Black hole matter accumulation

[PeterDonis]

If we use an object that is massive enough to make any potential impact on a BH then clearly we could not use the Schwazschild solution.

Yes, that's true. What does it have to do with the thought experiment I proposed? One could certainly have a black hole massive enough that sending, for example, a small probe, say a ton in size, falling inward towards the horizon would have a negligible effect on the spacetime curvature.

Frankly I do not understand this defensive approach. Einstein's general relativity is a masterpiece and clearly very useful but that does not mean that every single iota must be correct and that those who question parts which have never been empirically verified or perhaps can never be empirically verified are automatically idiots. When we take theories as dogmas then any potential for progress stops IMHO.

Once again, I have never said GR is perfect or dogma, nor have I said that those who question the claim that there is a region of spacetime inside the event horizon are idiots. In fact, I acknowledged in at least one post that the questions being asked are legitimate questions. They also have legitimate answers, which I and others have given. I have not claimed that GR is a theory of everything. I have given cogent reasons for believing that the particular prediction under discussion, that there is a region of spacetime inside the black hole horizon, is a robust one.

 

[PAllen]

Freely admitting to my having no training in GR, I nonetheless think all you have said here is not right. You for starters have pointedly not answered my questions re what a distant (coordinate) observer physically determines, and whether that is or is not accurately reflected by SM - as surely it ought to be if it is nothing more than an essentially meaningless construct. Your last sentence neatly sidesteps the issue as I see it, but here's hoping for an answer to my specific questions in #148!

What a distant observer sees is interesting, but just a matter of 'seeing'. My statement about local observations was not sidestep, but a point that in GR local physics paramount. Cosmologically, you can make good arguments for billion light year differences in distance and factors of c difference of speed for very distant objects, using different reasonable definitions.

As for #148, I won't touch it. I neither know about nor am interested in Yilmaz theories. I am interested in quantum gravity and string theoretic approaches (but have limited knowledge of them).

 

[Q-reeus]

What a distant observer sees is interesting, but just a matter of 'seeing'.

Cannot agree with that. Take the temporal component of SM. As you must well know, redshift follows directly from applying SM to the situation of emitter and receiver at differing potentials. Just 'optical illusion'? How about this thought experiment: Distant observer sends down a clock to the surface of planet X, waits a time T, and retrieves said clock. Repeats the procedure a second time, but now waits some different time T2 before retrieving as before. It is now an easy matter to subtract out any complications of the lowering and raising parts, and just figure out precisely the time dilation factor experienced by the lowered clock. Are you in agreeance or not that the so determined clock rate will have *physically* been depressed as per SM? Yes or no please! If yes, do you somehow think that the SM length measure would not have equally valid *physical* consequences, revealed by a suitable analogous procedure?

My statement about local observations was not sidestep, but a point that in GR local physics paramount.

I accept that's what you meant, but again, can't agree. Why should local physics be paramount - isn't the 'relative' part of General Relativity telling us this is at least equally about relating 'here' to 'there'? What else is SM designed for?

As for #148, I won't touch it. I neither know about nor am interested in Yilmaz theories. I am interested in quantum gravity and string theoretic approaches (but have limited knowledge of them).

My apologies for that typo - should have been #146, not #148. So feel free to comment on #146 (although a suitable response to the above would answer much of that anyway).

 

[Dale]

Well sorry if I made you work hard and I will try and do better, but I think 'incoherent' is more than a little exaggerated.

You are right, it is an exaggeration.

I hope you are not serious in saying that.

Completely. There is never any need to match coordinates.

Read my 'incoherent' entry #138 again, slowly.

No thanks. I am not going to make more effort reading it than you made writing it.

SM has, or at least is supposed to, accurately reflect the physical metric, referenced of course to coordinate measure. So is the flat Minkowski interior metric. Disagree? If not then please acknowledge the dilemma I raised is perfectly valid.

No disagreement, but what has that to do with your "dilemma"?

There is no need for different coordinate charts to be matched up to each other. There is not even a requirement that different coordinate charts cover the same region of the manifold. The only requirement for coordinate charts is that in any region of the manifold covered by two coordinate charts there needs to be a diffeomorphism between the two.

So your dilemma is a non-issue. They don't need to match up so there is no problem if they don't.

 

[PAllen]

Cannot agree with that. Take the temporal component of SM. As you must well know, redshift follows directly from applying SM to the situation of emitter and receiver at differing potentials. Just 'optical illusion'? How about this thought experiment: Distant observer sends down a clock to the surface of planet X, waits a time T, and retrieves said clock. Repeats the procedure a second time, but now waits some different time T2 before retrieving as before. It is now an easy matter to subtract out any complications of the lowering and raising parts, and just figure out precisely the time dilation factor experienced by the lowered clock. Are you in agreeance or not that the so determined clock rate will have *physically* been depressed as per SM? Yes or no please! If yes, do you somehow think that the SM length measure would not have equally valid *physical* consequences, revealed by a suitable analogous procedure?

Note that all coordinates (Kruskal, isotropic SM, whatever) agree on the result of this experiment, so it is not coordinate dependent. However, aspects of its interpretation are coordinate dependent. I will pose an argument I have with many people about 'where the missing time is in the twin paradox'? In my view, it truly has no location. The analogy I make is as follows: imagine a straight, vertical line between A and B, and also a wiggly line between A and B on a flat Euclidean plane. For the wiggly line, where is the extra length? I hope you see this is completely meaningless - I can line up the straight line with the bottom, middle, or top of the wiggly line and get whatever answer I want. I claim the twin paradox is semantically identical to this. Further, I claim, so is your example. What makes the time difference physical (invariant) is exactly the fact that the world lines of the clocks have two intersections. Then, and only then, is there an invariant statement to make: the proper time between intersections along one world line is less than the other.

I accept that's what you meant, but again, can't agree. Why should local physics be paramount - isn't the 'relative' part of General Relativity telling us this is at least equally about relating 'here' to 'there'? What else is SM designed for?

I completely disagree with your concept of GR. In my mind, it says SR concepts only apply locally to free falling frames. Further, it says global distances and 'velocity at a distance' and 'time at a distance' have no preferred definitions at all. Pick different (sensible!) ways of extending a local simultaneity globally, and get completely different answers to how far away something is. Same for each other concept.

 

[Passionflower]

Yes, that's true. What does it have to do with the thought experiment I proposed? One could certainly have a black hole massive enough that sending, for example, a small probe, say a ton in size, falling inward towards the horizon would have a negligible effect on the spacetime curvature.

OK you send that probe and then near the EH you will get almost no information from this probe because it is almost like being frozen.
Then what will you try to prove?

 

[Q-reeus]

Completely. There is never any need to match coordinates.

Really? See below.

No thanks. I am not going to make more effort reading it than you made writing it.

