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On the nature of the infinite fall toward the EH 
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#55
Dec512, 04:14 PM

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#56
Dec512, 04:20 PM

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A better way to say it might be: "for all events covered by both charts, all invariants at those events come out the same when computed in both charts". 


#57
Dec512, 04:51 PM

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1) We must add a couple of new axioms the theory: Universes containing naked singularities are prohibited (as a corollary, closed universes are prohibited because event horizons cannot technically be defined for them; the required new law I give next cannot be stated for a closed universe). Eternal WHBH are prohibited. (Much stronger than 'we think not physically plausible'). 2) We supplement the EFE with a new universal boundary law: The universe is bounded (chopped in spacetime) such that the world line of any particle or fluid element always has null paths extending from it to null infinity. Don't you find it contrived to muck up a beautiful, elegant theory with such additions? [Edit: An example of how strange this modified theory is shown by examining the history of a late infaller for an OS type collapse. It is a requirement of this theory that some matter disappear from existence at a certain finite local time. Freezing won't work. The reason is that a late infaller following the collapse has their world line chopped at the horizon, and this late infaller has encountered no matter on the way. There is no possible way to avoid this while keeping the EFE in any form. This means that all the orginal collapsing matter vanished, not just froze. To avoid this, we need to modify the EFE itself such that matter world lines in the collapsing body follow different spacetime trajectories than the EFE predicts, such that the dust boundary exists outside the EH when the later infaller encounters it (at the horizon). This new prediction cannot be accommodated without significant change to the EFE itself.] 


#58
Dec512, 04:59 PM

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If Alice's local "rate of t" were reduced to zero then then Alice would never know it; she would simply freeze and be oblivious to it for eternity. To be clear, I'm not saying this is what happens at the EH according to GR, I'm just pointing out that the usual refutation against the distant observer proclaiming that Alice freezes is that time does not slow down locally in her frame according to her experience; this on its own is not a valid refutation. 


#59
Dec512, 05:07 PM

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I think the weaker formulation is both more practical and more widely accepted. Consider, for example, the way that twodimensional hyperbolic coordinates allow me to make predictions only in one quadrant of a plane, whereas Cartesian coordinates work for the entire plane. No one would seriously argue that the broader Cartesian coordinates are illegitimate because they make predictions where hyperbolic coordinates don't. But this is basically the situation that we have when we write the Schwarzchild solution for the vacuum around a spherically symmetric nonrotationg massive body in either Schwarzchild coordinates or (for example) KS coordinates. We never get disagreeing predictions, but we do find regions of spacetime where the KS coordinates make predictions and the SC coordinates do not. Some of these predictions (both in and out of the region of overlap) may strike us as nonphysical, but that's not a problem with the coordinates. 


#60
Dec512, 05:12 PM

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#61
Dec512, 05:22 PM

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One can certainly say that Alice appears to freeze according to the coordinate time "t". But is this physically significant? It might be instructive to consider Zeno's paradox. I'll use the wiki definition of the paradox. At a zeno time of 1, Achilles is 50 meters behind the tortise. At a zeno time of 2, Achillies is 25 meters behind the tortise At a zeno time of n, Achillies is 100/(2^n) meters behind the tortise. Then, as n goes to infinity, Achillies is always behind the tortise. So, in "zeno time", Achilles never does catch up with the tortise, even as "zeno time" appoaches infinity. Are we therefore justified in claiming that Zeno was right, and that Achilles never catches the tortise? I don't think so, and I'd be more than a bit surprised if anyone really believed it. (I could imagine someone who likes to debate claiming they believed it as a debating tactic, I suppose  and to my view this would be a good time to stop debating and do something constructive). So in my opinion, the confusion arises by taking "zeno time", which is analogous to the Schwarzschild coordinate time "t", too seriously. While it is correct to say that as t> infinity Alice never reaches the event horizon, just as Achilles never reaches the tortise in zeno time, it still happens. It's just that that event hasn't been assigned a coordinate label. 


#62
Dec512, 05:30 PM

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Additionally... 


#63
Dec512, 05:35 PM

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#64
Dec512, 05:41 PM

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#65
Dec512, 05:44 PM

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#66
Dec512, 06:20 PM

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I notice that you aren't saying that you *do* postulate a preferred frame, are you in fact doing so? Are you saying that there isn't any other way to save your viewpoint? I'm getting a sort of debate feeling here, with this sudden lack of specificity, with all the "I could" and "I might". 


#67
Dec512, 06:35 PM

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#68
Dec512, 06:39 PM

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#69
Dec512, 10:24 PM

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To expand on this somewhat: for Alice's local "rate of time flow" to be reduced to zero, she would have to be traveling on a null worldline, not a timelike one. Since the SC chart is singular at the horizon, you can't actually compute directly what Alice's "local rate of t" there is in the SC chart. Instead, you have to do one of two things: (1) Switch to a chart that isn't singular at the horizon, such as the Painleve chart. In any such chart, it is easy to compute that Alice's worldline is still timelike at r = 2m, not null. So her "rate of time flow" does *not* go to zero at r = 2m. (2) Compute the tangent vector of Alice's worldline, in SC coordinates, as a function of r, for r > 2m, and then take the limit of the length of that tangent vector as r > 2m. If Alice's "rate of time flow" goes to zero at the horizon, this limit should be zero. It isn't; it's positive, indicating, again, that Alice's worldline is still timelike at the horizon. This is a good example of why you can't reason about a theory from popular presentations in English; you have to actually look at the math to properly determine what the theory predicts. Otherwise you will be refuting, not the actual theory, but your misinterpretation of the theory. 


#70
Dec512, 11:17 PM

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#71
Dec512, 11:32 PM

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If you're going to take this tack, you need to give some kind of physical basis for preferring the frame you choose. Can you give one? However, there is a twist if the quantum answer does turn out to be that horizons form, but not singularities. In that case, what happens to Alice after she falls below the horizon? Classically, she would get destroyed in the singularity, but it's possible that quantum effects below the horizon could alter that fate. As far as I know, nobody has come up with a model that would allow her to eventually escape back out when the black hole finally evaporates, but I don't know that anyone has ruled out that possibility either. Even if something like that last possibility pans out, however, it will still be true, if quantum effects allow the horizon to form, that there will be a *long* period of time for Bob between T_b, the last time at which he could keep Alice from falling below the horizon, and the first time when he sees any evidence of Alice escaping back out. (By "long" I mean times of the order of 10^70 years or more, IIRC, for holes of stellar mass or larger.) 


#72
Dec612, 02:26 AM

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