Schwartzschild exterior and interior solutions

[TrickyDicky]

KS don't have different boundary conditions. They are merely coordinate change followed by extension (which you can choose to make or not).

I don't believe there is any limitation of the coordinate definition of AF unimodular transforms. The criterion is simply:

If you can convert to coordinates with one timelike and 3 spacelike, that meet the coordinate conditions for AF, THEN the *geometry* is AF (a feature of geometry independent of coordinates).

Your only reference to unimodular transforms was to a t'Hooft document where George Jones indicated that what t'Hooft was saying was the idea that there is any limitation on coordinates was a mistake.

Let's check if we agree on anything:

Do you at least agree that there are two different definitions of asymptotically flat spacetime, one coordinate-dependent and one coordinate-free (the one currently used?

Do you agree that AF is a boundary condition at infinity of the vacuum solution of the Einstein equations?

Do you agree that the coordinate-dependent AF boundary condition restricts coordinate transformation to only those that are unimodular (g=1) while the coordinate free AF boundary condition doesn't have that restriction on coordinate substitutions?

Do you agree that in order to find the KS spacetime vacuum solution we must choose the coordinate free (current) AF definition because otherwise the U, V, change of coordinates instead of t,r, wouldn't be possible with the other choice of boundary condition that only permits unimodular coordinate transformations?

Please tell me which of those you don't agree with so that we can move on from there.

 

[PAllen]

Let's check if we agree on anything:

Do you at least agree that there are two different definitions of asymptotically flat spacetime, one coordinate-dependent and one coordinate-free (the one currently used?

Yes. However, I don't know that they are different in substance. I believe the coordinate free definition was not meant to give different answers, but simply to be applicable without having to find appropriate coordinates.

Do you agree that AF is a boundary condition at infinity of the vacuum solution of the Einstein equations?

It can be used as one. It can also simply be applied as a test of an arbitrary solution.
If the coordinate definition is used as a boundary condition, it will lead to *expression* of the solution in only certain types of coordinates. This is the real value of the coordinate indpendent definition - it does not artificially limit the expression of the solution.

If you find a solution using coordinate AF boundary conditions and transform to any other coordinates, you are still satisfying the same criterion of AF, and the same boundary conditions (though they might not be expressible in the new coordinates).

Do you agree that the coordinate-dependent AF boundary condition restricts coordinate transformation to only those that are unimodular (g=1) while the coordinate free AF boundary condition doesn't have that restriction on coordinate substitutions?

I disagree. It only restricts the initial expression of the solution, or the form of coordinate you must use to apply it as a test. It says nothing about other coordinates you may introduce to understand different aspects of the geometry (and coordinate transforms cannot change either the intrinsic geometry or topology).

Do you agree that in order to find the KS spacetime vacuum solution we must choose the coordinate free (current) AF definition because otherwise the U, V, change of coordinates instead of t,r, wouldn't be possible with the other choice of boundary condition that only permits unimodular coordinate transformations?

To find the KS solution directly (in KS coordinates) would require expressing the boundary condition in a coordinate independent way. I disagree with the rest of this statement. That is, I don't see the coordinate expression you find your solution in says anything about what other coordinates you may use for analysis.

Please tell me which of those you don't agree with so that we can move on from there.

 

[TrickyDicky]

From your answer I would say that there's no practical difference for you between the two definitions with respect to the vacuum solution. So no choice is really necessary.
Good, that is a way of seeing it, it makes me wonder,though, why the wikipedia page and the Townsend reference seems to imply that coordinate-dependent asymptotical flatness excludes black holes in the first place, if the old definition didn't exclude black holes, why change it?

 

[PAllen]

From your answer I would say that there's no practical difference for you between the two definitions with respect to the vacuum solution. So no choice is really necessary.
Good, that is a way of seeing it, it makes me wonder,though, why the wikipedia page and the Townsend reference seems to imply that coordinate-dependent asymptotical flatness excludes black holes in the first place, if the old definition didn't exclude black holes, why change it?

I do not read the wikipedia article as saying this at all. It says asymptotic simplicity (rather than the weak asymptotic simplicity) precludes black holes. I don't see anywhere it states or implies that coordinate definition of AF precludes black holes, or requires asymptotic simplicity. The definition was generalized for other reasons:

"Around 1962, Hermann Bondi, Rainer Sachs, and others began to study the general phenomenon of radiation from a compact source in general relativity, which requires more flexible definitions of asymptotic flatness" (from the Wikipedia article).

I can't find any reference at all to coordinate definition of AF in the Townsend paper. Can you provide a page number?

 

[TrickyDicky]

I do not read the wikipedia article as saying this at all. It says asymptotic simplicity (rather than the weak asymptotic simplicity) precludes black holes. I don't see anywhere it states or implies that coordinate definition of AF precludes black holes, or requires asymptotic simplicity. The definition was generalized for other reasons:

"Around 1962, Hermann Bondi, Rainer Sachs, and others began to study the general phenomenon of radiation from a compact source in general relativity, which requires more flexible definitions of asymptotic flatness" (from the Wikipedia article).
I can't find any reference at all to coordinate definition of AF in the Townsend paper. Can you provide a page number

Maybe there is not a direct quotation, but we can try to deduce it from what we know and what we read, right?
Can somebody help here? Does the coordinate-dependent definition of asymptotically flat spacetime exclude black holes or not?