Disarm me with some initial civility, and then this back-hander. Good thing you don't treat other respondents with such grace.

No disagreement, but what has that to do with your "dilemma"?

There is no need for different coordinate charts to be matched up to each other.

So tears, rips, and physically implausable jumps in potential dependence are OK in GR? Confirms my worst fears.

There is not even a requirement that different coordinate charts cover the same region of the manifold.

Not sure whether that means overlap, but either way, not the issue.

The only requirement for coordinate charts is that in any region of the manifold covered by two coordinate charts there needs to be a diffeomorphism between the two.

Meaning fit them together somehow - even if it is a physically nonsensical force-fit. That IS the issue.

So your dilemma is a non-issue. They don't need to match up so there is no problem if they don't.

So you assert. But if you care to follow my #146 and #153, and then go back to #138, something should light up internally. I sure hope so.

 

[Q-reeus]

Note that all coordinates (Kruskal, isotropic SM, whatever) agree on the result of this experiment, so it is not coordinate dependent.

Of course - but we have been discussing SM - stationary observers. Why throw in irrelevant issues?

However, aspects of its interpretation are coordinate dependent. I will pose an argument I have with many people about 'where the missing time is in the twin paradox'? In my view, it truly has no location. The analogy I make is as follows: imagine a straight, vertical line between A and B, and also a wiggly line between A and B on a flat Euclidean plane. For the wiggly line, where is the extra length? I hope you see this is completely meaningless - I can line up the straight line with the bottom, middle, or top of the wiggly line and get whatever answer I want. I claim the twin paradox is semantically identical to this. Further, I claim, so is your example. What makes the time difference physical (invariant) is exactly the fact that the world lines of the clocks have two intersections. Then, and only then, is there an invariant statement to make: the proper time between intersections along one world line is less than the other.

The wiggly vs straight line and Twin Paradox stuff is obviously true but equally not relevant. I will take it from the last sentence as a grudging admission though that there are perfectly valid *physical* consequences predicted as per SM, and that this will apply to both temporal and distance measure. That being so, it gets back to my argument in #138 one can of course mathematically force-fit a union between exterior SM and interior MM, but in so doing physical reasonableness is trashed. As I said there - one has to 'magically' destroy potential dependence of at least the radial metric component, and for consistency, that should apply to the temporal component also. Which is simply screaming loudly to me at least that SM and thus EFE's are plain wrong. But shucks, laymen should know their place, hey!

I completely disagree with your concept of GR.

Well perhaps you had better spell out what you see as my concept of GR - I don't like words being put in my mouth.

In my mind, it says SR concepts only apply locally to free falling frames.

And I have been mistakenly espousing SR concepts when it should have been GR? Where exactly?

Further, it says global distances and 'velocity at a distance' and 'time at a distance' have no preferred definitions at all. Pick different (sensible!) ways of extending a local simultaneity globally, and get completely different answers to how far away something is. Same for each other concept.

So the clock retrieval example I gave you is what, a hopelessly confusing problem for GR that 'cannot be sensibly defined or evaluated'? Holy cow. Could have sworn people have been performing precise tests of redshift etc, using SM.

 

[Dale]

So tears, rips, and physically implausable jumps in potential dependence are OK in GR? Confirms my worst fears.

In GR the scalar potential is replaced by the metric tensor. It is a more general object, covering situations that don't admit a scalar potential. The metric tensor doesn't have anything that could be described as a tear, rip, or jump, in the spacetime you describe, regardless of what coordinates you use.

Meaning fit them together somehow - even if it is a physically nonsensical force-fit. That IS the issue.

It is a non-issue. Use any coordinates you like for either region. As long as there is a diffeomorphism at the border they do not need to match.

 

[Q-reeus]

In GR the scalar potential is replaced by the metric tensor. It is a more general object, covering situations that don't admit a scalar potential.

Quite aware that spinning matter etc requires extra, tensorial components to describe. But let's get it right for a simple stationary shell, shall we?

The metric tensor doesn't have anything that could be described as a tear, rip, or jump, in the spacetime you describe, regardless of what coordinates you use.

Sure, and I was using a bit of hyperbole re rip'n'tear to emphasize a point - there is a need for a physically consistent transition.

It is a non-issue. Use any coordinates you like for either region. As long as there is a diffeomorphism at the border they do not need to match.

But it can't be any old diff. I don't think you have really grasped what the main point of #138 was. There are physical and precisely defined effects as per clock example in #153 (and if you really need it, I will cook up one involving length measure specifically). So please don't repeat the mantra SM is only a 'handy but arbitrary chart'. Rubbish. It is supposed to, and in weak gravity does a very good job of, accurately predicting measurable physical effects. Period. The fact that some of those effects are not locally observable is irrelevant to the issue raised.

Paragraph break - just for Dale. So, to drive it home again, recall from #138 that temporal and radial metric components have identical functional form exterior to the shell outer surface - in regime described by SC's. Recall we have established these coords accurately account for redshift etc. Now unless you want to claim that somehow nature-as-traffic-cop strangely and radically descriminates, there is a deep consistency issue to face here.

Would you have redshift disappear via a small hop from shell exterior to interior? If not (and of course it won't), why would the functionally identical (re grav potential) radial metric component go all haywire and inexplicably jump back to the 'infinity' value? You are truly comfortable with that? Seems much, much more reasonable to me that, having a consistent potential dependence everywhere exterior to the shell, this sensibly persists within. I don't personally believe in mathematical magic, whether or not that makes me 'fringe'.
Oh my, is it really 12 hours past my bedtime - must go.:zzz:

 

[PeterDonis]

OK you send that probe and then near the EH you will get almost no information from this probe because it is almost like being frozen.
Then what will you try to prove?

I drop the probe, freely falling into the hole. The probe is pre-programmed to send a radio signal back outward towards me every 1 second, by its clock. I can then calculate that I should receive the last signal I will ever receive from the probe at a certain time according to my clock. The next signal the probe sends after that "last" signal will be sent from below the horizon, according to the theory, so it will never reach me. If, then, I run such an experiment and I receive a radio signal from the probe at any time after the time at which the theory predicts I should receive the last signal, then GR's prediction is falsified.

Of course if GR's prediction is *true*, I will wait forever and never receive another signal from the probe after the "last" one I predicted I would receive. So at any finite time interval after receiving the "last" signal, a skeptic could always say, "Yes, you haven't received another signal from the probe *yet*, but you *might* receive one at some time in the future." At that point we're talking about the problem of induction, not anything specific to GR. It's true, technically, that I will never be able to *prove* GR correct this way; but at any time, I can certainly tell without ambiguity if the data to date are still consistent with the theory.

 

[Dale]

Sure, and I was using a bit of hyperbole re rip'n'tear to emphasize a point - there is a need for a physically consistent transition.

