Notions of simultaneity in strongly curved spacetime

[PAllen]

Going back to your OP, wouldn't the analysis of a white hole lead to the opposite conclusion?

Did you read a few sentences earlier:

"Now consider these for the Oppenheimer-Snyder spacetime (asymptotically flat; collapsing space time region; interior and exterior SC regions eventually). I choose this for qualitative plausibility and to avoid the white hole region (the notions certainly apply to full SC geometry)."

There is no white hole in this scenario. As noted, I could apply the definitions to a WH case, but then the results would be different. I wasn't interested in doing so, because I don't consider a WH plausible. GR itself requires white holes originate in the past without cause; while black holes are predicted (classically) to form from plausible starting conditions.

In case you were asking me to apply the concepts to the white hole case, I am not interested. There is more than enough confusion about collapse to BH; I don't want distraction from the white hole case.

 

[rjbeery]

No calculation shows that; even the classical calculations, that show an EH forming, don't show it forming prior to the Big Bang.

If an EH is shown to form at all it would be shown to occur after the BB by definition. I'm talking about starting with a black hole of mass M+A, where A is the mass of an object which has fallen past the EH, and calculating "when" from the distant observer's perspective that object crossed the EH. If the object takes a local eternity to cross the EH falling in then it takes a local eternity to cross the EH coming out.

 

[PAllen]

If an EH is shown to form at all it would be shown to occur after the BB by definition. I'm talking about starting with a black hole of mass M+A, where A is the mass of an object which has fallen past the EH, and calculating "when" from the distant observer's perspective that object crossed the EH. If the object takes a local eternity to cross the EH falling in then it takes a local eternity to cross the EH coming out.

I understand what you are asking except the part about coming out. Nothing comes out unless you are talking about quantum evaporation.

As for the rest:

- Classically, an infalling body merges with the pre-existing BH and expands its actual event horizon in finite (short) time locally for the infalling body; and reaches the singularity of the pre-existing BH in finite local time. The infaller does have an objective basis to correlate local and distant events, because they can keep receiving signals from outside until the moment they reach the singularity. They can see a specific distant clock time (in theory) as of the moment they reach the singularity.

- From a distant observers point of view, I keep repeating the question cannot be answered as worded; even similar questions in SR cannot be answered. You can answer when will a distant observe see the above happening? Then there is an answer: never. Because this physical answer is never, it follows that there is no objective answer to when A crossed the horizon for the distant observer. They can make an infinite number equally defensible answers, one of which is never.

 

[rjbeery]

GR itself requires white holes originate in the past without cause; while black holes are predicted (classically) to form from plausible starting conditions.

From a distant observers point of view, I keep repeating the question cannot be answered as worded; even similar questions in SR cannot be answered. You can answer when will a distant observe see the above happening? Then there is an answer: never. Because this physical answer is never, it follows that there is no objective answer to when A crossed the horizon for the distant observer. They can make an infinite number equally defensible answers, one of which is never.

Logic shows this is a contradiction. Take the BH mentioned above of mass M+A, where A is the mass of an object *having already passed* the EH from a distant observer's perspective. Note the time = T_0Now turn the clock back T_{-1}, T_{-2}, T_{-3}...until the object of mass A is no longer beyond the EH at T_{-x} (and I don't care if we're using Schwarzschild metric for the observer's calculations, for example, or we simply move backwards in time until he *sees* the object, as you said)

What are we left with? At T_{-x} we have an object outside of the BH, and at T_0 that object has crossed over the EH in finite time according to the distant observer. The conclusion is that observing the object crossing back out of the BH as we turn the clock backwards will never happen from the distant observer's perspective, certainly not within the finite age of the Universe.

 

[PAllen]

Logic shows this is a contradiction. Take the BH mentioned above of mass M+A, where A is the mass of an object *having already passed* the EH from a distant observer's perspective.

This is already a statement whose meaning is rejected by relativity. All else is irrelevant. Because the distant observer can never see it happening, they can never say the know it happened. Maybe it did, maybe it didn't.

Maybe I misunderstand your intent. It is absolutely possible for a distant observer to assign remote times in a consistent way such that they consider the object to have crossed the horizon in finite time. They can also consistently assign remote times so that never happens. It will never be possible to verify one assignment over another precisely because event horizon crossing will never be seen.

 

[rjbeery]

This is already a statement whose meaning is rejected by relativity. All else is irrelevant. Because the distant observer can never see it happening, they can never say the know it happened. Maybe it did, maybe it didn't.

Exactly. This is equivalent to saying "maybe the black hole exists, maybe it does not." Point being black holes do not even necessarily exist in Relativity. I'd be curious to see a time-reversed Kruskal analysis of objects crossing (back out of) the EH.

 

[PAllen]

Exactly. This is equivalent to saying "maybe the black hole exists, maybe it does not." Point being black holes do not even necessarily exist in Relativity. I'd be curious to see a time-reversed Kruskal analysis of objects crossing (back out of) the EH.

Yes, you can say a distant observer can never know an actual BH (rather than an 'almost BH') has formed. They can only compute that if GR is true, then for the matter at the 'almost BH' they do see, there is no alternative to a BH forming, in finite time for that matter. If GR is false in the near BH domain (as is likely to some degree), then there are alternatives.

However, because GR is unambiguous that a BH forms in finite time for the infalling matter, and new matter falls in in finite time for the infalling matter, it must be said that GR predicts black holes. That this prediction is not verifiable from the outside, doesn't make it not a prediction of GR. For example, QCD says quarks will never be detected in isolation. They are still a prediction of QCD.

The real 'way out' is that quantum gravity changes the classical GR predictions.

 

[PeterDonis]

I'd be curious to see a time-reversed Kruskal analysis of objects crossing (back out of) the EH.

If you are referring to a white hole, it's already present in the Kruskal diagram. The white hole is region IV on the Kruskal chart, as shown for example on the Wikipedia page:

http://en.wikipedia.org/wiki/Kruskal–Szekeres_coordinates

If you consider a timelike free-fall trajectory that starts at the past singularity (the hyperbola at the bottom of region IV), emerges from the white hole (i.e., crosses from region IV into region I), rises to some finite radius r at Kruskal time V = 0, then falls back into the black hole (crosses from region I into region II), and finally ends up at the future singularity (the hyperbola at the top of region II): such an object's trajectory is time-symmetric; the part before V = 0 is the exact time reverse of the part after V = 0.

If, however, you are referring to a spacetime where a BH forms from the collapse of a massive object, then evaporates away, I haven't seen a Kruskal-type diagram of that case, but I have seen Penrose diagrams of the most obvious way to model it (which not everyone agrees is the correct model, but it's a good starting point for discussion). See, for example, the diagram here:

http://en.wikipedia.org/wiki/Black_hole_information_paradox

Compare with the Penrose diagrams on this page:

http://www.pitt.edu/~jdnorton/teaching/HPS_041/chapters/black_holes_picture/index.html

The Penrose diagram corresponding to the Kruskal diagram I linked to above is in the section "Conformal Diagram of a Fully Extended, Schwarzschild Black Hole". The Penrose diagram corresponding to the classical GR model of a collapsing massive object (like a star) is in the section "A Conformal Diagram of a Black Hole formed from Collapsing Matter".

Note that in *none* of the diagrams, other than the Kruskal diagram and the Penrose diagram corresponding to it, does the white hole appear. In the evaporation diagram, Hawking radiation escapes as the hole evaporates, but there is still a black hole interior region and a singularity, and anything that gets inside the horizon is still doomed to be destroyed in the singularity, according to this model. The big open question is, if this model is *not* correct (which most physicists in the field now seem to think it is not, since it leads to the loss of quantum information), what replaces it? There are a lot of suggestions, but no good answer yet.

 

[rjbeery]

Yes, you can say a distant observer can never know an actual BH (rather than an 'almost BH') has formed. They can only compute that if GR is true, then for the matter at the 'almost BH' they do see, there is no alternative to a BH forming, in finite time for that matter. If GR is false in the near BH domain (as is likely to some degree), then there are alternatives.

However, because GR is unambiguous that a BH forms in finite time for the infalling matter, and new matter falls in in finite time for the infalling matter, it must be said that GR predicts black holes. That this prediction is not verifiable from the outside, doesn't make it not a prediction of GR. For example, QCD says quarks will never be detected in isolation. They are still a prediction of QCD.

The real 'way out' is that quantum gravity changes the classical GR predictions.

PAllen, I appreciate your maturity in acknowledging other (albeit subjective) points of view. The usual response is an emotional defense of BHs as a matter of fact...

 

[harrylin]

This discussion is growing a bit over my head, especially concerning time (my time, not Schwartzschild t, although it's almost the same ); I intended to quickly move on from a simple illustration to show that there is an issue, to a concrete physics discussion involving clocks and light rays. However it is interesting for me and perhaps also for invisible onlookers. I'll try to group things piece-wise and only discuss the essentials.

[..] there are no contridictions between maps. Classically, you just have different coverage by different maps.

Different coverage means to me in the context of relativity, from the same reference system - from the same "perspective". However, what I was referring to was the mapping between different reference systems, of a time τ to a time t>∞, as follows:

[..] [O-S] talk about t>∞. That doesn't make sense to me, which is what I had in mind with my remark that it looks like they didn't fully think it through. And that's not so strange, as their results were new.

[..] I think [O-S] didn't fully explore the question of what the region of spacetime with "t > infinity" would look like. But just contemplating the existence of such a region is not a contradiction. Check my latest post in the simultaneity thread. [..]

