# On the nature of the "infinite" fall toward the EH

by rjbeery
Tags: fall, infinite, nature
PF Patron
P: 4,463
 Quote by harrylin Good - although it isn't exactly a citation, that is in fact what I had in mind. Once more, compare: all coordinates are equally good, and none are physical per se (they are conventions) with "the time parameter t [..] is not suited to describe the physical problem at hand" For me it is a consistency requirement for a theory that all valid coordinate systems make the same predictions; both system should make the same predictions.
They do make the same predictions. Let's get to this: do you think it is somehow required in GR that if all the events (and all predictions about them) in one coordinate system are a subset of those in another coordinate system, the GR says only the coordinate system with least coverage counts? Rather than saying, woops, one coordinate system is as good as any other for what it covers, but you may have to use overlapping coordinate systems to cover the whole of existence.
PF Patron
P: 4,770
 Quote by PAllen They do make the same predictions.
I see another argument looming about what "prediction" means. :sigh:

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".
PF Patron
P: 4,463
 Quote by rjbeery Note the bolded portion. IMHO this concept isn't silly in any way, and I don't even need quantum approaches to accept it. All forms of measurement ultimately rely on c (or equivalently t). Change the "rate of flow of time" and a local observer would never know the difference. All of our instruments, including our physiological and cognitive processes, would also be changed accordingly. I'm not saying that is what happens, but I don't dismiss the idea prima facie.
Consider what this modification might look like, classically, and assuming we want to keep the coordinate independent nature of the equations of GR.

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 WH-BH 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 O-S 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.]
P: 252
 Quote by PeterDonis Do you realize what you've just done? You've just given a description of proper time. Suppose that, in addition to Alice and Bob, we have Charlie, who is hovering at a constant altitude close to, but above, the horizon. Charlie's "rate of flow of time" is slower than Bob's. How do we know? Because we can compute Charlie's proper time along his worldline, and verify that it "ticks" at exactly the same rate as his instruments, physiological and cognitive processes, etc. The *same* reasoning, and the *same* kind of computation, tells us that Alice experiences a finite amount of time to the horizon. Her instruments record a finite amount of time; her physiological and cognitive processes progress by a finite amount of time; etc. So if the reasoning applies to Charlie, it should apply to Alice as well.
This is true until "it isn't".

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.
P: 2,360
 Quote by harrylin For me it is a consistency requirement for a theory that all valid coordinate systems make the same predictions; both system should make the same predictions.
That's a stronger statement than "valid coordinate systems should never make different predictions"; the latter statement allows for the possibility that one of the coordinate systems makes no prediction in some regions.

I think the weaker formulation is both more practical and more widely accepted. Consider, for example, the way that two-dimensional 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 non-rotationg 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 non-physical, but that's not a problem with the coordinates.
P: 252
 Quote by PAllen Consider what this modification might look like, classically, and assuming we want to keep the coordinate independent nature of the equations of GR. 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 WH-BH 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?
Actually, I don't see these as mucking anything up. On a philosophical level, I think the concept of infinity has no physicality whatsoever and the Universe should be able to be described without it.
Emeritus
P: 7,204
 Quote by rjbeery This is true until "it isn't". 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.
The underlying thought process here is that there is some physically meaningful way to define a "local rate of time". Relativity doesn't necessarily say this. (I think one can make even stronger claims, but it'd start to detract from my point, so I'll refrain from now).

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.

 In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 metres, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 metres, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 metres. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise
Let's define a "zeno time" as follows. At a zeno time of 0, Achillies is 100 meters behind the tortise.

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.
P: 252
 Quote by pervect It might be instructive to consider Zeno's paradox. I'll use the wiki definition of the paradox. Let's define a "zeno time" as follows. At a zeno time of 0, Achillies is 100 meters behind the tortise. 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 (though I wouldn't necessarily claim that someone might try to claim otherwise as a debating tactic.) 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.
That's an interesting take, Pervect. I guess Zeno's Time paradox could be dealt with by postulating a minimal quantum "unit" of time.