Sure, for the metric. There is no such need for the coordinates.

But it can't be any old diff.

Yes, it can. That is one of the central features of Riemannian geometry.

There are physical and precisely defined effects as per clock example in #153 (and if you really need it, I will cook up one involving length measure specifically). So please don't repeat the mantra SM is only a 'handy but arbitrary chart'. Rubbish. It is supposed to, and in weak gravity does a very good job of, accurately predicting measurable physical effects.

Sure. So do all other coordinate systems. The measurable physical results are all tensors, so they are agreed upon by all coordinate charts. That is the whole point of writing the laws in a manifestly covariant form.

Would you have redshift disappear via a small hop from shell exterior to interior?

Of course not, the metric is continuous.

why would the functionally identical (re grav potential) radial metric component go all haywire and inexplicably jump back to the 'infinity' value?

Because you change coordinates. The continuity of the metric does not imply nor require continuity of the components of the metric as expressed in different coordinate systems.

Seems much, much more reasonable to me that, having a consistent potential dependence everywhere exterior to the shell, this sensibly persists within.

I agree. Again, the metric and the coordinates are not the same thing at all. The metric must be continuous, and every individual coordinate chart must also be continuous, but two different coordinate systems need not be continuous with each other anywhere.

 

[PeterDonis]

Would you have redshift disappear via a small hop from shell exterior to interior? If not (and of course it won't), why would the functionally identical (re grav potential) radial metric component go all haywire and inexplicably jump back to the 'infinity' value?

It doesn't. The metric inside the shell is "flat" in the sense that the metric coefficients are independent of the coordinates, so that it can be written in the Minkowski form. But the coordinates themselves are scaled differently than they are at infinity, so that what you are calling the "potential" matches up across the thin spherical shell, and the length contraction and time dilation factors do too.

Remember also that an "infinitely thin" spherical shell is unphysical if it has nonzero mass; there has to be *some* region of finite thickness where the stress-energy tensor is nonzero, and in that region the metric coefficients will change too.

 

[Passionflower]

I drop the probe, freely falling into the hole. The probe is pre-programmed to send a radio signal back outward towards me every 1 second, by its clock. I can then calculate that I should receive the last signal I will ever receive from the probe at a certain time according to my clock. The next signal the probe sends after that "last" signal will be sent from below the horizon, according to the theory, so it will never reach me. If, then, I run such an experiment and I receive a radio signal from the probe at any time after the time at which the theory predicts I should receive the last signal, then GR's prediction is falsified.

Yes but that is not what we are talking about right? Aren't we are talking about proving it passes the horizon into another region?

I do not believe anybody here in this topic questions the Sch. solution up to the EH. The question is what happens beyond the EH if anything at all. I do not understand how your experiment will prove anything related to that.

Making all kinds of general statements what happens beyond the EH is no longer science IMHO, we might as well claim that beyond the EH are stacked turtles. We cannot possibly know anything about it.

You could say "well we could in theory send someone" but what does that prove as the information from him dries up as soon as he is close to the EH. We simply cannot experimentally verify he crossed the EH.

I like Popper's idea about theories or parts of theories that cannot be falsified.

 

[PeterDonis]

I like Popper's idea about theories or parts of theories that cannot be falsified.

Um...I just described how the standard GR theory about a black hole spacetime can be falsified. I gave the exact experimental result that would do it. What more do you want?

Yes but that is not what we are talking about right? Aren't
we are talking about proving it passes the horizon into another region?

I'm confused. Are we talking about proof, or are we talking about falsification? I have not claimed I can *prove* the standard GR theory to be correct. In fact I have said explicitly that scientific theories can't be "proved" (though, as I noted just now, they can be falsified).

We can, however, gather indirect evidence and make cogent arguments that bear on the question. For example, we can in principle measure and report out to "infinity" the curvature components all the way down to just a smidgen above the horizon, and verify that they match the GR prediction. We can also observe objects traveling on infalling worldlines all the way to the horizon and see that their trajectories match the GR prediction. We can also, of course, observe that nothing ever comes *out* of the horizon. Further, we can take the limit of our measurements to calculate values for physical quantities *at* the horizon, and verify that, as GR predicts, they are finite and nonsingular.

At that point, as I've said before, we are faced with two choices:

(1) We can believe that, since physical objects have gone *into* the horizon, and none have come out, and there is no physical reason for them to have ceased to exist *at* the horizon, there must be a region of spacetime inside the horizon, where those objects went; or

(2) We can believe that somehow, spacetime just "stops" at the horizon, and all those objects we saw fall into it just stopped existing when they hit the horizon, even though there is no physical reason for them to have done so.

By taking option (1), are we adopting a belief about a region of spacetime that we can never directly observe? Yes, certainly. But, IMHO, it is better than the alternative--adopting a belief that, in order to avoid having to believe in a region of spacetime we can't directly observe, forces us to believe that physics all of a sudden starts working completely differently at the horizon, for no apparent reason.

 

[Passionflower]

We can, however, gather indirect evidence and make cogent arguments that bear on the question. For example, we can in principle measure and report out to "infinity" the curvature components all the way down to just a smidgen above the horizon, and verify that they match the GR prediction.

OK, I am with you here.

We can also observe objects traveling on infalling worldlines all the way to the horizon and see that their trajectories match the GR prediction.

Again, alright.

We can also, of course, observe that nothing ever comes *out* of the horizon.

Yes, correct.

Further, we can take the limit of our measurements to calculate values for physical quantities *at* the horizon, and verify that, as GR predicts, they are finite and nonsingular.

Sure we can calculate, but calculations and experiment are different things right?

At that point, as I've said before, we are faced with two choices:
(1) We can believe that, since physical objects have gone *into* the horizon,...

Ooops, they have gone into the horizon? All I think we can observe is that they freeze arbitrarily close to the horizon.

(2) We can believe that somehow, spacetime just "stops" at the horizon, and all those objects we saw fall into it just stopped existing when they hit the horizon, even though there is no physical reason for them to have done so.

I think there are more options, I am not saying these options are true but since you ask the most obvious other option is:

(3) Everything arbitrarily close to the EH is simply frozen and does not move, they are simply stuck there.

Now I am aware that the Sch. solution does not state that but so what? We are talking about empirical verification here not theory right?

Even if we would have a way of knowing something passes the EH then still we do not know what is beyond. Yes, we can calculate it using the Sch. solutions, calculations I do all the time and they work fine. But again we are talking here about experiment not theory.

 

[PeterDonis]

Sure we can calculate, but calculations and experiment are different things right?

Yes, but the calculation of a limiting case which is just a smidgen beyond the last point we can experimentally measure and send the results back to infinity is pretty close to a direct experimental result. We make these sorts of extrapolations all the time in physics and nobody bats an eye; maybe the results aren't exact but they're pretty close. So for this method to suddenly become egregiously wrong just because we're close to a black hole horizon--even though, as you know, locally you can't tell such a place from any other place in spacetime--physics would have to suddenly start working completely differently, for no apparent reason.