 

[PAllen]

A further thought on the wikipedia reference is that study of gravitational radiation from compact sources implies non-static solutions. Thus if I were to guess the motivation for coordinate free definitioin of AF (other than elegance) it would be to better deal with non-static solutions.

My intuition about the coordinate definition is that it is making requirements only on the behavior of a solution at infinity. I can hardly conceive of how it precludes any particular behavior in a finite region of spacetime unless such behavior somehow cannot be fit to the AF behavior at infinity.

Expert commentary would be very welcome here. However, I wonder that because no one uses the coordinate def. of AF anymore, it might be hard to locate an expert on its nuances.

 

[TrickyDicky]

My intuition about the coordinate definition is that it is making requirements only on the behavior of a solution at infinity. I can hardly conceive of how it precludes any particular behavior in a finite region of spacetime unless such behavior somehow cannot be fit to the AF behavior at infinity.

My understanding is that in the coordinate definition, the metric of Minkowski spacetime is approached as r → ∞, and this makes all null geodesics start and end in the fixed metric Minkowski boundary, in the case of black holes some worldlines go thru the event horizon towards the singularity where the fixed Minkowski metric is no longer valid, i.e. their endpoint can't be traced to the Minkowski coordinates,they can't be defined there, so coordinate-dependent asymptotic flatness would exclude black holes.

Expert commentary would be very welcome here. However, I wonder that because no one uses the coordinate def. of AF anymore, it might be hard to locate an expert on its nuances.

I think so too.

 

[PAllen]

My understanding is that in the coordinate definition, the metric of Minkowski spacetime is approached as r → ∞, and this makes all null geodesics start and end in the fixed metric Minkowski boundary, in the case of black holes some worldlines go thru the event horizon towards the singularity where the fixed Minkowski metric is no longer valid, i.e. their endpoint can't be traced to the Minkowski coordinates,they can't be defined there, so coordinate-dependent asymptotic flatness would exclude black holes.

At least going by the description in the wiki page you provided, I don't see this at all. It simply states you express the metric as minkowski metric plus <arbitrary deviation function>. Any metric can be put in this form. Then, it requires that this deviation, and its various derivatives go to zero as r->infinity with specified bounding orders in r. This definition suggests no limitations at all on how convoluted the deviation function is at any finite r. It suggests nothing to me about null geodesics within some finite r; for all I can see they could be circular. It could certainly be that there is some suble argument in favor of what you think, but my initial intuition is otherwise; and you haven't provided any real argument or reference.

 

[TrickyDicky]

It could certainly be that there is some suble argument in favor of what you think, but my initial intuition is otherwise; and you haven't provided any real argument or reference.

The argument is actually not subtle at all, it's pretty clear, but you are right that I'm not any good explaining it. I'll give it a try later.

 

[PAllen]

I propose a set of arguments that TrickyDicky's views about coordinate AF definition are incorrect. I state claims TrickyDicky believes follow from coordinate AF definition, and attempt to prove them false.

Claim 1: "all null geodesics start and end in the fixed metric Minkowski boundary"

This is disproven by applying the coordinate AF definition to the exterior Schwarzschild geometry over r > R (R being the event horizon). This manifold with one coordinate patch trivially satisfies coordinate AF (TrickyDicky says so himself). Yet claim 1 is demonstrably false: any radial null geodesics end approaching R and do not reach the r infinite Minkowsky boundary.

Claim 2: "coordinate-dependent asymptotic flatness would exclude black holes"

First, I should ask TrickyDicky whether he/she thinks a 2-sphere is manifold? If so, then one must admit the concept of multiple coordinate patches, each covering an open region, with overlap requirements and smooth mappings defined in the overlap regions. It is impossible to cover the 2-sphere in one coordinate patch.

Now I define a smooth manifold covering the Schwarzschild geometry from r>0 to r infinite (but not including the maximal extension typically done (but not required) via KS coordinates). I choose to use 3 coordinate patches, each avoiding any singular behavior of the metric. For r < R, the interior vacuum Schwarzschild solution; for r > R the exterior vacuum Schwarzschild solution. For r > .5R and < 1.5R I introduce KS coordinates, providing an open patch overlapping the prior patches. For r<R I use the corresponding U,V transform from Schwarzschild coordinates; similarly for r > R. For r=R, either U,V definition may be used, producing the same answer. The metric expressed in U,V is smooth and nonsingular throughout this region. This constitutes the smooth manifold I set out to define. Again, it trivially satisfies the coordinate definition of AF flatness (for which only the r>R patch is relevant). It clearly contains a black hole, invalidating claim 2.

---

I remain very interested in the alleged counter-argument.