As I clarified earlier, a "region of spacetime" is for me merely a mathematical tool for calculations of, as Einstein put it, "clocks and rods". t>∞ has as physical meaning a possible clock that indicates t>∞. That makes as little sense to me as v>∞.
On this point the discussion dropped outside of the speciality of GR into the realm of general philosophy of physics. Thus you would need to make a strong case with the following if your intention is to convince me (but I hope that that is not what you are trying to do):

The "time" you refer to is time according to a distant observer, i.e., an observer who is spatially separated from the infalling matter. Did you also calculate the proper time experienced by an observer who is *not* spatially separated from the infalling matter? That is, an observer who is falling in along with it? O-S show that that proper time is finite. Have you really stopped to think about what that means?
[..]
"it is impossible for a singularity to form in a finite time", are proofs that a singularity cannot form *in the region of spacetime where the distant observer's time coordinate is finite*. [..] In fact, it is easy to show that there *must* be another region of spacetime, below the horizon, that is not covered by the distant observer's time coordinate. That must be true *because* the proper time for an infalling observer to reach the horizon is finite. Have you considered this at all?

First of all, I can't find anything that explains how you map a time τ to a time t>∞, as O-S suggest, and make physical sense of it. You must have considered this, and you suggested that you did, but I do not see that you clarified that essential point. It is a simultaneity that looks completely impossible to me.

Secondly, of course I considered the fact that O-S map t->∞ to τ->a. I cannot understand how you can think that I didn't reflect on the only part on which everyone agrees. I did not make a plot of it, but I did not see an issue with that. In the O-S model, if a completely formed black hole exists (which, if I correctly read Schwartzschild, he deemed impossible!), an infalling observer will not reach the inside region and as measured in t, his clock time τ will nearly "freeze" to slowly never reach a certain value τ₀. As I picture it (for I have not seen a description of it), for the infalling observer the thus predicted effect will be very dramatic, with starlight in front of him reaching nearly infinite intensity as the universe speeds up around him and his observations come to a halt when this universe ends. In fact, it was a discussion based on a blog including that aspect with more than 100 posts that was the first thing that I read about this topic (http://blogs.discovermagazine.com/badastronomy/2007/06/19/news-do-black-holes-really-exist/)

Have you not been reading all the posts I and others have made explaining "how other people interpret this"? [..]

Sorry: I did not see any explanation for the inside region that made any sense to me, or that explains to me how it cannot contradict Schwarzschild's model. Perhaps some others who asked similar questions were convinced, but I did not see that happen (and of course, there is no use to try to convince anyone about which model is "right"; this is just a discussion of models). Perhaps there is another post that I overlooked?

 

[PAllen]

Different coverage means to me in the context of relativity, from the same reference system - from the same "perspective". However, what I was referring to was the mapping between different reference systems, of a time τ to a time t>∞, as follows:

I'm not quite sure what you mean by reference system. In GR there is no such thing a global frame of reference - there are only local frames of reference. As a result, you cannot discuss global issues in frames of reference in GR. Instead, for global issues you either use coordinate systems or coordinate free geometric methods (e.g. Plane geometry without coordinates).

Two coordinate systems are just two different sets of labels attached to an overall space time. It can happen that they don't cover all the same region of spacetime. However, they are just relabelings of the same geometry for coverage in common. You obviously can't use a particular coordinate system for a part of the geometry it doesn't cover.


As for coodinate infinities, let me try an example. Start with a flat plane with Euclidean metric (distance given by ds^2= dx^2 + dy^2). Now define coordinates u and v as:

u=1/x , v = 1/y ; the metric (distance formula) expressed in these will be different, such that all lengths, angles and areas computed in cartesian coordinates are the same with computed with u and v - using the transformed metric.

Note that u and v become infinite as you approach the x or y axis. However, no computation or measurement is different from cartesian coordinates (when you use the transformed metric). But you can't directly do a computation involving any point on or line crossing the x or y-axis in these coordinates. You can compute the length of a line approaching the x-axis and get a finite value limit value; you can continue it on the other side and get a finite value for its length, limiting from the other side.

The ininite value of u and v has no geometric meaning, because coordinates are interpreted through the metric.

The behavior of the t coordinate in SC coordinates is just like this. It has meaning only through the metric for computation of 'proper time' which is what a clock measures. If you compute proper time for an infalling clock, you get a finite value for it to reach the EH. If you continue it over the EH using, e.g. interior SC coordinates, you get an additional finite proper time from the EH to the singularity.

As I clarified earlier, a "region of spacetime" is for me merely a mathematical tool for calculations of, as Einstein put it, "clocks and rods". t>∞ has as physical meaning a possible clock that indicates t>∞. That makes as little sense to me as v>∞. On this point the discussion dropped outside of the speciality of GR into the realm of general philosophy of physics. Thus you would need to make a strong case with the following if your intention is to convince me:

No, t means nothing. It is not a reading on any clock. To get a reading on a clock, you have to specify the clock (world line) and compute proper time (clock time) along it.

You will find, that for a static clock (stationary with respect to the spherical symmetry), very far from the center, SC coordinate time matches clock time for that clock. It doesn't match clock time for other clocks. The closer you get the the EH, the less this t coordinate has anything to do with what clocks measure. Just like with my u coordinate above, u becoming infinite says nothing about what a ruler will measure.

First of all, I can't find anything that explains how you map a time τ to a time t>∞, as O-S suggest, and make physical sense of it. You must have considered this, and you suggested that you did, but I do not see that you clarified that essential point. It is a simultaneity that looks completely impossible to me.

Hopefully, my explanations above have helped a little. As for simultaneity, let's see if I can exploit my u,v example more. In a plane, I can propose, as an analog of simultaneity: both on a line parallel to the cartesian x axis. Then the points (x,y)=(-1,1) and (x,y)=(1,1) are 'simultaneous'. However, in u,v coordinates, the horizontal line connecting them goes through v=-∞ and v=∞. But I should still be able to call them simultaneous.

Secondly, of course I considered the fact that O-S map t->∞ to τ->a. I cannot understand how you can think that I didn't reflect on the only part on which everyone agrees. I did not make a plot of it, but I did not see an issue with that. In the O-S model, if a completely formed black hole exists (which, if I correctly read Schwartzschild, he deemed impossible!), an infalling observer will not reach the inside region and as measured in t, his clock time τ will nearly "freeze" to slowly never reach a certain value τ₀. As I picture it (for I have not seen a description of it), for the infalling observer the thus predicted effect will be very dramatic, with starlight in front of him reaching nearly infinite intensity as the universe speeds up around him and his observations come to a halt when this universe ends. As a matter of fact, it was a similar discussion on the other blog that was the first thing that I read about this.

This is not what SC or O-S geometry predicts. They predict that an infaller will see the external universe going at a relatively normal rate, with no extreme red or blueshift. There will be optical distortions, analogous to Einstein rings. The infaller sees perfectly SR physics locally, until they hit the singularity. If you declare their world line to end at some arbitrary point, (e.g. the EH), there is no possible local physics explanation for it.

Sorry: I did not see any explanation for the inside region that made any sense to me, or that explains to me how it cannot contradict Schwarzschild's model. Perhaps some others who asked similar questions were convinced, but I did not see that happen (and of course, nobody needs to convince anyone; this is just a discussion of models). Perhaps there is another post that I overlooked?

Well, we have tried and tried.

 

[PeterDonis]

Different coverage means to me in the context of relativity, from the same reference system - from the same "perspective".

But some perspectives may simply not be able to cover all of spacetime; they may be limited in scope. Do you admit this possibility?

As I clarified earlier, a "region of spacetime" is for me merely a mathematical tool for calculations of, as Einstein put it, "clocks and rods".

That's not quite how I'm using the term. A "spacetime" is a geometric object, like the surface of the Earth. A "region of spacetime" is a portion of that geometric object, like the western hemisphere on the Earth. It's not a "mathematical tool"; it's a part of a mathematical model, true, but I'm trying to convey the fact that the mathematical model is of something "real" and physical.

t>∞ has as physical meaning a possible clock that indicates t>∞.

No, it does *not*. Either you haven't been reading carefully or I (and PAllen) haven't been explicit enough. We are *not* saying that you can assign a "t" coordinate greater than infinity to events behind the horizon. We are saying you can't assign a "t" coordinate *at all* to events behind the horizon. (Strictly speaking, you can't assign one and have it correspond to "time" for the distant observer.) It's like saying you can't assign a real square root to a negative number; it simply can't be done.

This goes back to what I said above; you appear to be assuming that it must somehow be possible to assign a well-defined "t" value to every single event, everywhere in spacetime. You can't. That's just the way it is. If you want to describe events at or inside the horizon, you simply can't use the "t" that the distant observer uses. It just can't be done. If you can't admit or can't grok this possibility, then probably further discussion is useless unless/until you can. It's not easy, I agree; it took me quite some time to wrap my mind around it. But it's critical to understanding the standard classical GR model of black holes.

On this point the discussion dropped outside of the speciality of GR into the realm of general philosophy of physics.

That wasn't my intent, and I don't think it was the intent of PAllen. We are not trying to make philosophical points; we are trying to help you see the possibility of a kind of mathematical model that you hadn't seen before, and therefore of a kind of physical spacetime that you hadn't considered before. That model may or may not represent the actual spacetime of a black hole, because of the quantum issues that have been brought up many times in this and other threads. But it quite certainly does represent a *consistent* classical model of a black hole. That's what we're trying to help you see: that the model is consistent and represents something physically possible within the limits of classical theory.

First of all, I can't find anything that explains how you map a time τ to a time t>∞, as O-S suggest

You don't. See above. What you do is recognize that at the instant when an infalling observer crosses the horizon, his \tau is *finite*, not infinite; therefore we can construct a *different* coordinate chart that maps *finite* values of some "time" coordinate T to the finite values of his \tau that occur on his worldline after he has crossed the horizon, i.e., after the value \tau_0 that his clock reads at the instant he reaches the horizon. The simplest such chart is the Painleve chart, where the coordinate time T is simply equal to \tau. But there are others.

Those events inside the horizon, the ones with \tau > \tau_0, do *not* have well-defined "t" values at all, if "t" is the time coordinate of a distant observer. They simply can't be mapped in the distant observer's chart.

You must have considered this, and you suggested that you did, but I do not see that you clarified that essential point.