There are more options, I am not saying these are the case but since you ask the mos obvious option is:

(3) Everything arbitrarily close to the EH is simply frozen and does not move.

Technically you are right, this is a different alternative than (2), but in practice it works out the same. Remember that, as you know and have posted in other threads, the infalling observer himself does not see his own clock freezing; he will measure a finite amount of proper time from a given finite radius above the horizon, to the horizon. And then what happens? Saying that he actually "freezes" at that point (instead of the freezing being an illusion as seen by the observer far away, because of the delay in light getting out) is, practically speaking, the same as saying he ceases to exist; it's saying that physics suddenly starts working completely differently, for no apparent reason.

 

[Passionflower]

Yes, but the calculation of a limiting case which is just a smidgen beyond the last point we can experimentally measure and send the results back to infinity is pretty close to a direct experimental result. We make these sorts of extrapolations all the time in physics and nobody bats an eye; maybe the results aren't exact but they're pretty close. So for this method to suddenly become egregiously wrong just because we're close to a black hole horizon--even though, as you know, locally you can't tell such a place from any other place in spacetime--physics would have to suddenly start working completely differently, for no apparent reason.

Yes, OK, I think these are some solid arguments!

Remember that, as you know and have posted in other threads, the infalling observer himself does not see his own clock freezing;

Correct.

he will measure a finite amount of proper time from a given finite radius above the horizon, to the horizon.

Yes we can calculate that using the Sch. solution but do not know that is the case empirically? To the horizon or slightly above the horizon?

Remember, as you undoubtedly know, we cannot even empirically verify the correctness of the Sch. solution. For instance Pound-Rebka does not measure the discrepancy between delta r and delta rho as we simply lack the accuracy in instrumentation.

 

[PeterDonis]

Yes we can calculate that using the Sch. solution but do not know that is the case empirically? To the horizon or slightly above the horizon?

Remember, as you undoubtedly know, we cannot even empirically verify the correctness of the Sch. solution. For instance Pound-Rebka does not measure the discrepancy between delta r and delta rho as we simply lack the accuracy in instrumentation.

We do today, yes, but our instrumentation continues to get more accurate. I have no problem imagining that we will be able to detect the spatial as well as the temporal effects of spacetime curvature around the Earth in the future. Similarly, I have no problem imagining that at some future time we will be able to drop probes towards black holes, after synchronizing their clocks with ours at some reasonably large radius, and have the probes send back time-stamped radio signals that confirm the GR calculation of proper time elapsed in falling to the horizon.

 

[PAllen]

Of course - but we have been discussing SM - stationary observers. Why throw in irrelevant issues?

Because, at times, you have been treating one set of coordinates as specially meaningful. Some features you complain about don't exist in other coordinates. Therefore they are coordinate artifacts, not physical predictions.

The wiggly vs straight line and Twin Paradox stuff is obviously true but equally not relevant. I will take it from the last sentence as a grudging admission though that there are perfectly valid *physical* consequences predicted as per SM, and that this will apply to both temporal and distance measure. That being so, it gets back to my argument in #138 one can of course mathematically force-fit a union between exterior SM and interior MM, but in so doing physical reasonableness is trashed. As I said there - one has to 'magically' destroy potential dependence of at least the radial metric component, and for consistency, that should apply to the temporal component also. Which is simply screaming loudly to me at least that SM and thus EFE's are plain wrong. But shucks, laymen should know their place, hey!

The point of the wiggly line analogy was to emphasize the difference between physical predictions versus coordinate values. However you want to construct the clock paths, I could come up with coordinates that say the clock that shows more time at the end, for example, matched the other clock for 99% of the the time on this other clock; then, shot forward only as the lower clock was brought up. I will agree those would be strained coordinates, but the point is, it is only the comparison at the end that is a true physical prediction.

Well perhaps you had better spell out what you see as my concept of GR - I don't like words being put in my mouth.

In several places you talk about an observer making physical statements about what is true at a distance. In SR, this can be made to work because there exist global frames for inertial observers. In GR all such statements have no unique validity.

And I have been mistakenly espousing SR concepts when it should have been GR? Where exactly?

So the clock retrieval example I gave you is what, a hopelessly confusing problem for GR that 'cannot be sensibly defined or evaluated'? Holy cow. Could have sworn people have been performing precise tests of redshift etc, using SM.

No, it is trivial and can be computed in any coordinates.

As to your so call main problem, I just don't see the problem. The total solution for a shell is simply the Schwarzschild geometry outside, a non vacuum solution through the shell, and Minkowski inside. There would be no discontinuities. If the shell were transparent, and and light were emitted inside the shell of some frequency observed locally, it would red shifted when received by an outside observer slightly more than a similar signal from the surface of the shell (the difference being due to shell). Despite their being no formal, general, potential in GR, this experiment would behave very similar to the Newtonian case. Almost all of these properties follow directly from Birkhoff's Theorem, which is rigorously proven for GR.

 

[PAllen]

Yes there is such a beast as isotropic Schwarzschild coords - eg."Alternative (isotropic) formulations of the Schwarzschild metric" at http://en.wikipedia.org/wiki/Schwarzschild_metric, but one almost never hears of it being used. While it is true isotropy of length scale applies, it does so at the cost of imo a strange departure in dependence on potential between length scale and clock-rate as one goes from far out to nearer the source of gravity. One that does not apply between time and radial distance in standard SM coords. AS you unlike me are accomplished in the ins and outs of GR, please explain the justification and rationale for two distinct versions of SM (if there is one apart from Eddington's complaint that c was non-ispotropic in standard SM). SM is practically synonymous with the standard form, for which my entry #138 obviously refers to.

This is a truly fundamental misunderstanding. The isotropic coordinates are just different coordinates for the same geometry, as are the Kruskal, Eddington, Gullestrand, etc. They all represent the same metric as a geometric object, they all make the same physical predictions. It is no different that polar versus rectilinear versus logarithmic coordinates on a plan. Does drawing different lines on a plane change its geometry? No. However, the Euclidean metric expression can only be used with rectilinear coordinates. Using the standard tensor transformation law, you would derive different metric expressions for the other coordinates that would yield all the same geometric facts.

 

[Passionflower]

We do today, yes, but our instrumentation continues to get more accurate. I have no problem imagining that we will be able to detect the spatial as well as the temporal effects of spacetime curvature around the Earth in the future.

I have no problem with that either.

Similarly, I have no problem imagining that at some future time we will be able to drop probes towards black holes, after synchronizing their clocks with ours at some reasonably large radius, and have the probes send back time-stamped radio signals that confirm the GR calculation of proper time elapsed in falling to the horizon.

