On the nature of the infinite fall toward the EH

[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.

 

[PeterDonis]

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.

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?

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.

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.)

 

[harrylin]

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.

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:

[..] 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?

 

[harrylin]

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.

That looks to be a good summary of Rindler coordinates. However, you seem to hold that what in every other discipline are considered to be contradictory claims ("it will never happen" vs "it will happen") do not contradict each other. And we discussed that example in the thread that I linked as well as in earlier threads. As a result, for me it is not a counter example but an example if -as everyone does- we distinguish real fields from fictitious fields. That solves both paradoxes.

 

[pervect]

That looks to be a good summary of Rindler coordinates. However, you seem to hold that what in every other discipline are considered to be contradictory claims ("it will never happen" vs "it will happen") do not contradict each other.

It's no more of a paradox than the twin "paradox". In fact, it's more or less an extreme version of said paradox - A thinks it takes an infinite amount of time for something to happen, B thinks its' finite.

Similar "paradoxes" occur outside relativity, Zeno's paradox is very similar, and the answer is very similar as well. Basically one can map a finite interval of the real numbers (say 0-1) to an infinite interval (0-infinity) with a 1:1 mapping. Thus having an infinite expanse of coordinate time means nothing. Having an infinite amount of proper time does have physical significance, but the proper time here is fnite.

 

[Nugatory]

That looks to be a good summary of Rindler coordinates. However, you seem to hold that what in every other discipline are considered to be contradictory claims ("it will never happen" vs "it will happen") do not contradict each other. And we discussed that example in the thread that I linked as well as in earlier threads. As a result, for me it is not a counter example but an example if -as everyone does- we distinguish real fields from fictitious fields. That solves both paradoxes.

The claims being made are not "it will never happen" versus "it will happen".

The single claim being made is of the form "A light signal from point A will not reach point B".

 

[Dale]

Hi harrylin and rjbeery,

I would recommend that you read page 37 and 38 of Carroll's lecture notes on GR (it may be necessary to read earlier pages too if you don't understand some of the terminology used there, and of course I recommend reading the entire chapter 2).

http://arxiv.org/abs/gr-qc/9712019

A manifold, as used in GR, includes ALL of the possible coordinate charts covering the spacetime, and the charts only have to agree on tensors in their intersection.

I would also recommend paying attention to pervect's references to Zeno's paradox. It is always possible to make a t coordinate such that the arrow "never" reaches the target (where "never" means the limit of t goes to infinity as the arrow approaches the target). This is a perfectly valid coordinate system, and has a similar relationship to the proper time of a clock on the arrow as does Schwarzschild coordinates to an infalling clock. For any events covered by the Zeno coordinate chart all experimental measurements are the same as for any other coordinate chart. Do you therefore conclude that the target will not be wounded?

 

[Dale]

However, you seem to hold that what in every other discipline are considered to be contradictory claims ("it will never happen" vs "it will happen") do not contradict each other.

The claims are not contradictory because they are referring to different things. The first claim refers to the fact that the limit of the Schwarzschild t coordinate goes to infinity as the object crosses the horizon. The second claim refers to the fact that the coordinate time in other systems is finite as the object crosses the horizon. Since they are referring to coordinates of different coordinate systems there is no contradiction.

You need to be clear on what form of agreement between different coordinate systems is required by GR. Agreement means that they must make the same predictions about the outcome of any experimental measurement made in their intersection.

Coordinates themselves are not experimental measurements, but proper times are since they are experimentally measured by clocks. Both Schwarzschild and any other coordinate system agree that as a clock approaches the event horizon the proper time approaches a finite value.

 

[PeterDonis]

As a result, for me it is not a counter example but an example if -as everyone does- we distinguish real fields from fictitious fields. That solves both paradoxes.

As others have said, they are not "paradoxes", but I think by "solves both paradoxes" you mean "supports my interpretation of both scenarios, the Rindler one and the Schwarzschild one". You haven't justified that claim, though; see my post #102 in the other thread:

https://www.physicsforums.com/showpost.php?p=4185685&postcount=102

 

[stevendaryl]

Yes, that is what I meant; but not only on the patch, they must not contradict each other anywhere.

No, that's not correct. A coordinate patch doesn't say ANYTHING about what is happening off that patch.

For example, I have a street map of New York City, and I use to determine that there is no street called "Champs Elysees". Does that mean that no such street exists? No, there could very well be such a street, but not in New York City.

For the Schwarzschild geometry, there are two patches that are described by Schwarzschild coordinates:

1. r > 2GM/c², -∞ < t < ∞
2. 0 < r < 2GM/c², -∞ < t < ∞

Neither patch includes the event horizon. Does that mean that it is impossible to cross the event horizon? No, no more than the absence of a street called "Champs Elysees" in a map of New York City proves that there is no such street.

