On the nature of the infinite fall toward the EH

[harrylin]

Except ad nauseum, the t you are referring to is not a physical quantity in the theory at all. Its going to infinity has no physical meaning.

Reference please! - I wonder if this is the point where "the theory" becomes too poorly defined...

 

[PAllen]

Interesting - thanks!

And not so interesting when I posted exactly the same link for you over a week ago?

 

[PeterDonis]

in view of the IMHO correct non-acceptance of Einstein's twin paradox solution

You're going to have to elaborate on this, because I don't know what you're talking about; I'm quite familar with the Usenet Physics FAQ entry you linked to and I don't see anything like this. I suspect this is yet another issue of interpretation.

 

[harrylin]

And not so interesting when I posted exactly the same link for you over a week ago?

Sorry I did at the time not see a link to a refutation of the 2007 paper. Maybe you forgot to mention it, or maybe I overlooked your post.

 

[rjbeery]

All you have to do now is support your "almost black hole" with some detailed math.

We've already gone over in this thread (and several others) why GR doesn't predict this "almost black hole". So , you must be discussing some other non-GR theory.

Wait a minute, Pervect, you're moving goal posts. You said:

But let's move a bit onto the observational side and away from the math for a little bit.

There's clearly something very massive and rather dark at the center of our galaxy - we can see the orbits of stars around - something.

http://arxiv.org/abs/astro-ph/0210426 "Closest Star Seen Orbiting the Supermassive Black Hole at the Centre of the Milky Way"

Furthermore, it's very black.

I gave brief, qualitative description of an explanation for this. Now you're retreating back to GR? As I said before:

Actually, I'm trying to separate reality from idealized mathematical models.

Are you suggesting that GR tells the whole picture? Because on this point I'm certain you would be in the minority.

 

[PeterDonis]

Reference please!

How about every GR textbook that emphasizes, over and over, that coordinates have no physical meaning, and that you have to look at invariants to extract the physics? Look through MTW, for example, some time and count how many times they say this or something like it.

 

[stevendaryl]

The authors suggests next that a test particle will cross the horizon, although that can literally never happen according to Schwarzschild. And their claim that "the time parameter t [..] is not suited to describe the physical problem at hand" implies that the solution is not suited for describing physical reality until the end of time. More complete than t->∞ is physically impossible.

t is a label placed on a set of events in spacetime. There is no guarantee that a labeling scheme will give a label to every event. The counterexample that is easy to work with is Rindler coordinates. In two dimensional spacetime, we have coordinates X and T and a metric given by $ds^2 = g^2 X^2 dT^2 - dX^2$. Now, suppose at time $T=0$ you shine a flashlight in the negative-X direction. If I haven't screwed up, then the light signal will approach $X=0$ asymptotically according to:

$X = X_0 e^{-gT}$

So it never gets to $X=0$, and you might think it nonsense to ask what happens to the signal after crossing $X=0$. However, if we switch from Rindler coordinates back to Minkowsky coordinates, we see:

$x = X cosh(gT)$
$t = X sinh(gT)$

The path of the light signal is
$x = X_0 - t$

So $x=0$ after a finite amount of time, according to time coordinate $t$. The point $x=0, t=X_0$ in Minkowsky coordinates corresponds to the "point" $X=0, T=\infty$ in Rindler coordinates. The point $x < 0, t > X_0$ takes place after $T=\infty$.

You cannot simply say that because the time coordinate running to $\infty$, the description of events must be complete.

 

[PeterDonis]

Are you suggesting that GR tells the whole picture? Because on this point I'm certain you would be in the minority.

Well, let's start counting votes. I'm with pervect. Anybody else?

 

[Nugatory]

The authors suggests next that a test particle will cross the horizon, although that will literally never happen according to Schwarzschild.

When you say "according to Schwarzschild", do you mean "according to the Schwarzschild metric" or do you mean "according to Schwarzschild coordinates"? They are different things.

 

[PAllen]

Reference please! - I wonder if this is the point where "the theory" becomes too poorly defined...

General covariance = all coordinates are equally good, and none are physical per se (they are conventions). Any may equally be used to make predictions = compute invariants; invariants are coordinate independent. Einstein several times regretted that relativity wasn't called the theory of invariants instead.

General covariance was a key founding principle for Einstein, along with equivalence principle. In response to a critique by Kretschmann, he admitted it has little force a 'selector of valid theories'. But he never let go of as principle of GR, and nor does modern GR.

 

[PeterDonis]

More complete than t->∞ is physically impossible.

Why?

Here's what you should have said: Once you've taken t->infinity, going further *using that coordinate chart* is mathematically impossible, because coordinates are real numbers and there are no real numbers greater than infinity.

It's a very long way from that claim to the claim that you are making.

 

[harrylin]

You're going to have to elaborate on this, because I don't know what you're talking about; I'm quite familar with the Usenet Physics FAQ entry you linked to and I don't see anything like this. I suspect this is yet another issue of interpretation.

I referred to the IMHO correct non-acceptance of Einstein's "induced real gravitational fields".

 

[PeterDonis]

if the rate of Hawking radiation is inversely proportional to the BH's radius (which at this point the radius would be at a lower bound)

Hawking radiation, whatever the correct final theory of it may be, is irrelevant for assessing whether the object at the center of our galaxy is a black hole. It's important theoretically for the study of quantum gravity theories, but practically it's irrelevant. In fact, that goes for all of the "black hole candidates" we've detected thus far.

 

[PeterDonis]

I referred to the IMHO correct non-acceptance of Einstein's "induced real gravitational fields".