Well, I am hand waving but I think I see some potential problems, mainly to do with light travel time issues.

It would be nice to start a topic were we calculate stuff. Care to start one perhaps we can see people actually put down calculations and formulas? Should be relatively simply, a prove fall from zero and then some light travel times to the observer at a large r receiving signals.

It is unfortunate we have so many learned folks here who think they are experts (and some indeed are) but when it comes to showing numbers or formulas they are typically not around.

 

[Passionflower]

This is a truly fundamental misunderstanding. The isotropic coordinates are just different coordinates for the same geometry, as are the Kruskal, Eddington, Gullestrand, etc. They all represent the same metric as a geometric object, they all make the same physical predictions. It is no different that polar versus rectilinear versus logarithmic coordinates on a plan. Does drawing different lines on a plane change its geometry? No. However, the Euclidean metric expression can only be used with rectilinear coordinates. Using the standard tensor transformation law, you would derive different metric expressions for the other coordinates that would yield all the same geometric facts.

PAllen is correct here, different coordinates are like different fishnet stockings on a woman's leg, the leg does not change!

 

[PAllen]

I note that no one has commented on the ideas in my post #128, that these unverifiable issues become verifiable if there are exceptions to cosmic censorship; and there is plausible evidence that there may be exceptions.

 

[atyy]

I will open a new set of arguments on this topic. Somehow, I had forgotten developments I was following a number of years ago that have direct bearing on this debate. To wit, these issues may be quite testable and not academic.

As a preliminary observation, in a perfectly symmetric gravitational collapse (which implies, as has been noted, no white hole region), the actual predictions of GR (by definition coordinate independent) describe the horizon as purely optical phenomenon, not unlike gravitational lensing. We don't treat the lensed image as 'best description of reality'. Instead, understanding GR, we posit a more plausible reality behind the lensing. Similarly, GR taken literally asks us to understand the horizon as a frozen optical image, behind which perfectly normal physical processes occur (until the true singularity).

As a follow on to this, let us ask what would be seen if a globular cluster slowly coalesced to the point where it became a super massive black hole in a hypothetical dust free environment, with no stellar collisions occurring before horizon formation. My understanding of what we would see is a slowly compressing cluster brightening normally (same light, smaller area of image) until, at some point, motion slows down, emissions get redder and weaker (still looking like a cluster of stars), until, in finite time, the whole cluster has effectively vanished (all light so redshifted and emission rates so low, that no conceivable instrument can detect light from it anymore). Do we suppose that a globular cluster has vanished from the universe, or believe GR that perfectly ordinary physics is continuing that we just can't see? [and the only physics for what we can't see, consistent with GR, is further collapse].

Finally(!) my main point, alluded to in my intro, is that there is reasonable likelihood that the cosmic censorship hypothesis is simply false. In which case, the physics of what happens close to the true singularity may be accessible; at some point QG alternatives to the singularity may be testable. Then, one must ask, if there are cases we can see the physics of the final state of collapse, and others where an optical horizon prevents it, do we assume the latter represents fundamentally different physics, or do we believe GR that it is just an optical effect?

Here are some references on the doubtful nature of cosmic censorship, and the ideas of testing it:

http://prd.aps.org/abstract/PRD/v19/i8/p2239_1
http://arxiv.org/abs/gr-qc/9910108
http://arxiv.org/abs/0706.0132
http://arxiv.org/abs/gr-qc/0608136
http://arxiv.org/abs/gr-qc/0407109

You may like to read these discussions http://books.google.com/books?id=ZN...f+space+and+time+solvay&source=gbs_navlinks_s (just type in "cosmic censorship" in the seach box).

 

[?????]

A lot of talk here about how to experimentally verify what happens at the EH. This is the same old problem. Let me restate the problem by quoting an old Chinese proverb: If the grasshopper jumps halfway to the wall every time, how many jumps does it take him to get to the wall?

If the EH did exist, then this is a priori proof that you can never experimentally prove that it exists. The notion that some theoretical device could be lowered past the EH is irrelevant. If it could go past the EH, the rest of the universe would have lived out it lifetime and would no longer exist when the passage was finally made. Forget whether or not the coordinates match up. The Schwarzschild Radius EH is logically impossible on its face. Nothing can exist past the end of the life of the universe.

 

[Q-reeus]

...I agree. Again, the metric and the coordinates are not the same thing at all. The metric must be continuous, and every individual coordinate chart must also be continuous, but two different coordinate systems need not be continuous with each other anywhere.

I have never disagreed with that as trivially true for e.g. 'patching' between spherical and Cartesian co-ords, but it still ducks the point imo. The chart may not be the territory, but it had better darn well properly describe it in the regime intended. But I'll hammer that out in another thread.

 

[Q-reeus]

It doesn't. The metric inside the shell is "flat" in the sense that the metric coefficients are independent of the coordinates, so that it can be written in the Minkowski form. But the coordinates themselves are scaled differently than they are at infinity, so that what you are calling the "potential" matches up across the thin spherical shell, and the length contraction and time dilation factors do too.

Careful - this is in a sense basically agreeing with my argument. So you think there is finite length contraction in the interior Minkowski region. I believe that is the actual case, but it does not accord with a physically consistent transition from SM to MM, as per #138. In this case though, you are, like it or not, opting for an inexplicable contraction of the tangent (i.e. azimuthal) spatial components in traversing the shell from outer to inner radius. One cannot have it both ways. If the tangent spatial components remain independent of potential and thus invariant, interior flatness demands the radial component jump back to the potential-free value. Or vice versa as you imply, the contracted radial component changes but little, and the tangent components undergo a contraction jump. Makes no physical sense to me either way. An exterior metric ('chart' if you insist) that is isotropic in *all* components does - the transition is smooth, slight, and then and only then imo anomoly free.

Remember also that an "infinitely thin" spherical shell is unphysical if it has nonzero mass; there has to be *some* region of finite thickness where the stress-energy tensor is nonzero, and in that region the metric coefficients will change too.

Of course, and I have never suggested otherwise. A thin shell is simply perhaps the best way of manifesting the anomoly I maintain exists.

 

[Q-reeus]

Because, at times, you have been treating one set of coordinates as specially meaningful.

Well this gets down to whether SM (standard OR isotropic) is to be treated as a unique description of the spacetime surrounding a stationary spherically symmetric mass, as determined by coordinate values. You do not agree that Birkhoff's theorem http://en.wikipedia.org/wiki/Birkhoff's_theorem_(relativity) is saying just that? You do acknowledge that I have posed a problem pertinent to just that regime, and so for sure SM is 'specially meaningful' in that context? The issue should be thrashed out and settled in that regime and the transition issue to Minkowski regime.

Some features you complain about don't exist in other coordinates. Therefore they are coordinate artifacts, not physical predictions.