 Quote by Pervect The underlying thought process here is that there is some physically meaningful way to define a "local rate of time".
I can give physical meaning to this by simply postulating some arbitrary frame to be the preferred one. We then have a way to establish a true "local rate of time" as $$t_{local}\over t_{preferred}$$ as measured from the preferred frame.
PF Patron
P: 4,463
 Quote by PAllen See post #52 for context [Edit: An example of how strange this modified theory is shown by examining the history of a late infaller for an O-S 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.]
I wanted to re-post this edit separately, as it raises some crucial points.
P: 1,402
 Quote by harrylin Once more, compare: all coordinates are equally good, and none are physical per se (they are conventions) with "the time parameter t [..] is not suited to describe the physical problem at hand" For me it is a consistency requirement for a theory that all valid coordinate systems make the same predictions; both system should make the same predictions.
In general, a coordinate system is defined on a "patch": a small region of spacetime. Two different coordinate systems must make the same predictions on the overlap of the two patches. The point at which an infalling observer crosses the event horizon is not in patch described by the Schwarzschild coordinates.
P: 252
 Quote by PAllen I wanted to re-post this edit separately, as it raises some crucial points.
I don't agree with the edit. Consider the graphs y = 1/x, and y = 1/(x+1). Both lines approach zero without crossing with no problem.
Emeritus
P: 7,204
 Quote by rjbeery I can give physical meaning to this by simply postulating some arbitrary frame to be the preferred one. We then have a way to establish a true "local rate of time" as $$t_{local}\over t_{preferred}$$ as measured from the preferred frame.
It's not necessarily inconsistent with relativity to postulate some "preferred frame", but when your theory *requires* it, it's getting far, far, far outside the path of conventional SR.
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".
PF Patron
P: 4,463
 Quote by rjbeery I don't agree with the edit. Consider the graphs y = 1/x, and y = 1/(x+1). Both lines approach zero without crossing with no problem.
Hmm, I guess you could get that result, given that the chop produces a manifold without boundary (the EH is not in in the manifold), and you take your slices of constant time just the right way.
P: 252
 Quote by pervect It's not necessarily inconsistent with relativity to postulate some "preferred frame", but when your theory *requires* it, it's getting far, far, far outside the path of conventional SR. 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".
No this isn't my viewpoint; just poking holes in typical defenses of black holes. If it's true that after some time T_b, Alice cannot be saved by Bob under any circumstance as outlined in the OP, then I'm convinced that GR would allow for the formation of black holes as you and PAllen are saying. An extended discussion in this thread occurred when I brought up quantum effects, Hawking radiation, etc, and it sounds like the consensus on that is "no one knows enough to know the answer currently".
PF Patron
P: 4,770
 Quote by rjbeery 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.
This sounds good in English, but when you try to translate it into math, it turns out not to work. Which in turns means that the standard refutation, while it might not seem valid when expressed in English, *is* valid when expressed in math.

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.
PF Patron
P: 4,770
 Quote by PAllen I wanted to re-post this edit separately, as it raises some crucial points.
Yes, it does; this expresses what I was trying to get at by saying that the EFE predicts that spacetime, and Alice's worldline, continues below the horizon.
PF Patron
P: 4,770
 Quote by rjbeery Actually, I don't see these as mucking anything up. On a philosophical level, I think the concept of infinity has no physicality whatsoever and the Universe should be able to be described without it.
Using the proper coordinate chart, the entire black hole spacetime, including the portion below the horizon, can be described without using the concept of infinity. So your requirement is met.

 Quote by rjbeery I can give physical meaning to this by simply postulating some arbitrary frame to be the preferred one.
And I can simply postulate that some other arbitrary frame is the preferred one, such as Alice's (I assume you would postulate Bob's). Now what do we do?

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?

 Quote by rjbeery If it's true that after some time T_b, Alice cannot be saved by Bob under any circumstance as outlined in the OP, then I'm convinced that GR would allow for the formation of black holes as you and PAllen are saying.
And me. (Just sayin'. ) Classically (i.e., without any quantum effects included), this *is* what GR says.

 Quote by rjbeery An extended discussion in this thread occurred when I brought up quantum effects, Hawking radiation, etc, and it sounds like the consensus on that is "no one knows enough to know the answer currently".
Yes. However, as PAllen pointed out, the weight appears to be on the side of "horizons still form, but no singularities do". If that's the case, then the quantum answer to your criterion is the same as the classical answer: there is some time T_b after which Alice can't be saved by Bob, in the sense of being kept from falling below the horizon.

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.)
P: 3,180
 Quote by stevendaryl In general, a coordinate system is defined on a "patch": a small region of spacetime. Two different coordinate systems must make the same predictions on the overlap of the two patches. [..]
Yes, that is what I meant; but not only on the patch, they must not contradict each other anywhere.
 Quote by Nugatory That's a stronger statement than "valid coordinate systems should never make different predictions"; the latter statement allows for the possibility that one of the coordinate systems makes no prediction in some regions.[..]
OK - yes that's of course also fine to me. However, to me that doesn't seem to be the case here; see next.
 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 non-physical, but that's not a problem with the coordinates.
I don't follow that, and with the nature of Scwarzschild's infinite fall nothing strikes me as unphysical. Let's take an example:
 Quote by harrylin [..] Adler, Basin and Schiffer [..]: near[sic] r=2m [..] the time parameter t [..] is not suited to describe the physical problem at hand.
Contrary to their claim, for a hypothetical static universe with a far observer in rest relative tot the black hole I think that a signal that is sent straight out by the infalling observer at t=1010 can be received much later by the far observer - and similar for t=10100. Isn't that right?

 Related Discussions Computers 14 Introductory Physics Homework 1 Quantum Physics 2 Calculus & Beyond Homework 10 Special & General Relativity 5