I've tried to clarify it more above; but I see from your next comment that one more thing needs to be clarified:

It is a simultaneity that looks completely impossible to me.

That's because it is. There is *no* simultaneity that both (1) assigns "t" coordinates to events outside the horizon in such a way that t goes to infinity as the horizon is approached, *and* (2) assigns well-defined "t" coordinates from the same set of surfaces of simultaneity to events inside the horizon. If you are willing to take another look at the Kruskal chart, I can try to explain why (though I think I already tried to in a previous post in this thread or one of the others that's running). But first I need to know if you can grok the possibility of such a thing at all; that seems to me to be a major stumbling block at this point.

Secondly, of course I considered the fact that O-S map t->∞ to τ->a. I cannot understand how you can think that I didn't reflect on the only part on which everyone agrees.

If you agree with this, that's great. I wasn't sure, because if you realize this, it seems to me like a simple step to the reasoning I gave above (what you call a here, I called \tau_0 there). But of course that's just the way it seems to me; obviously it doesn't seem that way to you. But I think this is where attention needs to be focused.

In the O-S model, if a completely formed black hole exists (which, if I correctly read Schwartzschild, he deemed impossible!),

Schwarzschild may indeed have thought that. He was using still another coordinate chart, one in which his radial coordinate "R" went to *zero* at the horizon. But that would take us way too far afield.

an infalling observer will not reach the inside region and as measured in t, his clock time τ will nearly "freeze" to slowly never reach a certain value τ₀.

That's not really what O-S said. A finite value of \tau means a finite amount of time elapsed on the infalling observer's clock; there's no room there for his clock time to "slowly never reach a certain value". To the observer, if the infall time is 1 day (which was the order of magnitude of the value O-S calculated for the collapse of a sun-like star), he will experience 1 day, just like you will experience 1 day between now and this time tomorrow, and to him there will be nothing abnormal happening.

Sorry: I did not see any explanation for the inside region that made any sense to me, or that explains to me how it cannot contradict Schwarzschild's model. Perhaps some others who asked similar questions were convinced, but I did not see that happen (and of course, there is no use to try to convince anyone about which model is "right"; this is just a discussion of models). Perhaps there is another post that I overlooked?

I'm pretty sure you have read all the relevant posts; evidently they didn't make things click for you. I've given it another try above.

 

[PAllen]

No, it does *not*. Either you haven't been reading carefully or I (and PAllen) haven't been explicit enough. We are *not* saying that you can assign a "t" coordinate greater than infinity to events behind the horizon. We are saying you can't assign a "t" coordinate *at all* to events behind the horizon. (Strictly speaking, you can't assign one and have it correspond to "time" for the distant observer.) It's like saying you can't assign a real square root to a negative number; it simply can't be done.

This goes back to what I said above; you appear to be assuming that it must somehow be possible to assign a well-defined "t" value to every single event, everywhere in spacetime. You can't. That's just the way it is. If you want to describe events at or inside the horizon, you simply can't use the "t" that the distant observer uses. It just can't be done. If you can't admit or can't grok this possibility, then probably further discussion is useless unless/until you can. It's not easy, I agree; it took me quite some time to wrap my mind around it. But it's critical to understanding the standard classical GR model of black holes.

Here I would like to express a slightly different interpretation. Given a specific rule for relating t for one observer to other events, you may not be able to assign a t coordinate at all to all events. Specifically, I don't particularly like this statement: "Strictly speaking, you can't assign one and have it correspond to "time" for the distant observer." I don't agree this statement has well defined meaning. "Correspond" is just another word for simultaneity convention. If you insist simultaneity requires two way communication, this is true. However, I have proposed several simultaneity rules based on the one way causal connection from exterior to interior events, that, IMO assign a time to interior events corresponding to time for the distant observer. In effect, they simply delegate the correspondence between distant and interior events to the interior observer, who 'sees' the causal relation. This gets to the thrust of this thread as I conceived it:

If my wife gives birth to Judy and Jill, and Jill stays nearby and Judy goes to Africa, and I never hear from Judy again (unless I think Judy died), I have the expectation that there is simultaneity between events for Judy and for Jill. Their mutual causal connection to me gives me this expectation. Even more so if I believe Judy is getting my birthday cards (damn that she doesn't respond).

This concept can be formalized using the one of the procedures I outlined to say: I consider (though I can't verify it) that the singularity of that collapse formed at 3 pm today for me.

It almost seems you are saying there is a physically preferred chart for the distant observer. I don't accept this. I only accept that locally, there clear preference for Fermi-Normal coordinates; but globally? None. And the specific simultaneity I proposed a few times corresponds exactly, locally, to Fermi normal coordinates for an asymptotic observer - it diverges from this further away.

 

[PeterDonis]

Here I would like to express a slightly different interpretation. Given a specific rule for relating t for one observer to other events, you may not be able to assign a t coordinate at all to all events. Specifically, I don't particularly like this statement: "Strictly speaking, you can't assign one and have it correspond to "time" for the distant observer." I don't agree this statement has well defined meaning.

You're correct, I should have specified that by "time for the distant observer" I meant the "natural" time coordinate he would choose, i.e., Schwarzschild coordinate time. I meant that time coordinate specifically because that's the one that seems to be causing all the trouble. I fully agree that other choices of time coordinate are possible that match the distant observer's proper time (at least to a good enough approximation) and also assign finite time values to events on and inside the horizon. Painleve time itself is one example; as r goes to infinity, Painleve time and Schwarzschild coordinate time get closer and closer to each other.

It almost seems you are saying there is a physically preferred chart for the distant observer.

There is in a weak sense: Schwarzschild coordinate time is the only time coordinate in the exterior region with both of the following properties:

(1) The integral curves of the time coordinate are also integral curves of the timelike Killing vector field;

(2) The surfaces of constant time are orthogonal to these integral curves.

Painleve time has property #1, but not #2. Kruskal "time" has neither.

I agree this is a weak sense of "preferred", but it's important to note that it is these two properties together that make Schwarzschild coordinate time seem so "natural"; it *seems* like the "natural" extension of Fermi Normal coordinates along the distant observer's worldline, because it *seems* like properties #1 and #2 are the "right" ones for a "natural" coordinate chart to have. Only when you start looking close to the horizon do you start running into problems with this "natural" extension.

And the specific simultaneity I proposed a few times corresponds exactly, locally, to Fermi normal coordinates for an asymptotic observer

I'm not sure I agree with "exactly" here; I think the only global time coordinate that can correspond "exactly" to Fermi normal coordinates (by which I mean surfaces of constant global time are also *exactly* the same as surfaces of constant local Fermi normal coordinate time) is Schwarzschild coordinate time. This is because of property #2 above, which must be satisfied by any global time coordinate whose simultaneity corresponds exactly, locally, to the simultaneity of Fermi normal coordinates on the distant observer's worldline. I don't think the simultaneity you proposed satisfies that property, for the same reasons that Painleve coordinate time doesn't: any surface of simultaneity that crosses the horizon can't be orthogonal to integral curves of the timelike Killing vector field. The simultaneity you proposed is *approximately* the same far away from the hole, but the correspondence is not exact for any finite value of r.

 

[PAllen]

I agree this is a weak sense of "preferred", but it's important to note that it is these two properties together that make Schwarzschild coordinate time seem so "natural"; it *seems* like the "natural" extension of Fermi Normal coordinates along the distant observer's worldline, because it *seems* like properties #1 and #2 are the "right" ones for a "natural" coordinate chart to have. Only when you start looking close to the horizon do you start running into problems with this "natural" extension.

But I only see Fermi-Normal as natural locally in GR. Even in SR, I see it as natural globally only for inertial observers (where it becomes Minkowski coordinates). For a non-inertial observer in SR, at a distance, I see many other simultaneity conventions that are physically based that avoid numerous seeming absurdities of Fermi-Normal carried too far. For example, Radar simultaneity has the feature of approaching Fermi-Normal locally, but has far more natural global properties for a highly non-inertial observer. While it is not universal, the idea that Fermi-Normal coordinates are only natural locally is a common one in GR.

I'm not sure I agree with "exactly" here; I think the only global time coordinate that can correspond "exactly" to Fermi normal coordinates (by which I mean surfaces of constant global time are also *exactly* the same as surfaces of constant local Fermi normal coordinate time) is Schwarzschild coordinate time. This is because of property #2 above, which must be satisfied by any global time coordinate whose simultaneity corresponds exactly, locally, to the simultaneity of Fermi normal coordinates on the distant observer's worldline. I don't think the simultaneity you proposed satisfies that property, for the same reasons that Painleve coordinate time doesn't: any surface of simultaneity that crosses the horizon can't be orthogonal to integral curves of the timelike Killing vector field. The simultaneity you proposed is *approximately* the same far away from the hole, but the correspondence is not exact for any finite value of r.

A more accurate description, I agree, would be exactly matches in the limit for an asymptotic observer. Concretely, there exists a sufficiently distant observer where my proposed simultaneity matches Fermi-Normal to one part in 10^50 for one light year (for example). Formally, the relation is more like Radar locally converging to Fermi-Normal for arbitrary non-inertial observers in SR.

 

[zonde]

This is false. You detect readily in SC coorinates that there is a hole in space time. You integrate proper time along an infall trajectory and find that proper time stops at a finite value (unlike for various other world lines). You ask, what stops the clock?

No, I ask what SC coordinates (+ SC type interior coordinates) this event has got. And it doesn't got any because infinity is not a coordinate. And event without coordinates does not exist (if coordinate system covers the whole space-time).

There is no local physics to stop the clock - tidal gravity may be very small; curvature tensor components are finite. The infinite coordinate time is not a physical quantity in GR. Einstein spoke of rulers and clocks, as Harrylin likes to point out. This clock stops for no conceivable local reason. If you add SC interior coordinates, and use limiting calculations, you smoothly extend this world line to the real singularity (with infinite curvature). All of this is exactly as if you chopped a disk around the pole from a sphere - you would find geodesics ending for no reason.