 

[TrickyDicky]

Claim 1: "all null geodesics start and end in the fixed metric Minkowski boundary"

Forget this for a moment, I'm not the best at making geometrical postulates, let's center in easier ways to understand it

Claim 2: "coordinate-dependent asymptotic flatness would exclude black holes"

Every coordinate-dependent definition of an event horizon tells you that the event horizon lives at the boundary at infinity so it takes an infinite coordinate time to reach it for any observer.
Also coordinate-dependent AF implies that only unimodular coordinate transformations are allowed, so the Kruskal coordinate change can not be performed by definition of the coordinate-dependent AF.
So without further considerations it should be obvious that a coordinate-dependent set boundary at infinity excludes black holes because it forbids reaching them in finite time.
Therefore we need a coordinate-free definition of AF that allow us to talk about Locally defined asymptotic flatness, since the Black hole concept is only valid if we can use the Proper (local) time. This is what the Eddingto-Finkelstein and Kruskal coordinate change does by ignoring coordinate-dependent AF, and what Penrose, Geroch et al gave formal definitions and justifications to, using conformal geometry, by defining cordinate-free AF, shortly after Kruskal published his solution.

 

[atyy]

How would you reason starting from K-S coordinates? It is a solution of the vacuum field equations, and can be seen to contain a black hole without going to Schwarzschild coordinates at all.

 

[PAllen]

Every coordinate-dependent definition of an event horizon tells you that the event horizon lives at the boundary at infinity so it takes an infinite coordinate time to reach it for any observer.
Also coordinate-dependent AF implies that only unimodular coordinate transformations are allowed, so the Kruskal coordinate change can not be performed by definition of the coordinate-dependent AF.
So without further considerations it should be obvious that a coordinate-dependent set boundary at infinity excludes black holes because it forbids reaching them in finite time.
Therefore we need a coordinate-free definition of AF that allow us to talk about Locally defined asymptotic flatness, since the Black hole concept is only valid if we can use the Proper (local) time. This is what the Eddingto-Finkelstein and Kruskal coordinate change does by ignoring coordinate-dependent AF, and what Penrose, Geroch et al gave formal definitions and justifications to, using conformal geometry, by defining cordinate-free AF, shortly after Kruskal published his solution.

Please provide some reference to an established coordinate based definition of AF that says any of this. The wikipedia article you referenced says none of this. None of your other references even give any coordinate based definition of AF. Clearly, you can invent a personal definition that includes and excludes whatever you want, but that is of less interest to discuss. Even if you want to provide a personal definition, start with some semi-formal statement of it. Now, for specifics:

"Every coordinate-dependent definition of an event horizon tells you that the event horizon lives at the boundary at infinity so it takes an infinite coordinate time to reach it for any observer."

This is false. My proposed patch from r>.5R , r < 1.5R is a coordinate patch containing the event horizon and it is not a boundary at infinity. It is also false that takes infinite proper time for an observe to fall through the event horizon. As for coordinate time, I can make up coordinates where it takes infinite coordinate time to cross the street. What's that got to do with anything? It is true that external observer never sees anything cross the event horizon, but that has absolutely nothing to do with the coordinate definition of AF in your only reference (the wiki article). This is all stated in terms of limits as a radial coordinate expression goes to infinity. Time doesn't enter it all, neither does the perceptions of some particular class of observers.

"Also coordinate-dependent AF implies that only unimodular coordinate transformations are allowed, so the Kruskal coordinate change can not be performed by definition of the coordinate-dependent AF"

This is neither stated nor implied in any reference you have provided. It also makes no sense. One can require that special coordinates be used to apply a coordinate definition (as the wiki article states), but it is absurd to say that you cannot choose to do other analysis in any coordinate system you choose. So, in the patch covering r>R, I use coordinates compatible with the coordinate AF definition; that is all that is required because the coordinate AF definition only specifies requirements for limits as r->infinity.

It is obvious that these issues really cannot be discussed until you provide some reference justifying your conception of coordinate based AF. Nothing you have provided so far justifies any of these statements.

 

[PAllen]

How would you reason starting from K-S coordinates? It is a solution of the vacuum field equations, and can be seen to contain a black hole without going to Schwarzschild coordinates at all.

Actually, that is one thing TrickyDicky has answered. He is pushing the idea that the coordinate based definition of AF is substantively different from coordinate free definitions; and the coordinate criteria described in the wiki article he referred to are not applicable to KS coordinates.

 

[TrickyDicky]

If you read the wikipedia page you'd se that there is a coordinate-dep AF definition there.

"Every coordinate-dependent definition of an event horizon tells you that the event horizon lives at the boundary at infinity so it takes an infinite coordinate time to reach it for any observer."

This is false. My proposed patch from r>.5R , r < 1.5R is a coordinate patch containing the event horizon and it is not a boundary at infinity. It is also false that takes infinite proper time for an observe to fall through the event horizon. As for coordinate time, I can make up coordinates where it takes infinite coordinate time to cross the street. What's that got to do with anything? It is true that external observer never sees anything cross the event horizon, but that has absolutely nothing to do with the coordinate definition of AF in your only reference (the wiki article). This is all stated in terms of limits as a radial coordinate expression goes to infinity. Time doesn't enter it all, neither does the perceptions of some particular class of observers.

