Oppenheimer-Snyder model of star collapse

PeterDonis

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.

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

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, http://www.physicsforums.com/showthread.php?t=652839

PAllen

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there must be some solution (well, except for Penrose, who believes information is truly lost in a BH, and QM must be superseded).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

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

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:
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

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. 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.

Attached Thumbnails

 

harrylin

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

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 realise 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

Ok, good.

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.

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.
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. )

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

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.

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

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.

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.

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

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.
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?
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

This looks cool, thanks for the link!

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).

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

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:
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

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.

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.