Could you detail which such features?

The point of the wiggly line analogy was to emphasize the difference between physical predictions versus coordinate values. However you want to construct the clock paths, I could come up with coordinates that say the clock that shows more time at the end, for example, matched the other clock for 99% of the the time on this other clock; then, shot forward only as the lower clock was brought up. I will agree those would be strained coordinates, but the point is, it is only the comparison at the end that is a true physical prediction.

I agree it is a strained argument. By using my two-runs method, one can easily and unambiguously determine the relative clock rates - as precisely given by the SM temporal component. Which is a mathematical statement of how spacetime regions temporally relate, and no mere arbitrarily chosen 'chart'. How on Earth would GPS function as well as it does if this sort of thing was up for grabs?

In several places you talk about an observer making physical statements about what is true at a distance. In SR, this can be made to work because there exist global frames for inertial observers. In GR all such statements have no unique validity.

Ditto my remarks above re GPS.

As to your so call main problem, I just don't see the problem. The total solution for a shell is simply the Schwarzschild geometry outside, a non vacuum solution through the shell, and Minkowski inside. There would be no discontinuities.

That depends on what you mean by 'discontinuities'. Obviously there is a sharp transition in the potential gradient ("g"), but for sure the potential itself hardly alters, as is true imo of all the spatial and the temporal metric components.

If the shell were transparent, and and light were emitted inside the shell of some frequency observed locally, it would red shifted when received by an outside observer slightly more than a similar signal from the surface of the shell (the difference being due to shell). Despite their being no formal, general, potential in GR, this experiment would behave very similar to the Newtonian case. Almost all of these properties follow directly from Birkhoff's Theorem, which is rigorously proven for GR.

Well fine, you have committed to that the interior temporal metric component is depressed. Kindly commit to whether iyo the interior spatial metric components have depressed values, and would that be for all components equally, and equal to the temporal component?

 

[Q-reeus]

This is a truly fundamental misunderstanding. The isotropic coordinates are just different coordinates for the same geometry, as are the Kruskal, Eddington, Gullestrand, etc. They all represent the same metric as a geometric object, they all make the same physical predictions. It is no different that polar versus rectilinear versus logarithmic coordinates on a plan. Does drawing different lines on a plane change its geometry? No. However, the Euclidean metric expression can only be used with rectilinear coordinates. Using the standard tensor transformation law, you would derive different metric expressions for the other coordinates that would yield all the same geometric facts.

This seems to me to be trivializing the value and/or uniqueness of a coordinate system AS a true and faithful representation of some metric, within it's regime of applicability. How can standard SM and ISM for example be making the same predictions? Both claim to be a mathematically correct mapping of gravitationally depressed metric components to that of a distant coordinate observer. So when said observer checks for redshift, or determines whether a 'test sphere' remains perfectly spherical under the telescope (duly allowing for light bending etc), there is not a problem? If SSM predicts the test sphere will be viewed as oblate owing to radial length contraction, while ISM predicts the sphere will remain perfectly spherical (that is after all what 'isotropic' implies), I see a problem. And how is that sensibly resolved?

 

[Q-reeus]

PAllen is correct here, different coordinates are like different fishnet stockings on a woman's leg, the leg does not change!

Depends how tight the stockings are! Seriously though, I disagree but you will have seen that elsewhere by now.

 

[Q-reeus]

...Forget whether or not the coordinates match up. The Schwarzschild Radius EH is logically impossible on its face. Nothing can exist past the end of the life of the universe.

Actually, penta-questionmark, it very much matters whether the coordinates 'match up'. If as I maintain the 'mismatch' is a pathological feature of SM and thus the EFE's, EH's and BH's turn out to be literally non-entities so then just quit worrying about a non-issue, period.

 

[Dale]

The chart may not be the territory, but it had better darn well properly describe it in the regime intended.

And all charts do.

 

[Dale]

In this case though, you are, like it or not, opting for an inexplicable contraction of the tangent (i.e. azimuthal) spatial components in traversing the shell from outer to inner radius.

It is perfectly explicable. You are changing from anisotropic coordinates to isotropic coordinates.

An exterior metric ('chart' if you insist)

Here is further evidence of this confusion in your mind. There is a huge difference between the metric and a chart. They are not synonymous or interchangeable.

 

[PeterDonis]

Well, I am hand waving but I think I see some potential problems, mainly to do with light travel time issues.

The light travel time would affect how long it took the faraway observer to receive the time stamped radio signals, but that's why I specified they are time stamped; they contain information that tells the faraway observer what the infalling observer's clock reading was when the signal was emitted. That is what verifies that the GR calculation for proper time for an infalling observer is correct. The time of reception of the signal is not the primary piece of experimental data; it just affects how long it takes for the experimental data to be collected.

 

[Passionflower]

The light travel time would affect how long it took the faraway observer to receive the time stamped radio signals, but that's why I specified they are time stamped; they contain information that tells the faraway observer what the infalling observer's clock reading was when the signal was emitted. That is what verifies that the GR calculation for proper time for an infalling observer is correct. The time of reception of the signal is not the primary piece of experimental data; it just affects how long it takes for the experimental data to be collected.

Yes I realize that Peter, I was more hinting at how long it takes to get updates.

But why don't we make an attempt to calculate it?

And please the next on who comes around claiming it is simple should come with formulas. :)

 

[PeterDonis]

If it could go past the EH, the rest of the universe would have lived out it lifetime and would no longer exist when the passage was finally made. Forget whether or not the coordinates match up. The Schwarzschild Radius EH is logically impossible on its face. Nothing can exist past the end of the life of the universe.

This is an interesting comment. It is true that there is a sense in which the entire region of spacetime inside the horizon is "in the future" for every event outside the horizon. However, that does not mean it is logically impossible for the region inside the horizon to exist. First, a clarification: if the universe has an end point in time, i.e., if it is closed, then what I just said, and what you said in the above quote, is no longer true; the region of spacetime inside the EH of any black hole inside the closed universe gets caught up in the "big crunch" that ends the universe, just like everything else, so it is no longer true that, from the viewpoint of someone inside the EH, the outside universe "would have lived out its lifetime and would no longer exist".

Second, if we assume the universe exists forever into the future, so that a black hole can be "eternal" and the region inside remains hidden behind the EH forever, then the region inside the horizon is indeed "in the future" for any event outside the horizon--but that doesn't mean the region inside the horizon is logically impossible. If it were logically impossible, there could be no consistent mathematical models containing a region inside the horizon, and there are; any coordinate chart that is nonsingular at the horizon constitutes such a consistent mathematical model, and several have been named in this thread.

You could still claim that the region inside the horizon was *physically unreasonable*, but I've already laid out what that claim requires: it requires you to believe that physics suddenly starts working completely differently at the horizon, for no apparent reason.