Do you deny that r=2GM is a coordinate-singularity? Do you deny that one of the coordinates is the time coordinate? Do you deny that it takes infinite time for an observer to see anything reaching the coordinate-singulrity? The fact that one can construct a manifold that extends beyond a coordinate singularity (like KS) doesn't eliminate the coordinate-singularity at that point if one keeps using the Scwartzschild line element that is coordinate-dependent AF. (meaning all its components become the minkowski components when r tends to infinity.
Try to do the same thing with the Kruskal line element
http://upload.wikimedia.org/math/7/0/f/70fb9db32a241a97dd44a7dae8edfec8.png
and you'll see that the components doesn't become Minkowski when r tends to infinity.
Q.E.D.

"Also coordinate-dependent AF implies that only unimodular coordinate transformations are allowed, so the Kruskal coordinate change can not be performed by definition of the coordinate-dependent AF"

This is neither stated nor implied in any reference you have provided. It also makes no sense. One can require that special coordinates be used to apply a coordinate definition (as the wiki article states), but it is absurd to say that you cannot choose to do other analysis in any coordinate system you choose. So, in the patch covering r>R, I use coordinates compatible with the coordinate AF definition; that is all that is required because the coordinate AF definition only specifies requirements for limits as r->infinity.

No, your coordinates are not compatible with the coordinate-dependent AF definition, your coordinates are KS.

 

[JesseM]

Do you deny that r=2GM is a coordinate-singularity? Do you deny that one of the coordinates is the time coordinate? Do you deny that it takes infinite time for an observer to see anything reaching the coordinate-singulrity?

You would agree both of these are also true of the Rindler horizon in Rindler coordinates, yes?

Try to do the same thing with the Kruskal line element
http://upload.wikimedia.org/math/7/0/f/70fb9db32a241a97dd44a7dae8edfec8.png
and you'll see that the components doesn't become Minkowski when r tends to infinity.
Q.E.D.

If you're trying to verify whether the line element approaches the Minkowski line element I don't think you can calculate the limit using that form, because in the next line they say r is defined as a function of U and V...if you want to calculate the limit of the line element as the distance approaches infinity, I think you'd have to use the space coordinate in KS coordinates, which is U, not r. If you plug r=2GM(1 + W((U^2 - V^2)/e)) (where W is the Lambert W function) into the KS line element you get:

ds^2 = \frac{16G^2 M^2}{1 + W(\frac{U^2 - V^2}{e})} e^{-(1 + W(\frac{U^2 - V^2}{e}))} (-dV^2 + dU^2) + 4G^2M^2 (1 + W(\frac{U^2 - V^2}{e}))^2 d\Omega^2

...though I'm not sure how you would go about figuring the limit as U approaches infinity here, in order to check whether it approaches the Minkowski line element (in spherical coordinates, with V for time and U for the radial coordinate) ds^2 = -dV^2 + dU^2 + U^2 d\Omega^2

 

[PAllen]

If you read the wikipedia page you'd se that there is a coordinate-dep AF definition there.

Yes, and that is the definition I applied. You are adding all sorts of additional requirements, not stated there, that do not follow from it, and you don't provide justification for them.

Do you deny that r=2GM is a coordinate-singularity?

In some coordinate systems. So what? Rindler coordinates have coordinate singularity, though they describe flat Minkowski space.

Do you deny that one of the coordinates is the time coordinate? Do you deny that it takes infinite time for an observer to see anything reaching the coordinate-singulrity?

This is only true for some observers. For others, it takes only finite proper time to reach and cross the horizon. You can even see this by constructing Fermi-Normal coordinates for an infalling near horizon observer.

The fact that one can construct a manifold that extends beyond a coordinate singularity (like KS) doesn't eliminate the coordinate-singularity at that point if one keeps using the Scwartzschild line element that is coordinate-dependent AF.

A coordinate singularity has no physical significance per se, at all. An event horizon has observational significance for some observers and not others (whether we are talking about black holes or the Rindler horizon for accelerated observers).

(meaning all its components become the minkowski components when r tends to infinity.
Try to do the same thing with the Kruskal line element
http://upload.wikimedia.org/math/7/0/f/70fb9db32a241a97dd44a7dae8edfec8.png
and you'll see that the components doesn't become Minkowski when r tends to infinity.
Q.E.D.

*This* is nonsense because you have to apply a coordinate based criteria in coordinates meeting the requirements. KS don't meet the stated requirements (for coordinates in which to apply the test), so you must use other coordinates for the large r region, e.g. Schwarzschild as I have done.

That is, to apply the test, use appropriate coordinates. You can use any other coordinates you want for other purposes.

No, your coordinates are not compatible with the coordinate-dependent AF definition, your coordinates are KS.

No, I use KS coordinates in a limited region, not relevant to applying the AF criterion. The coordinate patch relevant for the AF test is Schwarzschild. I asked you if you reject that 2 sphere is a smooth manifold. Unless you do, then you must admit the approach of using overlapping coordinate patches with smooth mappings covering the overlapping regions.