You mean that SC coordinates has hole because there is no interior coordinates? Well, we add SC type interior coordinates (with simultaneity defined using round-trip of signal at light speed), but this worldline has nothing much to do with these coordinates if it already extends toward infinite future in SC exterior coordinates.

There is no physical observable, anywhere in SR or GR, that depends on simultaneity convention at all. This is part of what Pervect was saying above. Belief that simultaneity convention has physical consequence reflects complete, total, misunderstanding of SR and GR.

Don't know what to make about "physical observables" but surely there are physical quantities that depend on simultaneity convention.
Not sure about GR but I am certain about my understanding of SR.

As for flat spacetime, the Rindler example Dr. Greg has posted beautiful pictures of, is relevant. The belief that there is no hole in SC exterior coordinates is 100% equivalent to the belief that most of the universe doesn't exist because a uniformly accelerating rocket can't see it.

Hmm, I believe Rindler coordinates do not extend to infinity in every direction.

And isn't Rindler coordinates (and horizon) more an analogue of eternal BH rather than collapsing mass (forming BH)?

And besides you have to take into account that rocket can't remain in state of uniform acceleration for indefinite time. And observer on that rocket would not observe rather static picture of other matter in the same state of uniform acceleration.
But as far as we know we can remain in a state of gravitational acceleration for indefinite time and we observe a lot of matter in the same state.

 

[PeterDonis]

But I only see Fermi-Normal as natural locally in GR. Even in SR, I see it as natural globally only for inertial observers (where it becomes Minkowski coordinates). For a non-inertial observer in SR, at a distance, I see many other simultaneity conventions that are physically based that avoid numerous seeming absurdities of Fermi-Normal carried too far. For example, Radar simultaneity has the feature of approaching Fermi-Normal locally, but has far more natural global properties for a highly non-inertial observer. While it is not universal, the idea that Fermi-Normal coordinates are only natural locally is a common one in GR.

I understand all this, and I agree with it. I'm talking about what I think is a typical thought process among people who come to post on PF asking questions about black holes; it seems to me that to these people, SC coordinates are "privileged", and I'm speculating as to the reasons why.

A more accurate description, I agree, would be exactly matches in the limit for an asymptotic observer.

Ok, good, we're in agreement.

 

[PAllen]

No, I ask what SC coordinates (+ SC type interior coordinates) this event has got. And it doesn't got any because infinity is not a coordinate. And event without coordinates does not exist (if coordinate system covers the whole space-time).

The path of an outer edge infall particle has finite proper time integrated to the SC radius. If you declare it stops there, you have a hole in spacetime. You have a geodesic ending with finite 'interval', where curvature is finite.

If you imagine the surface of such particles infalling, and you don't allow it to proceed to Sc radius, you have, geometrically, a hole: proper time on this surface is finite, area is finite, but if you stop it from continuing, it is geometrically a hole. Geometry is defined by invariants, not coordinate quantities. Consider the example I gave to harrylin several posts back of a horizontal geodesic in the plane in (u,v) coordinates. u coordinate goes to infinity on both sides of coordinate singularity, but it is still nothing but a geodesic in the flat plane.

Don't know what to make about "physical observables" but surely there are physical quantities that depend on simultaneity convention.
Not sure about GR but I am certain about my understanding of SR.

Nope. Einstein was very clear that simultaneity is purely a convention, not an observable. There is no observation or measurement in SR that changes if you use a different one than the standard one (but you have to change the metric as well; it is no longer eg. diag(+1,-1,-1,-1) if you use a funky convention.

Hmm, I believe Rindler coordinates do not extend to infinity in every direction.

so what? The point is that the trajectory of an object dropped from the rocket has coordinate time approaching infinity as it approaches, say, x=0. Proper time is finite. If you take these as the 'natural' coordinates for a rocket, what do you make of this? If you use two way signals for simultaneity, the event of the dropped object reaching x=0 never becomes simultaneous to an event for the rocket. So, should the rocket conclude the universe ends, or consider using a different simultaneity convention to look at the further history of the dropped object? This is analagous to the choice of using different simultaneity that allows analysis of events smoothly over a horizon.

And isn't Rindler coordinates (and horizon) more an analogue of eternal BH rather than collapsing mass (forming BH)?

Again, so what? You asked for flat space analog of issues under discussion: coordinate infinities and simultaneity conventions.

And besides you have to take into account that rocket can't remain in state of uniform acceleration for indefinite time. And observer on that rocket would not observe rather static picture of other matter in the same state of uniform acceleration.
But as far as we know we can remain in a state of gravitational acceleration for indefinite time and we observe a lot of matter in the same state.

I don't see that this is relevant.

 

[harrylin]

[..] I'm talking about what I think is a typical thought process among people who come to post on PF asking questions about black holes; it seems to me that to these people, SC coordinates are "privileged", and I'm speculating as to the reasons why. [..]

I did not yet see what you speculated, and it will be most useful to tell you my thinking about this without knowing what you thought about the thinking of me and others. Then we can compare it to your speculation.

So here's my thought process: Schwartzschild's solution is the one that I heard about in the literature, and it happens to be the one that I happened to stumble on in the first papers that I read about this topic, dating from 2007 and 1939. It is obviously a valid reference system according to GR, and it is "privileged" in the same sense as inertial frames and centre of mass systems are "privileged": it allows for the most simple mathematics, so simple that one doesn't need to be an expert to understand it. Thus it is a natural choice in a public discussion about predictions of GR. And the way I understand the first paper that I read about this, it's probably all I will ever need to understand this topic.

 

[harrylin]

I may not be able to fully catch up with this thread - I'm reading this at work while I should be doing something very different ... But here's a quick unrelated point:

[..]Schwarzschild may indeed have thought that. He was using still another coordinate chart, one in which his radial coordinate "R" went to *zero* at the horizon. But that would take us way too far afield.

Thanks again - I quickly went through his 1916 papers and got puzzled by them, just on that issue!

 

[harrylin]

[...] It is absolutely possible for a distant observer to assign remote times in a consistent way such that they consider the object to have crossed the horizon in finite time. They can also consistently assign remote times so that never happens. It will never be possible to verify one assignment over another precisely because event horizon crossing will never be seen.

This is an essential clarification for me; it is exactly the kind of disagreement that I tried to illustrate with Earth maps. It doesn't solve the issue, but at least we agree that maps are used that disagree about events - which is a thing that was for me unheard of.

 

[harrylin]

another detail:

[..] for the infalling observer the thus predicted effect will be very dramatic, with starlight in front of him reaching nearly infinite intensity as the universe speeds up around him and his observations come to a halt when this universe ends. [..]

[..] This is not what SC or O-S geometry predicts. They predict that an infaller will see the external universe going at a relatively normal rate, with no extreme red or blueshift. [..]

Either I made a calculation error, or you made an interpretation error, or both.

I estimated the intensity of starlight that hits the eye of the infaller when looking forward - thus what he literally will see; the prediction of an event. In contrast, the "see" in your sentence is probably a prediction of what the infaller will calculate.

 

[PeterDonis]

So here's my thought process: Schwartzschild's solution is the one that I heard about in the literature, and it happens to be the one that I happened to stumble on in the first papers that I read about this topic, dating from 2007 and 1939.

A clarification about terminology: there are several concepts/objects that can be referred to as "Schwarzschild's solution", or "the Schwarzschild metric":

(1) The full, maximally extended vacuum, spherically symmetric solution to the Einstein Field Equation, considered as a geometric object independent of coordinates.

(2) Some portion of #1, such as the exterior region (region I on the Kruskal chart), or that plus the future black hole region (region II on the Kruskal chart), again considered as a geometric object or objects independent of coordinates.

(3) The Schwarzschild coordinate chart--strictly speaking, the *exterior* Schwarzschild coordinate chart--that covers region I only, considered as a particular way of describing the geometric object which is region I of #1 or #2 above.

(4) And for good measure, one can also use a portion of the Schwarzschild exterior chart to cover the vacuum spacetime outside of a massive body like the Earth or the Sun.

My reason for making these distinctions will be evident in a moment.

It is obviously a valid reference system according to GR, and it is "privileged" in the same sense as inertial frames and centre of mass systems are "privileged": it allows for the most simple mathematics, so simple that one doesn't need to be an expert to understand it. Thus it is a natural choice in a public discussion about predictions of GR. And the way I understand the first paper that I read about this, it's probably all I will ever need to understand this topic.

Here you are talking about #3 above; but you have to bear in mind a key point about GR: all of the actual physics in the theory is independent of which coordinate chart you decide to express it in; i.e., all of the physics must be capable of being expressed in terms of invariants, things that don't change when you change coordinate charts. You can use a particular chart as a starting point, so to speak, to get you to the invariants; but if you're not talking about invariants, at the end of the day, you're not talking about the actual physics.

For example, when you say that Schwarzschild coordinate time goes to infinity at the horizon, you are not talking about an invariant; there is no invariant physical quantity that goes to infinity at the horizon. So this is not a statement about any actual physics; it's only a statement about a particular coordinate chart, with no real physical content. In order to make this a statement about the actual physics, you would have to be able to translate it into a statement about invariants, and you can't; all invariant quantities are finite at the horizon.

Also, the converse of the statement I made above is *not* true: there is no requirement that any physical invariant must be expressible in *every* coordinate chart. It is perfectly possible to have physical quantities (such as, for example, the proper time on an infalling observer's worldline at an event inside the horizon) that can't be expressed in some coordinate charts, because those charts don't cover that portion of spacetime.

In view of the above, I would have to disagree that understanding things in terms of the Schwarzschild exterior coordinate chart is "all you will ever need to understand about this topic". By limiting yourself to understanding things in terms of that chart, you are limiting yourself to understanding things outside the horizon only. You can't understand what happens at or inside the horizon if the only tool you have is the Schwarzschild exterior chart, because the relevant physical quantities simply can't be expressed in that chart.