 

[PeterDonis]

Careful - this is in a sense basically agreeing with my argument. So you think there is finite length contraction in the interior Minkowski region. I believe that is the actual case, but it does not accord with a physically consistent transition from SM to MM

Why not? Remember that there is a shell of finite thickness in between, where the stress-energy tensor is nonzero. What does the "potential" look like from the outer to the inner surface of that shell?

If the tangent spatial components remain independent of potential and thus invariant

Do they inside the substance of the shell (i.e., between its outer and inner surfaces)?

 

[PeterDonis]

But why don't we make an attempt to calculate it?

I agree it would be an interesting calculation; if I have time in the next day or two I'll try to post one.

 

[PAllen]

This is an interesting comment. It is true that there is a sense in which the entire region of spacetime inside the horizon is "in the future" for every event outside the horizon. However, that does not mean it is logically impossible for the region inside the horizon to exist. First, a clarification: if the universe has an end point in time, i.e., if it is closed, then what I just said, and what you said in the above quote, is no longer true; the region of spacetime inside the EH of any black hole inside the closed universe gets caught up in the "big crunch" that ends the universe, just like everything else, so it is no longer true that, from the viewpoint of someone inside the EH, the outside universe "would have lived out its lifetime and would no longer exist".

Second, if we assume the universe exists forever into the future, so that a black hole can be "eternal" and the region inside remains hidden behind the EH forever, then the region inside the horizon is indeed "in the future" for any event outside the horizon--but that doesn't mean the region inside the horizon is logically impossible. If it were logically impossible, there could be no consistent mathematical models containing a region inside the horizon, and there are; any coordinate chart that is nonsingular at the horizon constitutes such a consistent mathematical model, and several have been named in this thread.

You could still claim that the region inside the horizon was *physically unreasonable*, but I've already laid out what that claim requires: it requires you to believe that physics suddenly starts working completely differently at the horizon, for no apparent reason.

One more thing: the interior region is in the infinite future from the point of view any external observer, using a typical simultaneity convention. From the point of view of an interior observer, their history from the horizon to the singularity corresponds to perfectly finite (and often short) period of external history. Further, nothing prevents an external observer from ignoring arguably optical effects and using Kruskal simultaneity to establish a relationship between interior events and exterior events. This would be analogous to an accelerating rocket choosing to ignore that light from distant objects can't catch them, and use Minkowski coordinates rather than Rindler coordinates. They simply say, I saw Earth until last light time T_final, I believe it's still there, evolving normally, rather than vanished or frozen.

 

[PAllen]

This seems to me to be trivializing the value and/or uniqueness of a coordinate system AS a true and faithful representation of some metric, within it's regime of applicability. How can standard SM and ISM for example be making the same predictions? Both claim to be a mathematically correct mapping of gravitationally depressed metric components to that of a distant coordinate observer. So when said observer checks for redshift, or determines whether a 'test sphere' remains perfectly spherical under the telescope (duly allowing for light bending etc), there is not a problem? If SSM predicts the test sphere will be viewed as oblate owing to radial length contraction, while ISM predicts the sphere will remain perfectly spherical (that is after all what 'isotropic' implies), I see a problem. And how is that sensibly resolved?

If this is the point you want to argue, I am sorry, there is no point in further discussion. There is no such thing as a unique correct coordinate system. There are some that make a class of calculations easier, or make certain aspects of reality more apparent (but others less apparent). In particular, Schwarzschild coordinates are extremely misleading for describing the local experience of a near horizon observer; yet they have many advantages for other purposes. You need to take this to the math forum and explain that differential geometry is invalid.

 

[Dale]

Well this gets down to whether SM (standard OR isotropic) is to be treated as a unique description of the spacetime surrounding a stationary spherically symmetric mass, as determined by coordinate values. You do not agree that Birkhoff's theorem http://en.wikipedia.org/wiki/Birkhoff's_theorem_(relativity) is saying just that?

Here again, you are confusing the metric with the coordinates. Birchoff's theorem shows that the Schwarzschild spacetime is the unique manifold for a spherically symmetric mass. It says nothing about Schwarzschild coordinates, which are not some uniquely valid coordinates.

Could you detail which such features?

Specifically the anisotropy. It is an artifact of the standard Schwarzschild coordinates only. When you change coordinates to an isotropic system, like Minkowski, then of course it disappears.

If you were sailing you might measure horizontal distance in nautical miles and vertical distances in fathoms. In such a coordinate system the metric would be anisotropic. When you got off the boat and started walking around on land you might measure all distances in meters. In such a coordinate system the metric would be isotropic and it would be scaled differently from the other metric.

Would you claim that there is some big tear in spacetime or some huge logical failure in physics because of those changes? If not, then perhaps you can understand why the rest of us are underwhelemed by the "gravity" of the problem.

 

[PAllen]

Well fine, you have committed to that the interior temporal metric component is depressed. Kindly commit to whether iyo the interior spatial metric components have depressed values, and would that be for all components equally, and equal to the temporal component?

Ignoring the shell, you can see how to fit Schwarzschild geometry to flat spacetime by looking at the isotropic coordinates. Using the conventions you can look up in wikipedia, if you simply 'freeze' r1 at the shell value, treating it constant inside, you have an interior metric of the form:

c^2 d tau ^2 = b^2 c^2 dt^2 - a^2 (dx^2 + dy^2 + dz^2)

where a and b are constants computed from shell r1 and other constants. This metric form obviously smoothly joins exterior metric, is diagonal, and flat (all metric derivatives are zero). Further, if we simply scale t, x, y, and z based on constants b and a, we recover the Minkowski metric inside. This re-scaling can be done globally, and now you have the natural point of view of an interior observer: I'm all Minkowski inside, the asymptotic infinity of the outside is also flat but scaled. In the original coordinates, it is the asymptotic infinity observer who is normal Minkowski, and the interior that is scaled Minkowski. The outside observer can also, of course, transform the outside to the more common Schwarzschild coordinates, to make some calculations easier, and some of their perceptions more apparent. No physical predictions are changed by any of these coordinate transforms. For that, I ask you to read any intro to differential geometry, as covered in typical intro to GR books (no need to study the full formality of math text treatment).

 

[Passionflower]

I agree it would be an interesting calculation; if I have time in the next day or two I'll try to post one.

That would be great as I think there are a few challenges.

For starters we cannot use an observer at infinity as it would take an infinite time for any signal to reach this observer. If we pick a very large r value we still would have to make sure this observer accelerates albeit very lightly.

I could propose a scenario:

Black hole: rs=1 (M = 1/2) to simplify our formulas.
Observer station S: Stationary at r=100,000 (if that is far enough? We could take something farther away or closer if we worry about the time delay)
Probe B: Free falling from r=100,000 and instantly decelerating to stationary at r=1.01

Of course feel free to change any of those numbers if you think they are unsuitable.