I believe I have strictly applied the wiki definition to my stated manifold.

 

[PAllen]

You would agree both of these are also true of the Rindler horizon in Rindler coordinates, yes?

If you're trying to verify whether the line element approaches the Minkowski line element I don't think you can calculate the limit using that form, because in the next line they say r is defined as a function of U and V...if you want to calculate the limit of the line element as the distance approaches infinity, I think you'd have to use the space coordinate in KS coordinates, which is U, not r. If you plug r=2GM(1 + W((U^2 - V^2)/e)) (where W is the Lambert W function) into the KS line element you get:

ds^2 = \frac{16G^2 M^2}{1 + W(\frac{U^2 - V^2}{e})} e^{-(1 + W(\frac{U^2 - V^2}{e}))} (-dV^2 + dU^2) + 4G^2M^2 (1 + W(\frac{U^2 - V^2}{e}))^2 d\Omega^2

...though I'm not sure how you would go about figuring the limit as U approaches infinity here, in order to check whether it approaches the Minkowski line element (in spherical coordinates, with V for time and U for the radial coordinate) ds^2 = -dV^2 + dU^2 + U^2 d\Omega^2

To arrive at a coordinate based AF to apply directly to KS coordinates, I think you would need to proceed as follows, and it would be complex:

1) Apply the U,V transform definitions to flat Minkowski space in standard coordinates to arrive at U,V expression of the appropriate limiting metric.

2) Express the actual KS line element as the metric in (1) + plus deviation function.

3) Take limits of the deviation function as r(U,V)->infinity. Messy because r(U,V) is messy. Particularly messy would be formulating the derivative convergence tests, because they should not be derivatives with respect to U,V, but instead with respect to functions of U,V characteristic of proper length in different spatial directions.

Much simpler is to transform to Schwarzschild coordinates for the purpose of applying the test.

 

[TrickyDicky]

You would agree both of these are also true of the Rindler horizon in Rindler coordinates, yes?

I'd say so but I have no clue why you keep bringing up the Rindler coordinates in this discussion.

If you're trying to verify whether the line element approaches the Minkowski line element I don't think you can calculate the limit using that form, because in the next line they say r is defined as a function of U and V...if you want to calculate the limit of the line element as the distance approaches infinity, I think you'd have to use the space coordinate in KS coordinates, which is U, not r. If you plug r=2GM(1 + W*((U^2 - V^2)/e)) (where W is the Lambert W function) into the KS line element you get:

ds^2 = \frac{16G^2 M^2}{1 + W*\frac{U^2 - V^2}{e}} e^{-(1 + W*\frac{U^2 - V^2}{e})} (-dV^2 + dU^2) + 4G^2M^2 (1 + W*\frac{U^2 - V^2}{e})^2 d\Omega^2

...though I'm not sure how you would go about figuring the limit as U approaches infinity here, in order to show it approaches the Minkowski line element (in spherical coordinates, with V for time and U for the radial coordinate) ds^2 = -dV^2 + dU^2 + U^2 d\Omega^2

You have actually a timelike variable V and a spacelike variable U and both are still undefined when r=2GM although this is disguised in the Kruskal diagram as the boundaries between the 4 regions of the diagram.
You'll see that the components of the Kruskal line element don't approach the minkowski metric when V tends to infinity, in fact when V increases I'd say the BH singularity is hit.
http://upload.wikimedia.org/wikipedia/commons/c/c1/KruskalKoords.jpg

 

[TrickyDicky]

Much simpler is to transform to Schwarzschild coordinates for the purpose of applying the test.

This is absurd, you are agreeing with me and you don't even realize it.

 

[JesseM]

I'd say so but I have no clue why you keep bringing up the Rindler coordinates in this discussion.

Well, I have no clue why you are bringing up the fact that it takes an infinite coordinate time for anything to reach r=2GM in Schwarzschild coordinates, as this seems to have nothing to do with the issue of asymptotic flatness. Are you trying to say that this somehow makes it suspicious that Schwarzschild coordinates with the Schwarzschild line element could be equivalent to a region of KS coordinates with the KS line element, since in KS coordinates the horizon is crossed in finite coordinate time? I thought you were, which is why I brought up Rindler coordinates. If not, what is your point in bringing this up?

You have actually a timelike variable V and a spacelike variable U and both are still undefined when r=2GM

They might be undefined there if you start with Schwarzschild coordinates and define V and U in terms of r and t, but isn't it equally valid to start with KS coordinates and the KS line element (where nothing problematic happens at the event horizon) and define the Schwarzschild coordinates r and t in terms of V and U, with the understanding that the Schwarzschild coordinates only cover a patch which does not include the event horizon itself? Isn't it just a historical accident that, in the study of black holes, the equations of the Schwarzschild metric were discovered to be a solution to the EFEs prior to the equations of the KS metric?

You'll see that the components of the Kruskal line element don't approach the minkowski metric when V tends to infinity, in fact when V increases I'd say the BH singularity is hit.