Finally, a few words about the way, if any, in which the Schwarzschild exterior chart is "privileged". This chart is indeed a "valid reference system" in that region, and it has the attractive property, as I said before, of having surfaces of simultaneity that match up exactly with the surfaces of simultaneity of observers who are static--i.e., who "hover" at a constant radius above the horizon. This allows you to simplify your view of the physics, so that the connection with your intuitive understanding of Newtonian gravity is evident (for example, concepts like "potential energy" can be defined). So the math in terms of this chart does "look simple"; but the price you pay for that is, as I just said, only being able to express physical quantities in the region outside the horizon.

It's tempting to think that, since things look so nice and intuitive outside the horizon when expressed in the Schwarzschild exterior chart, it must be sufficient to express *all* of the physics everywhere in the spacetime, and therefore the region outside the horizon, since it's the region where the Schwarzschild exterior chart works, must *be* the entire spacetime. But it isn't; when you actually work through the solution of the Einstein Field Equation, in either case #1 above (mathematically simple, because the entire spacetime is vacuum, but physically unreasonable) or case #2 above (more complex because there is collapsing matter present in a portion of the spacetime, but physically more reasonable, though still highly idealized), you find that any solution that only includes the region outside the horizon is incomplete; the EFE itself tells you that that region cannot be the entire spacetime. That's why you can't get a complete understanding by just using the Schwarzschild exterior chart.

This is an essential clarification for me; it is exactly the kind of disagreement that I tried to illustrate with Earth maps. It doesn't solve the issue, but at least we agree that maps are used that disagree about events - which is a thing that was for me unheard of.

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. But this is only a "disagreement" in the same sense as a Mercator projection "disagrees" with, say, a polar projection of the Earth's surface; the former can't assign coordinates to the North Pole, while the latter can. Yet both charts can assign coordinates to, say, Big Ben in London. So someone in London could choose either chart to map his surroundings, but one choice would allow him to map the North Pole on the same chart, while the other wouldn't.

These all seem like innocuous statements about coordinate charts on geometric manifolds to me. Apparently they seem like huge issues to you; I'm not sure why. Why is it such a big deal that some charts can cover points or regions that others can't?

 

[PAllen]

This is an essential clarification for me; it is exactly the kind of disagreement that I tried to illustrate with Earth maps. It doesn't solve the issue, but at least we agree that maps are used that disagree about events - which is a thing that was for me unheard of.

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. So I assign an independent time range (a different chart) to these events which are still part of the universe (if the laws are true). So one choice is to use two charts to cover the universe, one for events I will see, eventually, one for those I will never see.

The other choice, equally consistent, is to construct a map which covers the whole universe in the one map, assigning coordinates both to events I never see and those I will see.

If you believe the physical laws, there is no disagreement at all about what events happen, or exist; or about what any instrument will measure, including a hypothesized instrument I can't communicate with.

 

[PAllen]

another detail:

Either I made a calculation error, or you made an interpretation error, or both.

I estimated the intensity of starlight that hits the eye of the infaller when looking forward - thus what he literally will see; the prediction of an event. In contrast, the "see" in your sentence is probably a prediction of what the infaller will calculate.

You made a calculation error. The way I described it is a textbook calculation you can look up. I am talking about the directly observed doppler for an infaller who has crossed the horizon and is looking out. I suspect you applied the gravitational redshift for a static observer (which cannot even exist inside). But there is a huge difference between what an infaller (from a good distance) sees near and beyond the horizon, versus what a static, near horizon observer sees (note, the static observer is experiencing acceleration measurable with an accelerometer approaching infinite near the horizon; the free faller is experiencing no proper (measurable) acceleration).

An infaller from a good distance away is passing a near horizon hovering observer at a speed approaching c. If the hovering observer is seeing extreme blue shift (correct), the passing infaller is seeing the hovering clock extremely slow, and almost all the blueshift disappears. They continue to see the outside quite normally (in time rate and frequency; there are optical distortions), until they hit the singularity. [Actually, I believe, if the infaller falls from far enough away, they see moderate redshift at and beyond the horizon.]

 

[harrylin]

You made a calculation error. The way I described it is a textbook calculation you can look up. I am talking about the directly observed doppler for an infaller who has crossed the horizon and is looking out. [..]

Sorry, you even doubly misunderstood my description! At Peter's request I attempted a description for what an infaller experiences who is looking forward towards starlight as he is going towards the horizon, according to the prediction of a Schwartzschild "distant observer".

Of course, I could still be mistaken. I simply multiplied the gravitational time dilation with "SR Doppler", which is itself Doppler times SR time dilation :
- gravitational time dilation f->∞ for r->r₀
- SR Doppler f->∞ for v->c

 

[PAllen]

Sorry, you even doubly misunderstood my description! At Peter's request I attempted a description for what an infaller experiences who is looking forward towards starlight as he is going towards the horizon, according to the prediction of a Schwartzschild "distant observer".

Of course, I could still be mistaken. I simply multiplied the gravitational time dilation with "SR Doppler", which is itself Doppler times SR time dilation :
- gravitational time dilation f->∞ for r->r₀
- SR Doppler f->∞ for v->c

Ok, our track record for failing to describe things in a mutually meaningful way continues. To me, as you are close the BH, most of the stars are behind you. Of course, beaming will shift things toward the front. In the very front, you are staring at the growing BH. For free fall from a great distance, you will see stars behind you moderately redshifted and stars in front of you highly blueshifted. The blue shift in front will never be infinite, because more and more of it is transverse + beaming (because the BH blocks the very front). Once inside the horizon, you can only see the outside in back of you, and it looks moderately red shifted. You will also see Einstein ring in front of you (before you get too close).

 

[harrylin]

I think that it is time to enquire more about observable events, in relation to possible mappings of simultaneity.

[...] All the maps agrees on every computation of an observable, for the events they have in common. [..] Kruskal, GP, Lemaitre, etc. are simply maps that cover more events. Every computed measurement in them agrees with SC for the events included in both. SC assigns infinite coordinate values at a boundary of its coverage, the others do not, but all measurements right up to this edge agree in all coordinates (that infaller's clocks pass finite time reaching the edge; that distant observers never see/detect anything reaching the edge = EH). [..]

Please elaborate that with one or two simple examples.

Voyager 35 is sent to a black hole, which for simplicity we assume to be eternal static and in rest wrt the solar system. And of course, the Voyager is indestructible and always in operation.

1. A time code is emitted from Earth that can be received by Voyager. A Schwartzschild observer calculates that for t->∞, Voyager's clock goes to τ=42. He predicts that no signal can be received by Voyager at τ>=42.

2. Voyager also emits a time code. A Schwartzschild observer predicts that the Earth will never loose its signal.

What will a Kruskal observer predict for those cases, and why?

 

[PeterDonis]

1. A time code is emitted from Earth that can be received by Voyager. A Schwartzschild observer calculates that for t->∞, Voyager's clock goes to τ=42. He predicts that no signal can be received by Voyager at τ>=42.

2. Voyager also emits a time code. A Schwartzschild observer predicts that the Earth will never lose its signal.

This is not what the Schwarzschild observer will predict if he is using standard classical GR. (Note: technically the Schwarzschild observer has to change charts to make some of the predictions I'm going to give; but there is nothing stopping him from doing that. See the end of this post, however, for some further comments on that.) What he will predict is:

0. Voyager crosses the horizon at τ=42. At some later proper time by Voyager's clock, let's say τ=48, Voyager hits the singularity and is destroyed. But no value of t can be assigned to any event on Voyager's worldline with τ>=42, assuming the Schwarzschild observer is using the simultaneity convention for "t" of the exterior Schwarzschild coordinate chart. (As PAllen has pointed out, he could choose other simultaneity conventions; but we'll focus on this one because it's the one you are considering "privileged", so it's good to illustrate its limitations.)

1. Time codes emitted from Earth are received by Voyager just fine at τ=42, and indeed all the way up to τ=48. Voyager is destroyed at τ=48, so obviously it can't receive any signals after that; its worldline stops at τ=48. Signals sent from Earth towards Voyager that don't reach Voyager before τ=48 will hit the singularity instead.

2. No signal sent *from* Voyager at τ>=42 can ever be received by Earth; and signals sent at τ<42, but closer and closer to τ=42, will be received by Earth at later and later times t->∞. In other words, as the Voyager time codes Earth receives get closer and closer to τ=42, the signals carrying those codes will be received at Earth times t->∞. Earth will never receive any Voyager time code with τ>=42.

What will a Kruskal observer predict for those cases, and why?

All of the predictions above are about invariant, physical observables, and so they are the same regardless of which chart you use. If I have time soon I'll try and draw an illustration of what the worldlines of Earth, Voyager, and the light signals going back and forth look like in the Kruskal chart.

Now for the further comments I promised. The Schwarzschild observer could frame predictions similar to the ones you gave, like this:

1. A time code is emitted from Earth that can be received by Voyager. A Schwartzschild observer calculates that for t->∞, Voyager's clock goes to τ=42. He predicts that no signal can be received by Voyager at τ>=42 if Voyager's worldline does not continue past τ=42.

2. Voyager also emits a time code. A Schwartzschild observer predicts that the Earth will never lose its signal if Voyager's worldline does not continue past τ=42.

However, you have not shown that the clause in bold is actually true; and as I've said before, if you look at the solution to the EFE that applies here, it tells you that the clause in bold is *false*. But even without looking at the gory details of the EFE, it should be obvious that Voyager's worldline can't just stop at τ=42. No physical quantity is singular there. Saying that Voyager's worldline suddenly stops there, for no apparent reason, would be like saying your worldline suddenly stops 42 minutes from now, for no apparent reason. Did you get hit by something and destroyed? No. Did you get torn apart by tidal forces increasing without bound? No. Then why does your worldline stop? No reason.

That's not physically reasonable. Voyager's worldline has a finite length up to τ=42, and worldlines don't just stop at a finite length unless some physical quantity, some invariant, is singular there. That doesn't happen at τ=42, so the only physically reasonable conclusion is that Voyager's worldline continues on *past* τ=42.