I see. What does that have to do with "proper time advocacy"? Or were you just saying you are a "contrarian" in more than one area?

[Edit: I should clarify that, while I would certainly call non-acceptance of proper time as physical "contrarian", I wouldn't say the same about non-acceptance of "induced real gravitational fields". The presence of tidal gravity is a clear physical observable that distinguishes "real" gravitational fields from what the Usenet Physics FAQ calls "pseudo" ones, and that distinction is accepted in GR. So GR does not accept "induced real gravitational fields"--"induced" fields are "pseudo" fields with zero tidal gravity and would not be considered "real".]

 

[rjbeery]

Imagine a classical theory that matches GR except as follows:

"local physics ceases to be governed by SR, and instead local physics freezes, whenever the normal progress of local physics would lead to formation of a horizon."

would be indistinguishable by a distant observer from classical GR (which has, built into its mathematical and physical foundations, that local physics is always, everywhere, governed by SR).

This modified GR, would, indeed, predict a 'frozen' star or frozen stellar cluster (for large galactic central clusters) that is externally indistinguishable from a BH.

There are some quantum approaches proposed, which rationalize this modification (which is pretty silly classically). Krauss et. al. is one; there are others. I believe the majority view remains that quantum effects do not forestall the formation of an event horizon (though its behavior is not strictly classical); nor do quantum effects prevent that matter crosses the EH. However, quantum effects are presumed to prevent any singularity and avoid the information paradox.

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.

 

[harrylin]

General covariance = all coordinates are equally good, and none are physical per se (they are conventions). Any may equally be used to make predictions = compute invariants; invariants are coordinate independent. Einstein several times regretted that relativity wasn't called the theory of invariants instead.

General covariance was a key founding principle for Einstein, along with equivalence principle. In response to a critique by Kretschmann, he admitted it has little force a 'selector of valid theories'. But he never let go of as principle of GR, and nor does modern GR.

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.

 

[PeterDonis]

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.

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.

 

[pervect]

I've come up with a somewhat simpler approach for presenting the solution for the EF geodesic equations.

Let r, v be the Eddington-Finklesein (EF) coordinates, which are presumed to be functions of proper time $\lambda$. Then let:

$$\dot{r} = \frac{dr}{d\lambda} \hspace{3cm} \dot{v} = \frac{dv}{d\lambda}$$

The Ingoing Eddington Finklestein metric (gemoetrized) is
http://en.wikipedia.org/w/index.php?title=Eddington–Finkelstein_coordinates&oldid=516198830

$$-(1-2m/r) dv^2 + 2\,dv\,dr$$

The Christoffel symbols are:

$$\Gamma^{v}{}_{vv} = \frac{m}{r^2}$$
$$\Gamma^{r}{}_{vv} = \frac{m(1-2m/r)}{r^2}$$
$$\Gamma^{r}{}_{vr} = \Gamma^{r}{}_{rv} = -\frac{m}{r^2}$$So we can write the geodesic equations as

$$\ddot{v} + \Gamma^{v}{}_{vv} \dot{v}^2 = \frac{d \dot{v}}{dr} \dot{r} + \frac{m}{r^2} \dot{v}^2 = 0$$

$$\ddot{r} + \Gamma^{r}{}_{vv} \dot{v}^2 + 2 \,\Gamma^{r}{}_{vr} \dot{v} \dot{r} = \frac{d \dot{r}}{dr} \dot{r} + (1-2m/r)\frac{m}{r^2} \dot{v}^2 - 2 \frac{m}{r^2} \dot{r}\dot{v} = 0$$

Note that I've used the chain rule to write
$$\ddot{v} = \frac{d^2 v}{d \lambda^2} = \frac{d \dot{v}}{d \lambda} = \frac{d \dot{v}}{dr}\frac{dr}{d\lambda} = \frac{d \dot{v}}{dr} \dot{r}$$

Then we can write the solution in infalling EF coordinates for m=2 as:

$$\dot{r} = - \sqrt{\frac{4}{r}}$$
$$\dot{v} = \frac{\sqrt{r}}{2+\sqrt{r}}$$

And just plug them into the geodesic equations above to demonstrate that they are a solution.

Note that $\dot{r}$ is negative, the first post had a sign error for the equivalent expression.

The math here is only mildly obnoxious compared to the previous expressions, though I've skipped over a lot of textbook stuff like computing the EF metric (the Wiki does that), and computing the Christoffel symbols for said metric.

[add]If you want r as a function of $\lambda$, it remains
$$r = \left(9 \lambda^2\right)^{\frac{1}{3}}$$

 

[PAllen]

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.

As long as you agree not to call it GR, and not to promote it in these forums (see the rules), that is fine by me. In fact, it would be quite instructive to work out how to make such a theory precise (I just gave a hand wave description of it; I know how I would start trying to make it precise, if I cared to, but don't know what logical or mathematical conundrums might crop up).

 

[PAllen]

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.

 

[PeterDonis]

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".

 

[PAllen]

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.]

 

[rjbeery]

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.

 

[Nugatory]

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 Schwarzschild solution for the vacuum around a spherically symmetric non-rotationg massive body in either Schwarzschild 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.

 

[rjbeery]

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.

 

[pervect]

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.

 

[rjbeery]

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.

Additionally...

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.

 

[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.

 

[stevendaryl]

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.

 

[rjbeery]

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.

 

[pervect]

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".

 

[PAllen]

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.

 

[rjbeery]

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".

 

[PeterDonis]

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.

 

[PeterDonis]

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.