Would that be a good starting point for you?
If so, what do you propose next? The probe to send messages at fixed intervals? Or alternatively we could have a small rocket being sent back to the observer station as well.

I am flexible but not with "we assume the observer station is at infinity and does not need to accelerate" because obviously then it would take forever to get any signal from the probe to the observer station.

Makes sense?
Others want to pitch in as well?

 

[PeterDonis]

I could propose a scenario:

Black hole: rs=1 (M = 1/2) to simplify our formulas.
Observer station S: Stationary at r=100,000 (if that is far enough? We could take something farther away or closer if we worry about the time delay)
Probe B: Free falling from r=100,000 and instantly decelerating to stationary at r=1.01

In general this looks reasonable, except that I would want the probe to be in free-fall always, no acceleration to a stop above the horizon. The key thing to determine is, if the probe emits a light signal at fixed intervals of its own proper time, what is the last r value > 1 (i.e., above the horizon) where a signal is emitted?

I don't know about the exact numerical values; I would first work the problem leaving them undetermined, calling the observer's radius r_O and the probe's radius r_P (the latter would of course be a function of time). I agree that r_O has to be finite.

Setting rs=1 is fine as that basically just scales the r and t coordinates as r / 2M, t / 2M, which is often done in the GR formulas anyway (a lot of the analysis in MTW is done this way, IIRC).

 

[Passionflower]

In general this looks reasonable, except that I would want the probe to be in free-fall always, no acceleration to a stop above the horizon.

No problem, but it will make the calculations slightly more difficult.

The key thing to determine is, if the probe emits a light signal at fixed intervals of its own proper time, what is the last r value > 1 (i.e., above the horizon) where a signal is emitted?

Well that is not that hard. I think think that only depends on the r-value we pick for the stationary space station. The probe traveling from the space station to the EH will send a given number of signals, unless the last signal is exactly at r=rs all signals will eventually be received by the space station. (However with the caveat that the signals will be dimmer as well and if the signal is below the Planck value the space station would not pick it up but I assume you want a classical answer to the question).

Also I hope you agree the space station must accelerate because otherwise it starts to move towards r=rs as well. And I suspect it will make a difference when we consider limit conditions.

Setting rs=1 is fine as that basically just scales the r and t coordinates as r / 2M, t / 2M, which is often done in the GR formulas anyway (a lot of the analysis in MTW is done this way, IIRC).

Yes, that makes reading those formulas a lot easier.

 

[Passionflower]

One more thing: the interior region is in the infinite future from the point of view any external observer, using a typical simultaneity convention. .

Could you explain this a bit more.

Lets have two observers at r=100,000 (rs=1) one is free falling and another one is stationary, what is exactly in the infinite future according to you for each observer?

 

[PeterDonis]

Also I hope you agree the space station must accelerate because otherwise it starts to move towards r=rs as well.

Yes. The space station is supposed to "hover" at r = r_O.

 

[PeterDonis]

Lets have two observers at r=100,000 (rs=1) one is free falling and another one is stationary, what is exactly in the infinite future according to you for each observer?

I'll pitch in here since I used the "infinite future" bit too. The crucial thing is not a particular observer's state of motion, per se, but what simultaneity convention is used--what set of spacelike lines (or hypersurfaces, if we include the angular coordinates) count as "lines of simultaneity".

If the simultaneity convention is that of exterior Schwarzschild coordinates, then the horizon and the interior region are in the "infinite future" for both the observers you mention. That's because all the lines of simultaneity with finite values of the Schwarzschild t coordinate are in the exterior region; on a Kruskal chart, they are lines emanating from the "center point" where the two null "horizon" lines cross, and spreading out into the exterior region at various angles up to 45 degrees. The limit of those lines is the future horizon itself, which has a "t coordinate" of plus infinity, loosely speaking.

If the simultaneity convention is that of Kruskal coordinates, however, then the lines of simultaneity are simply horizontal lines on the Kruskal chart, and those cover the future interior region as well as the exterior. We could also use lines of constant Painleve "time" or ingoing Eddington-Finkelstein "time" as lines of simultaneity; those look more complicated on a Kruskal chart but they also cover the future interior region as well as the exterior. So in all these cases events in the future interior region have finite "time" coordinates. So with this convention, both of the observers you mention would *not* see the horizon and the future interior as being in the "infinite future".

If we wanted to draw a distinction between the "time" perceived by the two observers, we could say that the Schwarzschild lines of simultaneity are the "natural" ones for the hovering observer, while the Painleve lines of simultaneity are "natural" to the infalling observer. In *that* case, then the future interior region would be in the "infinite future" for the hovering observer but not for the infalling one. But each observer could, in principle, choose a simultaneity convention other than the "natural" one for his state of motion. The physics is the same.

 

[Q-reeus]

Originally Posted by Q-reeus:
"In this case though, you are, like it or not, opting for an inexplicable contraction of the tangent (i.e. azimuthal) spatial components in traversing the shell from outer to inner radius."
It is perfectly explicable. You are changing from anisotropic coordinates to isotropic coordinates.

So you have elected to answer on PeterDonis's behalf here. Alright, but firstly note I was above making a basically rhetorical comment to the effect such azimuthal variations would be 'voodoo'. But regardless, to answer your comment, how on Earth do you arrive at your conclusion of anisotropic -> isotropic? As far as I'm concerned, one sticks with spherical coordinate system, but one finds that the metric components either do or do not undergo physically real change in traversing the shell (as predicted by SM, that is). No chopping and changing of coordinate system.

Originally Posted by Q-reeus: "An exterior metric ('chart' if you insist)"

Here is further evidence of this confusion in your mind. There is a huge difference between the metric and a chart. They are not synonymous or interchangeable.

That criticism has been repeated now so often, decided to do a little searching and found this: http://casa.colorado.edu/~ajsh/schwp.html

"Schwarzschild metric

Schwarzschild's geometry is described by the metric (in units where the speed of light is one, c = 1)
ds² = - (1-r_s/r)dt²+(1-r_s/r)^-1dr²+r²do² .
The quantity ds denotes the invariant spacetime interval, an absolute measure of the distance between two events in space and time, t is a `universal' time coordinate, r is the circumferential radius, defined so that the circumference of a sphere at radius r is 2pi*r, and do is an interval of spherical solid angle."

Well is this right or wrong then, because seems clear enough SM here is described entirely in a slightly compact form of SC's, just as I thought was so.
And that exprerssion is clearly showing anisotropy of spatial components - of the metric, just as I thought it should.
Much earlier on I pointed out that the physically significant redshift formula lifts straight out of SC's. But you will maintain it is meaningless because coordinates are just chalk lines drawn on the ground and in no way tell us what the 'real metric' is all about? This response btw is to cover your #192 also.