But then where does it go? It can't rise back up again to a larger radius; to do that it would have to move faster than light. It can't even *stay* at the same radius; to do that it would have to move at the speed of light, and it's an ordinary object and can't do that. The only possibility is for it to continue *inward*, and that means there has to be a region of spacetime inside the horizon, where the portion of Voyager's worldline with τ>=42 goes. Only when Voyager reaches the curvature singularity at r = 0, at τ=48, will its worldline actually stop, because there, a physical quantity, the curvature, *does* become singular.

 

[PAllen]

I think that it is time to enquire more about observable events, in relation to possible mappings of simultaneity.
Please elaborate that with one or two simple examples.

Voyager 35 is sent to a black hole, which for simplicity we assume to be eternal static and in rest wrt the solar system. And of course, the Voyager is indestructible and always in operation.

1. A time code is emitted from Earth that can be received by Voyager. A Schwartzschild observer calculates that for t->∞, Voyager's clock goes to τ=42. He predicts that no signal can be received by Voyager at τ>=42.

Not quite. After the clock reaches 42, you need to switch to interior SC coordinates. Then the trajectory continues for another short finite period of proper time before the voyager reaches the singularity. Voyager continues to get signals from outside until reaching the singularity. All of this can be calculated in pure SC coordinates (note the interior is the same coordinates and metric as the exterior; you just have to use limiting operations to 'step over' the horizon.

2. Voyager also emits a time code. A Schwartzschild observer predicts that the Earth will never loose its signal.

The last signal sent by voyager infinitesimally before crossing the horizon will be received from Earth in the infinite future. No signal voyager sends from past the horizon will reach earth. Not sure how much this agrees with what you wrote - as usual, I am not quite sure how to interpret your phrasing.

What will a Kruskal observer predict for those cases, and why?

All other coordinates predict exactly the same thing. By construction, computing observables (invariants) in different coordinates is as tightly guaranteed to produce the same result as (1/2)*(1/2) = (1/4)*(1). This is simply because the metric is transformed in such a way along with coordinate transform of e.g. world lines as to make this a pure mathematical identity.

 

[harrylin]

These surprising clarifications were very helpful for me to understand what the two of you were telling me in concert - it brought to light an important point.

For my account of what "Schwartzschild" would calculate, I used the Oppenheimer-Snyder "map" and directions (the O-S map of O-S, as explained by O-S). I held it for quite possible that I made a calculation or interpretation error. However, it now looks to me that you use a force-fitted "Schwartzschild" map that has been made to comply with other maps - perhaps based on a textbook treatment (is it perhaps de facto a MTW map?!).

BTW, the question "is this really GR?" is of course completely independent of "do I find this reasonable?".

I have in mind to dig deeper, but as this is very much the topic of O-S, I will do so (later) in the appropriate thread - https://www.physicsforums.com/showthread.php?t=651362

Thanks again!

 

[PAllen]

These surprising clarifications were very helpful for me to understand what the two of you were telling me in concert - it brought to light an important point.

For my account of what "Schwartzschild" would calculate, I used the Oppenheimer-Snyder "map" and directions (the O-S map of O-S, as explained by O-S). I held it for quite possible that I made a calculation or interpretation error. However, it now looks to me that you use a force-fitted "Schwartzschild" map that has been made to comply with other maps - perhaps based on a textbook treatment (is it perhaps de facto a MTW map?!).

BTW, the question "is this really GR?" is of course completely independent of "do I find this reasonable?".

I have in mind to dig deeper, but as this is very much the topic of O-S, I will do so (later) in the appropriate thread - https://www.physicsforums.com/showthread.php?t=651362

Thanks again!

Just a quick comment here - the SC metric you've seen (as derived from the EFE) applies inside the horizon. That is, if you look at it, it works just fine for r < Rs; it only doesn't work (without limits for the finite invariants) on Rs itself. So it is not added later - it is the same solution, and EFE are telling us it applies everywhere - the derivation applies up to r=0.

So, if you look back at my (u,v) versus (x,y) example, it is as if you got the full (u,v) solution; you just have to deal with technical problems at x and y-axis coordinate (but not invariant) discontinuity.

 

[PeterDonis]

For my account of what "Schwartzschild" would calculate, I used the Oppenheimer-Snyder "map" and directions (the O-S map of O-S, as explained by O-S).

Yes, I understand that. My rewrite of those predictions, with the key clause added in bold, was to make the point that the map and directions provided by O-S are incomplete; they tell you how to calculate things as long as tau < 42, but they do *not* say that Voyager stops existing at tau = 42. Nor do they say that it continues to exist at tau >= 42, but in a region of spacetime that needs a different chart to map it. They do not address that question either way. They do say that t -> infinity as tau -> 42; but they do not show (nor, I think, do they claim to show) that t -> infinity represents a *physical* limitation; they only show that it represents a limitation of SC exterior coordinates. (Showing that it represents a physical limitation would require showing that some invariant quantity goes to infinity there, and O-S certainly don't do that; and in the light of further knowledge since then, we know there isn't one.)

However, it now looks to me that you use a force-fitted "Schwartzschild" map that has been made to comply with other maps - perhaps based on a textbook treatment (is it perhaps de facto a MTW map?!).

No, we are just addressing the question that the O-S map does not address: what happens to Voyager at tau = 42? Based on the answer to that question given by the Einstein Field Equation, we are taking the incomplete O-S map and adding a new region, and an expanded set of directions, onto it to make it physically complete.

BTW, the question "is this really GR?" is of course completely independent of "do I find this reasonable?".

Understood; and the answer is yes, it is "really GR".

 

[PeterDonis]

I have in mind to dig deeper, but as this is very much the topic of O-S, I will do so (later) in the appropriate thread - https://www.physicsforums.com/showthread.php?t=651362

I would find this helpful; I was having trouble remembering which thread I was posting in.

 

[PAllen]

Having come this far, I want to emphasize a very physical reason for not being satisfied with an object's history stopping at an arbitrary time in its history. This is that a key physical foundation of GR is the principle of equivalence; one aspect of this, is that sufficiently locally, GR = SR, everywhere, every when. So we have an infall clock whose mechanics is following normal SR physics locally at all times; tidal gravity is irrelevant due to its small size (esp. for a supermassive BH); the relation between its time rate and some distant clock when sending messages is irrelevant locally. So you posit at 2:59:59 pm, it is operating the same as any similarly constructed clock; but at 3 pm it stops dead for no reason explicable with SR physics. This is a gross violation of the principle of equivalence: a free fall clock near the horizon has a behavior completely different than free fall clocks everywhere else.

---

A related mathematical point is that the EFE are system of 10 equations that satisfy 4 identities. They are thus sufficient only to determine 6 independent functions. But 10 are needed to specify a metric expressed in some coordinates. This means you need to pose 4 'coordinate conditions' to fully determine the metric expression. Given the same boundary and initial conditions, these coordinate conditions don't change the physics of the solution - they just determine its coordinate expression. So you arrive as SC geometry by saying you want a vacuum solution that is spherically symmetric. This is enough to uniquely determine the geometry. One type of coordinate conditions gives you SC coordinates; another can lead directly to Kruskal. You can verify that SC coordinates compute all the same physics and geometry as Kruskal except that they have coordinate discontinuity at the horizon. This mathematical structure leave little alternative but to see the SC coordinates as two patches and a boundary problem that correspond to two parts of the Kruskal coordinates that cover the whole geometry without coordinate difficulties.

Historically, I think all of the following being well understood did not occur until the mid 1960s:

- spherical symmetry + vacuum uniquely determine a solution (this part was known, in various forms a long time)

- using right coordinate conditions, you can directly get the Kruskal coordinates from
the EFE; all other known coordinates are subsets of these. Historically, Kruskal coordinates arose by geodesic extension of incomplete charts; but sometime in the mid 60s (I think) it was realized they actually follow directly from the EFE as the unique complete spherically symmetric solution if you impose the right coordinate conditions.

 

[zonde]

The path of an outer edge infall particle has finite proper time integrated to the SC radius. If you declare it stops there, you have a hole in spacetime. You have a geodesic ending with finite 'interval', where curvature is finite.

PAllen, do you realize that in Minkowski geometry zero space-time distance (zero proper time) between two points does not mean it's the same point?
This is very different from traditional geometry where zero distance between two points does mean it's the same point.

If you imagine the surface of such particles infalling, and you don't allow it to proceed to Sc radius, you have, geometrically, a hole: proper time on this surface is finite, area is finite, but if you stop it from continuing, it is geometrically a hole. Geometry is defined by invariants, not coordinate quantities. Consider the example I gave to harrylin several posts back of a horizontal geodesic in the plane in (u,v) coordinates. u coordinate goes to infinity on both sides of coordinate singularity, but it is still nothing but a geodesic in the flat plane.

I found your example. I just have to think what I have to say about it. It is just mock transformation when you undo all the consequences of transformation using transformed metric.

Nope. Einstein was very clear that simultaneity is purely a convention, not an observable. There is no observation or measurement in SR that changes if you use a different one than the standard one (but you have to change the metric as well; it is no longer eg. diag(+1,-1,-1,-1) if you use a funky convention.

There is one statement in SR that gives it physical content - it is principle of relativity.
But principle of relativity applies to certain class of inertial coordinate systems. This class of inertial coordinate systems is defined using particular simultaneity convention.
So you can't really speak about SR with different simultaneity convention as this particular simultaneity convention is integral part of the theory (and it's predictions).

If you want you can say that relativity principle gives physical content to particular simultaneity convention.

 

[harrylin]

Just a quick comment here - the SC metric you've seen (as derived from the EFE) applies inside the horizon. That is, if you look at it, it works just fine for r < Rs; it only doesn't work (without limits for the finite invariants) on Rs itself. So it is not added later - it is the same solution, and EFE are telling us it applies everywhere - the derivation applies up to r=0.

