, Adam's coordinates cover the entire spacetime, but Eve's only cover the wedge to the right of the horizon), then Eve's "point of view" will be limited in a way that Adam's is not.

in modern GR many people reject "induced gravitation" and agree that we can discern the difference between gravitation and acceleration.

References, please? In "modern GR", people recognize that the word "gravitation" can refer to multiple things. If it refers to "acceleration due to gravity", then "modern GR" agrees with "1916 GR" that "gravitation" can be turned into "acceleration" by changing coordinates, so both are "relative" in that sense. I don't know of anyone in "modern GR" who claims we can distinguish between "gravitation" in this particular sense and acceleration.

But if "gravitation" refers to "tidal gravity", then "gravitation" in that sense is *not* relative; it is spacetime curvature, which is a coordinate-independent thing. "Modern GR" *does* claim that "gravitation" in the sense of spacetime curvature can't be removed by choosing coordinates. However, it can be made negligible in a sufficiently small patch of spacetime by choosing coordinates appropriately. "1916 GR" said the same thing, so again "modern GR" is no different than "1916 GR" in this sense.

We thus distinguish in that example that Eve's acceleration is real, and that her gravitational field is only apparent because the effect is not caused by the nearby presence of matter. For that reason I think that we should prefer Adam's interpretation. Similarly, in case of a real gravitational field that we ascribe to the presence of matter, it is Eve's interpretation that we should prefer.

But Eve's interpretation doesn't cover all of the spacetime. That's obvious in the scenario given on Egan's web page, but you still don't appear to realize that exactly the *same* reasoning applies to the case of a black hole.

In the Adam-Eve scenario, Eve can easily compute that the proper time along Adam's worldline from when he steps off the ship to when he reaches the Rindler horizon is finite. She can also easily compute that there is nothing physically present at the Rindler horizon that would cause Adam's worldline to end there. Finally, she can compute that, once Adam reaches the Rindler horizon, he can't get back out into the region of spacetime "above" it, because to do so he would have to move faster than light. So Eve can conclude that there must be a region of spacetime beyond the Rindler horizon, where Adam's worldline goes, even though she can't see it (light rays from it can never reach her).

If Eve were hovering above a black hole, and Adam stepped off the ship and fell in, *exactly* the same reasoning would apply. You can even draw a spacetime diagram of that scenario that looks almost identical to Egan's diagram; I did it in a recent post in the other thread that we have running on this topic. So just as in the case of Egan's scenario, in the black hole scenario we can see that Adam's coordinates cover a region of spacetime that Eve's don't. *That* is the reason that Adam's interpretation is "preferred", to the extent that it is--in the region of spacetime that both Adam's and Eve's coordinates cover, neither one is "preferred"; they can both be used to describe events and calculate physical quantities, and both will give the same answers. But Eve's is limited in coverage in a way that Adam's is not.

Now, it is still not clear to me if O-S used what Einstein called a Gaussian coordinate system

Every coordinate system that I've ever seen in any relativity paper is a Gaussian coordinate system by Einstein's definition; it's a very general definition. All of the coordinates used in the O-S paper are certainly Gaussian.

So, I want to make sure that their model is self-consistent. I guess that it is; the only difference between their inner and outer region modelling is the presence of matter - correct?

Yes. The key constraint that needs to be enforced to make the model consistent is basically that the metric and its derivatives match at the boundary; the technical term is "junction conditions". (I'm not sure that specific term appears in the paper; I think it was coined later on. But I think they talk about matching at the boundary.)