On the nature of the infinite fall toward the EH

[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.

 

[PeterDonis]

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".

You know it's complete when you can't analytically extend the manifold any further. One way of testing this, as you note later in your post, is to test whether the patch you are looking at is geodesically 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?

The manifold described by this patch is geodesically incomplete, which means it can be analytically extended.

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?

Because not having it would mean that objects moving on geodesic worldlines would just "disappear" at a finite value of their proper time, without any physical reason. This violates energy-momentum conservation: where does the energy and momentum carried by the object go?

One could also give a similar argument using spacetime itself: if a spacetime were not geodesically complete, then the law that the covariant divergence of the stress-energy tensor must be zero would be violated at the boundary at which geodesics were incomplete. (In the case of Schwarzschild spacetime, the SET is identically zero because the spacetime is vacuum, but that does not prevent one from computing its covariant divergence.)

 

[PAllen]

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 responded to one aspect of this - that it is a silly requirement that would deny 2-spheres from being legitimate objects. However, in the case of the BH, it is even sillier in that there are many coordinate systems that cover the interior and exterior in one coordinate patch with no infinities except at the central singularity: Kruskal, Eddington-Finkelstein, Gullestrand-Panlieve, Lemaitre.

 

[stevendaryl]

Because not having it would mean that objects moving on geodesic worldlines would just "disappear" at a finite value of their proper time, without any physical reason. This violates energy-momentum conservation: where does the energy and momentum carried by the object go?

That's aesthetically unpleasant, but it's not really a problem. You can amend it to say that the differential form of the law of conservation of energy-momentum applies only in the interior.

One could also give a similar argument using spacetime itself: if a spacetime were not geodesically complete, then the law that the covariant divergence of the stress-energy tensor must be zero would be violated at the boundary at which geodesics were incomplete. (In the case of Schwarzschild spacetime, the SET is identically zero because the spacetime is vacuum, but that does not prevent one from computing its covariant divergence.)

Well, isn't that a little circular? The law is not a first-principle, but is PROVABLE using the assumption of geodesic completeness. If you don't assume geodesic completeness, then that law isn't provable. But it's still provable in the interior.

 

[rjbeery]

Stevendaryl responded to one aspect of this - that it is a silly requirement that would deny 2-spheres from being legitimate objects. However, in the case of the BH, it is even sillier in that there are many coordinate systems that cover the interior and exterior in one coordinate patch with no infinities except at the central singularity: Kruskal, Eddington-Finkelstein, Gullestrand-Panlieve, Lemaitre.

I can deny that 2-spheres are anything but idealized mathematical models, or I can account for them in three dimensions.

 

[PeterDonis]

That's aesthetically unpleasant, but it's not really a problem. You can amend it to say that the differential form of the law of conservation of energy-momentum applies only in the interior.

You can't do that without also modifying the rest of the theory; the law of energy-momentum conservation is not an independent assumption. See below.

The law is not a first-principle, but is PROVABLE using the assumption of geodesic completeness. If you don't assume geodesic completeness, then that law isn't provable.

Huh? The law is a mathematical identity, the Bianchi identity, that is satisfied by the Einstein tensor; therefore, by the Einstein Field Equation, it is also satisfied by the stress-energy tensor. There is no assumption of geodesic completeness that I'm aware of that is required to prove the Bianchi identity or to derive the EFE.

Unless you mean that one could simply declare by fiat that we don't allow derivatives to be defined at all on the boundary (since the Bianchi identity involves derivatives of the metric). But I'm not sure you can even get away with that without violating other continuity requirements on the manifold; in other words, you'd have to declare by fiat that the manifold structure of spacetime is not applicable at the boundary. I would have to think about that some more.

 

[Dale]

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, it differs substantially. Coordinate velocities are frame variant quantities and can easily exceed c. A null tangent vector is frame invariant and is only possible for massless particles.

I can easily come up with a coordinate system where my coordinate velocity sitting here typing this response is c, but there is no coordinate system where my worldline is null. [EDIT: and why settle for c, I can make a coordinate system where my v>>c, woohoo FTL travel solved!]

What coordinate velocity would Bob assign to Alice as she crossed the EH?