So, if you look back at my (u,v) versus (x,y) example, it is as if you got the full (u,v) solution; you just have to deal with technical problems at x and y-axis coordinate (but not invariant) discontinuity.

And a quick comment on that quick comment: I don't see the qualitative difference with "The light speed limit doesn't exist; the tachyon space works just fine, it is the same solution. You just have to deal with technical problems to get through c".

 

[martinbn]

Having come this far, I want to emphasize a very physical reason for not being satisfied with an object's history stopping at an arbitrary time in its history. This is that a key physical foundation of GR is the principle of equivalence; one aspect of this, is that sufficiently locally, GR = SR, everywhere, every when. So we have an infall clock whose mechanics is following normal SR physics locally at all times; tidal gravity is irrelevant due to its small size (esp. for a supermassive BH); the relation between its time rate and some distant clock when sending messages is irrelevant locally. So you posit at 2:59:59 pm, it is operating the same as any similarly constructed clock; but at 3 pm it stops dead for no reason explicable with SR physics. This is a gross violation of the principle of equivalence: a free fall clock near the horizon has a behavior completely different than free fall clocks everywhere else.

So far I was following your explanations, but here I am I little confused. Why do you say that the clock will stop? Surely passing the horizon will not stop the clock and someone with the clock will see it ticking after 3:00pm, no?

A related mathematical point is that the EFE are system of 10 equations that satisfy 4 identities.

What do you mean by this? Equations that satisfy identities!

 

[harrylin]

A clarification about terminology: there are several concepts/objects that can be referred to as "Schwarzschild's solution", or "the Schwarzschild metric" [..] My reason for making these distinctions will be evident in a moment.

As I elaborate in a parallel thread, I make a similar distinction between different "flavours" of GR.

[..] this is not a statement about any actual physics; it's only a statement about a particular coordinate chart, with no real physical content.

In fact that chart is an equation. You next suggest that another equation has more physical content than the equation which was derived from it, using reasonable physical assumptions. I suspect that the one who derived that equation would disagree with you for reasons that I will briefly* mention.

[..] when you actually work through the solution of the Einstein Field Equation, in either case #1 above ([..] physically unreasonable) or case #2 above (more complex [..] but physically more reasonable, though still highly idealized), you find that any solution that only includes the region outside the horizon is incomplete; the EFE itself tells you that that region cannot be the entire spacetime. [..] These all seem like innocuous statements about coordinate charts on geometric manifolds to me. Apparently they seem like huge issues to you; I'm not sure why. Why is it such a big deal that some charts can cover points or regions that others can't?

[..] technically the Schwarzschild observer has to change charts to make some of the predictions I'm going to give; but there is nothing stopping him from doing that. [..] why does your worldline stop? No reason.
That's not physically reasonable. [..]

[..] Based on the answer to that question given by the Einstein Field Equation, we are taking the incomplete O-S map and adding a new region, and an expanded set of directions, onto it to make it physically complete. [..]
Understood; and the answer is yes, it is "really GR".

If I correctly understood the explanations, those equations lead to white holes when blindly followed through without physical concerns; following your arguments, white holes are "really GR". Is it a big deal to you?

However, and as we discussed earlier, we agree that considerations of what makes physical sense should play a role. Most theories do contain more than mere equations, and GR is no exception. For me a theory consists of its physical foundations (both those postulated and those clearly mentioned). Those foundations are all kept except if they lead to contradictions; and an equation is only physically valid insofar as those physical foundations are not violated. An obvious one is the Einstein equivalence principle, but there are also the physical reality of gravitational fields and the requirement that theorems must describe the relation between measurable bodies and clocks.

Consequently I expect that Einstein would reject Peter's argument and say that Peter denies the physical reality of gravitational fields. Einstein would argue that the clock's worldline never really stops, but does not reach beyond 42 due to the physical reality of the gravitational field. I think that Kruskal's white holes and his inside solution for black holes are not compatible with Einstein's GR.

Having come this far, I want to emphasize a very physical reason for not being satisfied with an object's history stopping at an arbitrary time in its history. This is that a key physical foundation of GR is the principle of equivalence; one aspect of this, is that sufficiently locally, GR = SR, everywhere, every when. So we have an infall clock whose mechanics is following normal SR physics locally at all times; tidal gravity is irrelevant due to its small size (esp. for a supermassive BH); the relation between its time rate and some distant clock when sending messages is irrelevant locally. So you posit at 2:59:59 pm, it is operating the same as any similarly constructed clock; but at 3 pm it stops dead for no reason explicable with SR physics.
This is a gross violation of the principle of equivalence: a free fall clock near the horizon has a behavior completely different than free fall clocks everywhere else.

PAllen, it looks to me that you are mixing up reference frames. As far as I can see, in no valid GR reference system is the clock suddenly stopping dead.
As described from S, the clock never stops ticking. I guess that for such an extreme case the validity of SR probably shrinks to nothing. And as described from S', dramatic things happen upto 3 pm but no stopping of clocks is observed.

[ADDENDUM: It may look a little weird if you believe that the universe is eternal. But in case you believe that the universe is not eternal, as is commonly thought, then the universe ends at for example 2:59:58.]

And don't you think that Einstein would have exclaimed the same, if it was really a "gross violation of the principle of equivalence"? Instead he commented that "there arises the question whether it is possible to build up a field containing such singularities". (E. 1939.)

The Einstein principle of equivalence:
"K' [..] has a uniformly accelerated motion relative to K [..] [This] can be explained in as good a manner in the following way. The reference-system K' has no acceleration. In the space-time region considered there is a gravitation-field which generates the accelerated motion relative to K'."
- https://en.wikisource.org/wiki/The_Foundation_of_the_Generalised_Theory_of_Relativity

*Regretfully this forum has been stripped from philosophy on the grounds that the mentors don't want to spend time on monitoring such discussions; I will respect that by not elaborating much on philosophy of science.

 

[harrylin]

[..] Surely passing the horizon will not stop the clock and someone with the clock will see it ticking after 3:00pm, no? [..]

According to the Schwartzschild equations as used by Einstein and Oppenheimer, that clock will only reach the horizon (and indicate 3:00pm) at t=∞. In common physics that means "never"; however some smart person (in fact, who?) invented a different interpretation!

 

[PeterDonis]

In fact that chart is an equation. You next suggest that another equation has more physical content than the equation which was derived from it, using reasonable physical assumptions.

No, I don't. I suggest that the first chart/equation (exterior Schwarzschild) does not cover a particular portion of the spacetime that the second chart/equation (Kruskal) does.

However, underlying all of this is just one equation, the EFE. That equation is what's really at issue here. See below.

If I correctly understood the explanations, those equations lead to white holes when blindly followed through without physical concerns; following your arguments, white holes are "really GR".

You didn't correctly understand the explanations. The EFE leads to white holes only if we assume the spacetime is vacuum everywhere (and spherically symmetric, but that's a minor point for this discussion). Nobody thinks that assumption is physically reasonable. If the spacetime is not vacuum everywhere--for example, if there is collapsing matter present--then the EFE does *not* predict white holes. So white holes are part of the set of all possible mathematical solutions of the EFE, but they are not part of the set of physically reasonable solutions of the EFE.

Just an "equation" isn't enough; you have to add constraints--initial/boundary conditions--to get a particular solution. Which solution of the equation you get--i.e., which spacetime geometry models the physical situation you're interested in--depends on the constraints.

However, and as we discussed earlier, we agree that considerations of what makes physical sense should play a role. Most theories do contain more than mere equations, and GR is no exception. For me a theory consists of its physical foundations (both those postulated and those clearly mentioned).

Of course. See above.

Those foundations are all kept except if they lead to contradictions; and an equation is only physically valid insofar as those physical foundations are not violated. An obvious one is the Einstein equivalence principle, but there are also the physical reality of gravitational fields and the requirement that theorems must describe the relation between measurable bodies and clocks.

Sure.

Consequently I expect that Einstein would reject Peter's argument and say that Peter denies the physical reality of gravitational fields.

Einstein *did* reject arguments of this type. Einstein was wrong.

Einstein would argue that the clock's worldline never really stops, but does not reach beyond 42 due to the physical reality of the gravitational field.

What is "the gravitational field"? What mathematical object in the theory does it correspond to? Before we can even evaluate this claim, we have to know what it refers to. But let's try it with some examples:

(1) The "gravitational field" is the metric. The metric (the coordinate-free geometric object, not its expression in particular coordinates) is perfectly finite and continuous at the horizon, and for reasons that both PAllen and I have explained, it can't "just stop" at the horizon without violating the EFE.

(2) The "gravitational field" is the Riemann curvature tensor. Like the metric, this is perfectly finite and continuous at the horizon.

(3) The "gravitational field" is the proper acceleration experienced by a "hovering" observer (an observer who stays at the same radius and does not move at all in a tangential direction). This *does* increase without bound as you get closer and closer to the horizon. However, there is *no* "hovering" observer *at* the horizon, because the horizon is a null surface: i.e., a line with constant r = 2M and constant theta, phi is not a timelike line; it's a null line (the path of a light ray--a radially outgoing light ray). So there is no observer who experiences infinite proper acceleration, and this definition of "gravitational field" simply doesn't apply at or inside the horizon.

As far as I can see, the only possible basis you could have for claiming that "the physical reality of the gravitational field" means that the clock's worldline stops as tau->42, would be #3. However, #3 doesn't apply to infalling observers; it only applies to accelerated, "hovering" observers. Infalling observers don't feel any acceleration, so there's nothing stopping them from falling through the horizon. The "gravitational field" in the sense of #3 is simply not felt by them at all.

Note that in all these cases, the physical "field" has to correspond to something invariant in the mathematical model, *not* something that only exists in a particular coordinate chart. That is something Einstein would have *agreed* with. Note also that none of the definitions of "gravitational field" I gave above used Schwarzschild coordinate time, or the fact that t->infinity as you approach the horizon. Einstein simply didn't understand that claims about t->infinity as you approach the horizon were claims about something that only exists in a particular coordinate chart.