Are you claiming it's part of the definition of asymptotic flatness that they should approach the Minkowski metric in the limit as V approaches infinity as well as the limit as U approaches infinity? It doesn't seem like this would be true of the Schwarzschild metric either, if you pick some fixed finite r and take the limit as t approaches infinity it wouldn't approach the Minkowski metric.

 

[PAllen]

This is absurd, you are agreeing with me and you don't even realize it.

No, the disagreement seems fundamental. I say the coordinate AF test is a test for a geometric attribute of a manifold that you apply in an appropriate coordinate system. Being geometric attribute, it is simply true of manifold if established with *any* appropriate coordinate system. You seem to say it is a feature of coordinate systems and that different coordinate systems on the same manifold may or may not have this attribute. I claim this is absurd.

 

[TrickyDicky]

Well, I have no clue why you are bringing up the fact that it takes an infinite coordinate time for anything to reach r=2GM in Schwarzschild coordinates, as this seems to have nothing to do with the issue of asymptotic flatness. Are you trying to say that this somehow makes it suspicious that Schwarzschild coordinates with the Schwarzschild line element could be equivalent to a region of KS coordinates with the KS line element, since in KS coordinates the horizon is crossed in finite coordinate time? I thought you were, which is why I brought up Rindler coordinates. If not, what is your point in bringing this up?

I brought it up to show the obvious fact that a coordinate singularity(like r=2GM) is found in the Schwartzschild line element because this line element applies a coordinate dependent AF definition. To understand this one looks at the component (1-2GM/r) and checks that this mathematical expression fulfills both that when r tends to infinity the component approaches 1 (aka coordinate dependent AF) and that when r=2GM the component is undefined (singular).
If you don't have the restraint that the quotient 2GM/r has to go to zero at radial infinity,(that is that the metric component must approach 1) you don't have to use the line element with the coordinate singularity r=2GM, and you are free to build a different line element like the Kruskal, without that constraint of course the Kruskal line element could have been first historically, that's my point. That Schwartzschild developed his line element mistakenly based on a coordinate-dependent AF- well he actually did it as the Hooft quote says because he had a preliminar version of the Einstein equations that weren't generally covariant yet but only permted unimodular transformations (wich once again amounts to cordinate-dependent AF or in ther words is equivalent to demand that the components of the line element at infinity must approach the Minkowski metric (unimodular).

Isn't it just a historical accident that, in the study of black holes, the equations of the Schwarzschild metric were discovered to be a solution to the EFEs prior to the equations of the KS metric?

Yes as I explain above

Are you claiming it's part of the definition of asymptotic flatness that they should approach the Minkowski metric in the limit as V approaches infinity as well as the limit as U approaches infinity? It doesn't seem like this would be true of the Schwarzschild metric either, if you pick some fixed finite r and take the limit as t approaches infinity it wouldn't approach the Minkowski metric.

No

 

[JesseM]

I brought it up to show the obvious fact that a coordinate singularity(like r=2GM) is found in the Schwartzschild line element because this line element applies a coordinate dependent AF definition.

I don't understand why you say "because", they seem like totally unrelated statements, the coordinate singularity at r=2GM isn't somehow caused by the fact that the Schwarzschild line element approaches the Minkowski metric at r approaches infinity. The statement that the line element "applies" a particular definition of AF also seems pretty meaningless, it may be possible for us to define AF in terms of the line element of a particular coordinate system (though I'd like to see a reference if you are claiming that two coordinate systems + line elements which are geometrically identical in terms of ds along all worldlines can disagree about AF, as PAllen says this seems wrong). But the line element itself does not force us to define AF in any particular way.

To understand this one looks at the component (1-2GM/r) and checks that this mathematical expression fulfills both that when r tends to infinity the component approaches 1 (aka coordinate dependent AF) and that when r=2GM the component is undefined (singular).

Huh? Why should asymptotic flatness have anything to do with whether r=2GM is singular or not? I would assume that if you adopt a coordinate-based definition of AF it would depend only on what happens as the r coordinate approaches infinity, the behavior at any finite r should be completely irrelevant to whether a spacetime exhibits AF or not.

If you don't have the restraint that the quotient 2GM/r has to go to zero at radial infinity,(that is that the metric component must approach 1) you don't have to use the line element with the coordinate singularity r=2GM

I still don't see your point, why would the fact that the metric component approaches 1 at radial infinity in one particular coordinate system mean you "have to" use that particular coordinate system? You might be able to find a different coordinate system that also approaches flatness at radial infinity but which doesn't have the same coordinate singularity, no? For example you might consider ingoing Eddington-Finkelstein coordinates where if you use the line element at the bottom of the wiki page, it does clearly approach the Minkowski metric as r approaches infinity, but infalling particles do cross the horizon in finite coordinate time (though outgoing particles from the white hole region of the KS diagram have been traveling outwards from the horizon for an infinite coordinate time).

edit: one interesting thing about these coordinates is that if you instead use the line element at the middle of the page, where the timelike coordinate is v rather than t' as at the bottom, it seems like this line element doesn't approach the Minkowski line element as r approaches infinity because there's an extra term of 2dvdr...would you say that the version of Eddington-Finkelstein coordinates on the middle of the page is not asymptotically flat while the version on the bottom is, even though the only difference between them is that the middle one uses the substitution v=t+r* while the one at the bottom uses t'=t+r*-r ? As usual, if you are claiming that a coordinate system + line element can fail to be asymptotically flat solely because the line element does not approach Minkowski as the radial coordinate approaches infinity, I'd like to see a reference for that claim.

and you are free to build a different line element like the Kruskal, without that constraint of course the Kruskal line element could have been first historically, that's my point.