That depends entirely on the coordinate chart selected. However, note that you could not select Schwarzschild coordinates for this since they don't cover the EH. The closest you could do in Schwarzschild coordinates is the limit of Alice's velocity as she approached the EH.

 

[stevendaryl]

You can't do that without also modifying the rest of the theory; the law of energy-momentum conservation is not an independent assumption. See below.

Huh? The law is a mathematical identity, the Bianchi identity, that is satisfied by the Einstein tensor; therefore, by the Einstein Field Equation, it is also satisfied by the stress-energy tensor. There is no assumption of geodesic completeness that I'm aware of that is required to prove the Bianchi identity or to derive the EFE.

I feel that what you're saying is circular. Yes, you can prove that the differential form of the conservation of energy-momentum holds, but it doesn't imply anything about geodesics continuing.

We have a "patch" P, with a boundary B. We propose the (quite weird, I admit) rule that any geodesic that intersects B ceases to exist on the "far" side of B. How can that rule possibly violate a tensor identity?

Unless you mean that one could simply declare by fiat that we don't allow derivatives to be defined at all on the boundary (since the Bianchi identity involves derivatives of the metric). But I'm not sure you can even get away with that without violating other continuity requirements on the manifold; in other words, you'd have to declare by fiat that the manifold structure of spacetime is not applicable at the boundary. I would have to think about that some more.

Right, it would be a different kind of manifold. Derivatives are only defined in the interior.

 

[Dale]

I believe reality should be describable mathematically using a single coordinate system (no patches, no infinities, etc).

My kids believe in Santa Claus.

Do you have any evidence supporting your belief? If so, which coordinate system is the "one"?

 

[PeterDonis]

The closest you could do in Schwarzschild coordinates is the limit of Alice's velocity as she approached the EH.

And just for extra fun, the limit as r -> 2m of Alice's coordinate velocity, dr/dt, is *zero*, not c, in Schwarzschild coordinates. Alice's coordinate velocity goes to c at r = 2m in *Painleve* coordinates. None of which changes anything physically, but we might as well get all the coordinate velocities out on the table for what it's worth.

 

[stevendaryl]

I can deny that 2-spheres are anything but idealized mathematical models, or I can account for them in three dimensions.

Right, you can declare that the only geometry is Euclidean geometry, but there is no reason to do that.

 

[Dale]

Right, it would be a different kind of manifold. Derivatives are only defined in the interior.

Manifolds have open boundaries, so I think that you can take derivatives all the way to the edge. That said, I don't know anything about geodesic completeness.

 

[Dale]

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.

I can deny that 2-spheres are anything but idealized mathematical models, or I can account for them in three dimensions.

Stevendaryl's statement is true for any manifold which is topologically the same as a sphere, which could very well be true for the universe as a whole. Also, the embedding space works for a 2-sphere but doesn't help in GR since we don't know of any 5th or higher dimensions in which to account for spacetime manifolds.

 

[PAllen]

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

Whatever he wants. Pick any value, and there is a coordinate system the produces that value. Remember, there is no specific coordinate system Bob must use. You can adopt, as a reasonable rule, that Bob should use a coordinate system that matches his local inertial frame near near each event on his world line. But since there is no such thing as a global inertial frame, that still leaves great freedom for how coordinates are assigned further and further from Bob's world line. It is, indeed, easy to construct a coordinate system that approaches local inertial coordinates near Bob's world line and assigns any coordinate velocity you want to Alice at horizon crossing. As with any such coordinate question, yours has no physical meaning.

A physical question would be e.g. what redshift does Bob see for Alice as Alice approaches the horizon. And the coordinate independent answer is obviously redshift factor approaches infinite.

 

[stevendaryl]

Manifolds have open boundaries, so I think that you can take derivatives all the way to the edge. That said, I don't know anything about geodesic completeness.

That's a good point. That's a counter-argument to PeterDonis' claim that the EFE implies geodesic completeness. If the manifold is an open set, then the EFE would be satisfied at every point in the manifold, whether or not there is geodesic completeness. Similarly, the Bianchi identities would be satisfied at every point. So I don't think that anything would be violated by simply declaring that nothing exists outside the manifold.

 

[martinbn]

About geodesic completeness, the theorems of Hawking and Penrose show that "quite often" there are geodesicly incomplete unextendable manifolds.