Note that the patches described above do not have any overlap. So the Schwarzschild coordinates do not provide a complete set of patches. You need at least one more patch to describe the event horizon. The Kruskal-whatever coordinates describe the event horizon, and you can use them to describe the transition between interior and exterior patches of the Schwarzschild coordinates.

 

[stevendaryl]

That looks to be a good summary of Rindler coordinates. However, you seem to hold that what in every other discipline are considered to be contradictory claims ("it will never happen" vs "it will happen") do not contradict each other.

A coordinate system on a patch cannot make a claim of the form "such and such will never happen". It can only make a claim of the form "such and such does not happen on this coordinate patch". It's just like the case with a map of New York City. The map cannot be used to prove that there is no street called "Champs Elysees". It can only be used to prove that there is no such street in New York City.

 

[stevendaryl]

That looks to be a good summary of Rindler coordinates. However, you seem to hold that what in every other discipline are considered to be contradictory claims ("it will never happen" vs "it will happen") do not contradict each other. And we discussed that example in the thread that I linked as well as in earlier threads. As a result, for me it is not a counter example but an example if -as everyone does- we distinguish real fields from fictitious fields. That solves both paradoxes.

Nothing being discussed has anything to do with real fields versus fictitious fields. I didn't say anything about fields at all. I was talking about coordinate systems. Not every coordinate system describes the entire manifold. For the 2-D plane, we can describe the entire plane using Cartesian coordinates (x,y). If we are only interested in the region |x| > |y|, we can use an alternative coordinate system R,θ, where

R = √(x² - y²)
θ = arctanh(y/x)

R and θ are perfectly good coordinates, as long as we recognize that there are points that are not described by those coordinates.

 

[rjbeery]

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.

I'm sorry but I've never been convinced that switching charts solves everything. If Bob was a PROPER (non-hovering) distant observer (as is usually the case in these scenarios) then he is located 'at infinity'. I know that Kruskal and I also suspect Painleve coordinates are not able to deal with observers at infinity, so this argument it a bit like pushing the problem under the rug.

Also, if we had a preferred frame from which Bob was measuring Alice's coordinate acceleration and declared her velocity to be absolute, he would indeed conclude that it reached c at the EH. Does that differ substantially from him declaring her to be on a null worldline?

 

[stevendaryl]

The claims being made are not "it will never happen" versus "it will happen".

The single claim being made is of the form "A light signal from point A will not reach point B".

Harrylin is (I think) talking about the infalling observer crossing the event horizon. That event is not described by Schwarzschild coordinates. I think it's getting off track to bring up the fact that the light signal from this event will never reach the distant observer. That's true, but people don't normally assume that "it's impossible for me to see X" means "it's impossible for X to happen". I don't think anyone would have a problem with an explanation of the form: "At time such and such, the infalling observer crosses the event horizon. The light signal sent at that point is bent back down by the enormous gravity, so it never reaches distant observers." Nobody would interpret that to mean that the event never happens. The weird thing about Schwarzschild coordinates is that there IS no time "such and such" at which the infalling observer crosses the event horizon. Literally, there is no time t in Schwarzschild coordinates in which this happens. The point I've been making is that that simply means the Schwarzschild coordinates are incomplete---they don't describe everything that happens.

The two facts A = "There is no time t in Schwarzschild coordinates such that the infalling observer crosses the event horizon at time t" and B = "The light signal sent from the moment the infalling observer crosses the event horizon never reaches the distant observer" are distinct facts. They are related, of course, but they're not the same.

 

[stevendaryl]

I'm sorry but I've never been convinced that switching charts solves everything. If Bob was a PROPER (non-hovering) distant observer (as is usually the case in these scenarios) then he is located 'at infinity'. I know that Kruskal and I also suspect Painleve coordinates are not able to deal with observers at infinity, so this argument it a bit like pushing the problem under the rug.

What? You can't ACTUALLY be infinitely far from a black hole. When people talk about an observer being "infinitely far away", they really mean far enough away that the gravity from the black hole is neglibible, for the purposes of calculation. It's never exactly zero.

 

[PeterDonis]

I know that Kruskal and I also suspect Painleve coordinates are not able to deal with observers at infinity

Huh? Where are you getting that? Strictly speaking, there is no "observer at infinity" in any chart, including Schwarzschild, because "infinity" isn't a valid coordinate value. But you can take limits as coordinates go to infinity in any chart. In the Painleve chart, for instance, the limit of the line element as r -> infinity is identical to the same limit in the Schwarzschild chart. In the Kruskal chart, "r" isn't a coordinate, but it's a function of coordinates, and you can certainly take the limit as that function goes to infinity, which again, gives an identical limit to that in the Schwarzschild chart. So I don't understand what you're basing your claim on.