You haven't actually shown that Kruskal line element doesn't approach Minkowski as the radial coordinate U approaches infinity (though it probably doesn't, if for no other reason than it has all those extra constant factors like G and M), your argument was based on considering a line element which involves both r and U but you failed to take into account that r and U are dependent so you can't take the limit as r approaches infinity while holding U constant. If you express the line element purely in terms of U and V it becomes the complicated expression I gave in post #76, I'm not sure what the limit as U approaches infinity would be and from your non-response I suspect you aren't sure either.

Are you claiming it's part of the definition of asymptotic flatness that they should approach the Minkowski metric in the limit as V approaches infinity as well as the limit as U approaches infinity? It doesn't seem like this would be true of the Schwarzschild metric either, if you pick some fixed finite r and take the limit as t approaches infinity it wouldn't approach the Minkowski metric.

No

Er, then why did you bring up the fact that "the components of the Kruskal line element don't approach the minkowski metric when V tends to infinity"? What was your point there?

 

[TrickyDicky]

Er, then why did you bring up the fact that "the components of the Kruskal line element don't approach the minkowski metric when V tends to infinity"? What was your point there?

My point was showing that it's not coordinate-dependent AF.

So if you agree that Kruskal line element is not coordinate-dependent AF, we can move on and say that there is actually a choice of boundary condition for the vacuum solution, between the coordinate-dependent and the coordinate-free AF, right?
And that currently the latter is chosen.

 

[PAllen]

My point was showing that it's not coordinate-dependent AF.

So if you agree that Kruskal line element is not coordinate-dependent AF, we can move on and say that there is actually a choice of boundary condition for the vacuum solution, between the coordinate-dependent and the coordinate-free AF, right?
And that currently the latter is chosen.

I would put it differently.

If you use coordinate dependent AF boundary conditions, you will arrive at Schwarzschild coordinates (or similar). You are then free to change to any other coordinates you like, and they still meet the same AF boundary conditions (which are a feature of the geometry, irrespective of their being applied using particular coordinates). Further, you can extend or change the geometry any bounded amount with changing the fact that it satisfies the original boundary conditions at infinity.

 

[JesseM]

My point was showing that it's not coordinate-dependent AF.

Again, what does the limit as the timelike coordinate goes to infinity have to do with the question of whether it's "coordinate-dependent AF"? I thought you were defining "coordinate-dependent AF" solely in terms of whether the line element approaches the Minkowski line element in the limit as the radial coordinate goes to infinity.

So if you agree that Kruskal line element is not coordinate-dependent AF we can move on and say that there is actually a choice of boundary condition for the vacuum solution, between the coordinate-dependent and the coordinate-free AF, right?

I don't know what you mean by "choice of boundary condition". Boundary conditions are normally understood to have some physical meaning, I've never heard of anyone calling a purely coordinate-dependent notion a "boundary condition". And do you think there are any actual physicists who use this sort of coordinate-dependent notion of "asymptotic flatness", or would you that this is an idiosyncratic personal definition you have invented yourself? (it may be that physicists find it useful to point out that a particular metric's line element approaches the Minkowski line element as a way of showing that the metric is asymptotically flat, but that doesn't mean they're suggesting this is a necessary condition for a metric to be asymptotically flat or that it's a definition of what they mean by asymptotically flat)

Also, just to be clear, do you agree that one version of ingoing Eddington-Finkelstein coordinates does not match your definition of "coordinate-dependent AF" while the other does? If so, do you actually question whether the two are physically equivalent in some sense, or do you agree this is just a coordinate issue with no physical implications?

 

[TrickyDicky]

Again, what does the limit as the timelike coordinate goes to infinity have to do with the question of whether it's "coordinate-dependent AF"? I thought you were defining "coordinate-dependent AF" solely in terms of whether the line element approaches the Minkowski line element in the limit as the radial coordinate goes to infinity.

It was just an example to stress the coordinate nature of the singularity that can be removed once you change the coordinate dependent restriction on the line element.
If the example is not a fortunate one to understand that is a different thing and my own fault.