I think that Kruskal's white holes and his inside solution for black holes are not compatible with Einstein's GR.

See above. You are equivocating on different meanings of "Einstein's GR". White holes are mathematically compatible, but not physically reasonable. Black hole interiors are both mathematically compatible *and* physically reasonable.

And don't you think that Einstein would have exclaimed the same, if it was really a "gross violation of the principle of equivalence"? Instead he commented that "there arises the question whether it is possible to build up a field containing such singularities". (E. 1939.)

As I've said before, Einstein's paper only considered the stationary case--i.e., he only considered systems of matter in stable equilibrium. All his paper proves is that *if* a system has a radius less than 9/8 of the Schwarzschild radius corresponding to its mass, the matter can't be in stable equilibrium. A collapsing object that forms a black hole meets this criterion: the collapsing matter is not in stable equilibrium. So Einstein's conclusion doesn't apply to it.

 

[PAllen]

I found your example. I just have to think what I have to say about it. It is just mock transformation when you undo all the consequences of transformation using transformed metric.

I only have time for one quick comment:

But that is the way all coordinate transforms work in differential geometry! That is the whole point! The the metric is transformed along with coordinates so all geometric properties (lengths, angle, intervals, etc.) are preserved as mathematical identities.

 

[martinbn]

According to the Schwartzschild equations as used by Einstein and Oppenheimer, that clock will only reach the horizon (and indicate 3:00pm) at t=∞. In common physics that means "never"; however some smart person (in fact, who?) invented a different interpretation!

That is not true, the clock will pass the horizon, and its proper time will go after 3:00pm.

 

[harrylin]

[..] Einstein *did* reject arguments of this type. Einstein was wrong.

Just give me the physics paper that proves that your philosophy is right, and Einstein's was wrong.

What is "the gravitational field"? [..]

Perhaps your beef with Einstein could be summarized as follows:

Peter: What is "the gravitational field"? It is not a real mathematical object
Einstein: What is a "region of spacetime"? It is not a real physical object.

In my experience it can be interesting to poll opinions, and to inform onlookers about different points of view; however discussions of that type are useless.

 

[harrylin]

That is not true, the clock will pass the horizon, and its proper time will go after 3:00pm.

Not in "Schwartzschild" time t of the Schwartzschild equation. Did you calculate it?
Once more: According to the Schwartzschild equations as used by Einstein and Oppenheimer, that clock will only reach the horizon (and indicate 3:00pm) at t=∞. In common physics that means "never"; however some smart person (in fact, who?) invented a different interpretation. In that different interpretation, which I still don't fully understand, the clock will pass the horizon despite Schwartzschild's t=∞.

For details, see the ongoing discussion: https://www.physicsforums.com/showthread.php?t=651362
incl. an extract of Oppenheimer-Snyder: https://www.physicsforums.com/showpost.php?p=4162425&postcount=50

 

[martinbn]

Not in "Schwartzschild" time t of the Schwartzschild equation. Did you calculate it?
For details, see the ongoing discussion: https://www.physicsforums.com/showthread.php?t=651362

This makes no sense. The clock shows its proper time, nothing else, and it will not stop when it reaches 3:00pm.

 

[harrylin]

This makes no sense. The clock shows its proper time, nothing else, and it will not stop when it reaches 3:00pm.

A so-called "asymptotic observer" predicts that it will slow down so much that it will not reach 3:00pm before the end of this universe. However, a "Kruskal observer" says that that is true from the viewpoint of the asymptotic observer but predicts that the clock will nevertheless continue to tick beyond 3:00pm.

PeterDonis and PAllen say that these predictions do not contradict each other. And that still makes no sense to me, despite lengthy efforts of them to explain this to me.

 

[pervect]

And a quick comment on that quick comment: I don't see the qualitative difference with "The light speed limit doesn't exist; the tachyon space works just fine, it is the same solution. You just have to deal with technical problems to get through c".

It does require closer inspection to see if the apparent singularity in the equations of motion is removable or not.

What do I mean by a removable singularity?

http://en.wikipedia.org/w/index.php?title=Removable_singularity&oldid=507006469

n complex analysis, a removable singularity (sometimes called a cosmetic singularity) of a holomorphic function is a point at which the function is undefined, but it is possible to define the function at that point in such a way that the function is regular in a neighbourhood of that point.

For instance, the function

f(z) = \frac{\sin z}{z}

has a singularity at z = 0. This singularity can be removed by defining f(0) := 1, which is the limit of f as z tends to 0. The resulting function is holomorphic.

It's been known for a very long time that in the black hole case that the singularity is removable.

IT does takes a bit of work to decide if the apparent singularity is the result of a poor coordinate choice , or is an inherent feature of the equations.

It might be helpful to give a quick example of how this happens. Consider the equations for spatial geodesics on the surface of the Earth. (Why geodesics? Because that's how GR determines equation of motion. So this is an easy-to-understand application of the issues involved in finding geodesics).

If you let lattitude be represented by \psi and longitude by \phi, then you can write the metrc ds^2 = R^2 (d \psi^2 + cos^2(\psi) d\phi^2) and come up with the equations for the geodesic (which we know SHOULD be a great circle) for \psi(t) and \phi(t)

<br /> \frac{d^2 \psi}{dt^2} + \frac{1}{2} \sin 2 \psi \left( \frac{d \phi}{dt} \right)^2<br /><br /> \frac{d^2 \phi}{dt^2} - 2 \tan \psi \left(\frac{d\phi}{dt}\right) \left( \frac{d\psi}{dt} \right) = 0<br />

Now, one solution of these equations is \phi = constant. It makes both equations zero. It's also half of a great circle. But, if we look more closely, we see that we have a term of the form 0*infinity in the second equation as we approach the north pole, because of the presence of \tan \psi when \psi reaches 90 degrees.

THis apparent singularity is mathematical, not physical. If you're drawing a great circle around a sphere, there's no physical reason to stop at the north pole.

Of course we already know what the answer is - we need to join two half circles together. In particular, we know we need to splice together two half circles, 180 degrees apart in lattitude, though as far as I know all the solution techniques (change of variable, etc) are equivalent to not using lattitude and longitude coordinates at the north pole, because the coordinates are ill-behaved there.

The same is in the black hole case, though to justify it you need to either do the math yourself, or read a textbook where someone else has. Note that you probably won't find this sort of thing in papers so much, it's assumed everyone knows it in the literature. Where you're more likely to find an explanation in a textbook or lecture notes.

Which brings me to the next point.

We don't have textooks online, but we've got several good sets of lecture notes.

What does Carroll's lecture notes have to say on the topic?
He defines the geodesic equation of motion - they're pretty complex looking, and I wouldn't be surprised if you didn't want to solve them yourself. But what does Caroll have to say about solving them?

I'll give you a link http://preposterousuniverse.com/grnotes/grnotes-seven.pdf , and a page reference (pg 182) in that link.

Then I'll give you some question

1) Does Carroll support your thesis? Or does he disagree with it?
2) What do other textbooks and online lecture notes have to say?

And for my own information
3) Do you think you know the difference between "absolute time" and "non-absolute time"
4) Do you think your argument about "time slowing down at the event horizon" depends on the existence of "absolute" time?

 

[Nugatory]

A so-called "asymptotic observer" predicts that it will slow down so much that it will not reach 3:00pm before the end of this universe. However, a "Kruskal observer" says that that is true from the viewpoint of the asymptotic observer but predicts that the clock will nevertheless continue to tick beyond 3:00pm.

PeterDonis and PAllen say that these predictions do not contradict each other. And that still makes no sense to me, despite lengthy efforts of them to explain this to me.

Here's an analogy that may help it make a bit more sense.
Consider an ordinary boring constant-velocity special relativity problem: You are rest and you watch me passing by at some reasonable fraction of the speed of light, so you observe that my clock is ticking more slowly than yours. If the universe has a a finite age, it is certainly possible for me to observe a time on my clock that you will claim will never be reached - all that necessary is that:
1) I get to read my clock on my worldline before it terminates at the end of the universe.
2) Your worldline terminates at the end of the universe before it intersects the line of (your) simultaneity through the event of me reading my clock.

But, you will say, I'm cheating by introducing this arbitrary "end of the universe" to cut off your worldline (actually, you introduced it - I'm just abusing it )before it can intersect the relevant line of simultaneity. If I didn't do that, then no matter how much of my time passes before I read my clock, you'd be able to extend your worldline to intersect the line of simultaneity. That is true enough, but then again the entire concept of "line of simultaneity" only really makes sense in flat space.

The bit about a "Kruskal observer" is a red herring. The geometry around a static non-charged non-rotating mass is the Schwarzschild geometry, no matter what coordinates we use, and the only meaningful notion of time that we have is proper time along a time-like worldline. The Kruskal coordinates allow us to calculate the proper time along the infalling clock's worldline as it crosses the Schwarzschild radius, whereas the the Schwarzschild coordinates (as opposed to geometry) do not. So it's not that the "Schwarzschild observer" and the "Kruskal observer" are producing conflicting observations, it is that the Kruskal coordinates are producing a prediction for the infalling observer's worldline and the Schwarzschild coordinates are not.

 

[zonde]

I only have time for one quick comment:

But that is the way all coordinate transforms work in differential geometry! That is the whole point! The the metric is transformed along with coordinates so all geometric properties (lengths, angle, intervals, etc.) are preserved as mathematical identities.

Okay, I have kind of working hypothesis about how this works.
We have global coordinate system where we know how to get from one place to another i.e. it provides connection, but this global coordinate system does not tell anything useful about distances and angles and such. And then we have another coordinate system that tells us distances and angles but it works only locally, meaning that if we have two adjacent patches with local coordinate systems we don't know how to glue them together.
So we take take global coordinate system with metric that will give us geometric values in accord with local coordinate systems.

Something like that. Only I don't know how to check if this is right.