Also, if we had a preferred frame from which Bob was measuring Alice's coordinate acceleration and declared her velocity to be absolute, he would indeed conclude that it reached c at the EH. Does that differ substantially from him declaring her to be on a null worldline?

Yes, because "null worldline" has an invariant definition in terms of the tangent vector, which has a direct physical interpretation, and declaring by fiat that some frame is "preferred" or some coordinate velocity is "absolute" does not change the tangent vector. The words "preferred" and "absolute" don't have any physical meaning; they're just labels that you are choosing to slap on things you would like to be privileged that aren't.

 

[rjbeery]

Nothing being discussed has anything to do with real fields versus fictitious fields. I didn't say anything about fields at all. I was talking about coordinate systems. Not every coordinate system describes the entire manifold. For the 2-D plane, we can describe the entire plane using Cartesian coordinates (x,y). If we are only interested in the region |x| > |y|, we can use an alternative coordinate system R,θ, where

R = √(x² - y²)
θ = arctanh(y/x)

R and θ are perfectly good coordinates, as long as we recognize that there are points that are not described by those coordinates.

Ohh Stevendaryl, you just hit on one of my biggest philosophical reasons for having a distaste for black holes. I believe reality should be describable mathematically using a single coordinate system (no patches, no infinities, etc). THAT would be beautiful to me.

 

[stevendaryl]

Yes, because "null worldline" has an invariant definition in terms of the tangent vector, which has a direct physical interpretation, and declaring by fiat that some frame is "preferred" or some coordinate velocity is "absolute" does not change the tangent vector. The words "preferred" and "absolute" don't have any physical meaning; they're just labels that you are choosing to slap on things you would like to be privileged that aren't.

I'm on your side here, but I think that there is an unresolved issue (at least for me) as to how you know that a solution to GR equations is "complete".

Suppose we take the "patch" consisting of the region of the Schwarzschild geometry

• r > 2GM/c²
• -∞ < t < +∞

and we declare that that's our universe. There is nothing else. There is no interior. What is wrong with that?

We can certainly say what's weird about it, which is that we can transform to Kruskal-whatever coordinates, and we see that the universe just stops at some boundary (I don't know what the boundary would be in terms of KS coordinates) for no good reason. But why do you need a good reason? What are the rules for completeness of a solution?

Something that someone has mention is completeness of geodesics. If there is a geodesic leading to a boundary, and nothing singular happens at that boundary, then we need to describe what happens on the other side of the boundary, to "complete" the geodesic. But is that just an aesthetic consideration, or is there some reason we must have geodesic completeness?

 

[rjbeery]

Yes, because "null worldline" has an invariant definition in terms of the tangent vector, which has a direct physical interpretation, and declaring by fiat that some frame is "preferred" or some coordinate velocity is "absolute" does not change the tangent vector. The words "preferred" and "absolute" don't have any physical meaning; they're just labels that you are choosing to slap on things you would like to be privileged that aren't.

Forget the the preferred frame, it's just confusing things. What coordinate velocity would Bob assign to Alice as she crossed the EH?

 

[stevendaryl]

Ohh Stevendaryl, you just hit on one of my biggest philosophical reasons for having a distaste for black holes. I believe reality should be describable mathematically using a single coordinate system (no patches, no infinities, etc). THAT would be beautiful to me.

That's more of a wish than a physical principle, it seems to me. For example, the surface of a sphere cannot be described by a single patch. But there is nothing weird about the surface of a sphere. Almost everywhere, you can use latitude and longitude to navigate around a sphere. But right at the North Pole, they become useless (because every direction is "South"). That's inconvenient, but it's not really a problem. If you lived at the North Pole, you would just use a different coordinate system to navigate, instead of latitude and longitude.

 

[stevendaryl]

Forget the the preferred frame, it's just confusing things. What coordinate velocity would Bob assign to Alice as she crossed the EH?

As has been pointed out, Bob's coordinate system does not describe the event of Alice crossing the event horizon. So it's sort of meaningless to ask what coordinate velocity Bob would give to an event that isn't in his coordinate system.

It's similar to using polar coordinates (r,θ) for the 2-D plane. If you have an object that is going right through the center (r=0), what is the coordinate velocity dθ/dt? It's undefined. θ switches suddenly jumps by pi when the object crosses the center. What that means is not that something catastrophic happens at r=0, but that polar coordinates are not usable at the center. They are only useful in the "patch" -pi < θ < pi, with r > 0.