I don't know what you mean by "choice of boundary condition". Boundary conditions are normally understood to have some physical meaning, I've never heard of anyone calling a purely coordinate-dependent notion a "boundary condition". And do you think there are any actual physicists who use this sort of coordinate-dependent notion of "asymptotic flatness", or would you that this is an idiosyncratic personal definition you have invented yourself? (it may be that physicists find it useful to point out that a particular metric's line element approaches the Minkowski line element as a way of showing that the metric is asymptotically flat, but that doesn't mean they're suggesting this is a necessary condition for a metric to be asymptotically flat or that it's a definition of what they mean by asymptotically flat)

You are right that relativists currently only used the coordinate-free definition and that I seem to be the only one (that I know of) stressing that the coordinate-dependent definition implies an alternative assumption to the current one, and that we should compare the physical implications of the two assumptions, and apply the Occam's razor.
When you say:"Boundary conditions are normally understood to have some physical meaning" , that perfectly summarizes my point in this thread.

Also, just to be clear, do you agree that one version of ingoing Eddington-Finkelstein coordinates does not match your definition of "coordinate-dependent AF" while the other does? If so, do you actually question whether the two are physically equivalent in some sense, or do you agree this is just a coordinate issue with no physical implications?

Yes, and I really don't know if it is just a coordinate issue or it has physical meaning.

 

[PAllen]

I think the core issue is that TrickyDicky has personal definition of coordinate based AF criteria, with several distinctive features not present in the practice of GR from 1920 to 1963 (when conformal definitions were introduced). Note that I learned GR from books that used exclusively coordinate based index gymnastics, did not use any language of forms, only the last 2 books I got introduced formal manifold theory at all (and these were not the books I studied in depth). Yet I understand the coordinate AF definition completely differently than TrickyDicky does.

The distinctive features of his definition (none of which are stated or implied in the wiki definition) are:

1) AF is a feature of coordinates, rather than a feature of geometry.

2) Changing coordinates can change physics or geometry (this, in my opinion, violates the fundamental assumptions of GR). Thus, he can pose the question of whether one coordinate system is AF while another isn't, while I find this an inconceivable question based on reading only *old* books that predated conformal definitions.

3) There is special significance to unimodular coordinate transforms. This may have been true for one obsolete version of GR, but from 1917 on, for 40+ years before 'modern' terminology and techniques were introduced, there was no great significance attached such transforms over any others.

4) Coordinate singularities are significant. Anyone familiar with coordinates on a 2-sphere discovers that any attempt to used one coordinate patch produces a coordinate singularity, yet this has no geometric significance, and you can move it wherever you want or eliminate it using two coordinate patches.

5) There is something inadmissable about using multiple coordinate patches when applying the coordinate based AF definition.

6) The behavior at r<=R is relevant to applying a limiting condition as r->infinity.

I believe none of these beliefs were part of the practice of GR from 1920 to 1960.

 

[TrickyDicky]

I think the core issue is that TrickyDicky has personal definition of coordinate based AF criteria, with several distinctive features not present in the practice of GR from 1920 to 1963 (when conformal definitions were introduced). Note that I learned GR from books that used exclusively coordinate based index gymnastics, did not use any language of forms, only the last 2 books I got introduced formal manifold theory at all (and these were not the books I studied in depth). Yet I understand the coordinate AF definition completely differently than TrickyDicky does.

The distinctive features of his definition (none of which are stated or implied in the wiki definition) are:

1) AF is a feature of coordinates, rather than a feature of geometry.

2) Changing coordinates can change physics or geometry (this, in my opinion, violates the fundamental assumptions of GR). Thus, he can pose the question of whether one coordinate system is AF while another isn't, while I find this an inconceivable question based on reading only *old* books that predated conformal definitions.

3) There is special significance to unimodular coordinate transforms. This may have been true for one obsolete version of GR, but from 1917 on, for 40+ years before 'modern' terminology and techniques were introduced, there was no great significance attached such transforms over any others.

4) Coordinate singularities are significant. Anyone familiar with coordinates on a 2-sphere discovers that any attempt to used one coordinate patch produces a coordinate singularity, yet this has no geometric significance, and you can move it wherever you want or eliminate it using two coordinate patches.

5) There is something inadmissable about using multiple coordinate patches when applying the coordinate based AF definition.

6) The behavior at r<=R is relevant to applying a limiting condition as r->infinity.

I believe none of these beliefs were part of the practice of GR from 1920 to 1960.

1) This is your misunderstanding, all the time I've just pointed to an existing difference between 2 types of definition of AF

2)this also misinterprets what I've been saying, changing coordinates can only change the geometry if the change introduces some further modification such as a conformal transformation, and I cite from a current text-book: General relativity from Hobson et al. page 51: "A conformal transformation is not a change of coordinates but an actual change in the geometry of a manifold" I've maintained that the introduction of the tortoise coordinate r to get Kruskal line element is equivalent to a conformal transformation of the Scwartzschild line element that is actually possible given the general covariance of the GR equations, but that doesn't fulfill coordinate-dependent AF. It is known that the Einstein field equations per se, without further conditions allow many different geometries, as the collection of cosmological solutions that historically have been derived from them since 1915 makes evident.

4) coordinate singularities might or might not be significant, I thinks this is the common understanding.

6)I don't know what exactly you are calling r and R here.