## Notions of simultaneity in strongly curved spacetime

 Quote by PeterDonis 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.

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 Quote by rjbeery 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.

 Quote by PAllen GR itself requires white holes originate in the past without cause; while black holes are predicted (classically) to form from plausible starting conditions.
 Quote by PAllen 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 = $$T_0$$Now 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 Schwarzchild 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.

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 Quote by rjbeery 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.

 Quote by PAllen 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.

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 Quote by rjbeery 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.

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 Quote by rjbeery 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...es_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://www.pitt.edu/~jdnorton/teachi...ure/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.

 Quote by PAllen 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...

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.
 Quote by PAllen [..] 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:
Quote by harrylin
[..] [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.
 Quote by PeterDonis [..] 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):
 Quote by PeterDonis 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/ba...-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?

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 Quote by harrylin 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.
 Quote by harrylin 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.
 Quote by harrylin 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.
 Quote by harrylin 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.
 Quote by harrylin 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.

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 Quote by harrylin 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?

 Quote by harrylin 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.

 Quote by harrylin 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.

 Quote by harrylin 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.

 Quote by harrylin 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.

 Quote by harrylin 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:

 Quote by harrylin 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.

 Quote by harrylin 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.

 Quote by harrylin 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.

 Quote by harrylin 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.

 Quote by harrylin 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.

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 Quote by PeterDonis 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.

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 Quote by PAllen 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.

 Quote by PAllen 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.

 Quote by PAllen 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.

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 Quote by PeterDonis 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.
 Quote by PeterDonis 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.

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 Quote by PAllen 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).

 Quote by PAllen 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.

 Quote by PAllen 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.

 Quote by PAllen 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.

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 Quote by PAllen 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.

 Quote by PAllen A more accurate description, I agree, would be exactly matches in the limit for an asymptotic observer.
Ok, good, we're in agreement.

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 Quote by zonde 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.
 Quote by zonde 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.
 Quote by zonde 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.
 Quote by zonde 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.
 Quote by zonde 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.