Schwartzschild exterior and interior solutions

[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.

 

[PAllen]

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

But both definitions define an intrinsic feature of geometry. Whether they allow exactly the same geometries, I don't know, but they qualify geometries, not coordinate systems. The coordinate definition can only be applied in certain types of coordinates, but if it is true, the geometric fact is true no matter what coordinate transforms are used.

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.

The transform to introduce Kruskal is not a conformal transform. It does not change geometry. You can choose to extend the geometry, or not, but the transform itself is just a coordinate transform. You have not provided any reference or argument that it is a conformal transform. References you have provided discuss applying a conformal tansform *after* the coordinate transform to produce e.g. Penrose-Carter diagrams.

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.

R is the event horizon radius, r the Schwazschild coordinate for this geometry.

 

[JesseM]

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.

It seems to me this may just be a semantic issue where the answer would just depend on how you chose to define "physical meaning". In non-quantum relativistic theories I think of the "physical truth" as just being frame-independent facts about each point in spacetime, like what two clocks read at the moment they pass each other, or what readings show on a particular measuring-instrument after it takes a reading, so two coordinate systems with line elements would be "physically equivalent" if they predicted exactly the same set of local facts of this type. Originally I thought your point was that you were suspicious that spacetime actually continued past the r=2GM boundary of exterior Schwarzschild coordinates (as suggested by the title you chose for the thread), but if you have exactly the same objections to viewing the two forms of ingoing Eddington-Finkelstein coordinates as "physically equivalent" then I really don't know what you mean by that term, since both forms of the coordinate system cover exactly the same region of spacetime, and they don't disagree about coordinate singularities.

If we lived in the spacetime described by the second form of EF coordinates at the bottom of the wiki article (the one whose line element approaches the Minkowski line element as r approaches infinity), it would be a trivial matter to just relabel each physical event with the coordinates of the first form of EF coordinates (the one whose line element has the extra 2dvdr in its line element so it doesn't approach the Minkowski line element), obviously a simply relabeling would not change the local facts about what occurs at any point in the spacetime. So in what sense are the two forms of EF not "physically equivalent", how can they have a different "physical meaning"? Can you provide clear definitions of what you mean by these terms? And if you can, do you think there is any compelling reason why physicists "should" adopt your definitions (as opposed to my definition of 'physically equivalent' above, for example), or would you agree that this is a basically arbitrary choice of which definitions we find more aesthetically appealing and so the whole thing boils down to semantics?

 

[TrickyDicky]

The transform to introduce Kruskal is not a conformal transform. It does not change geometry. You can choose to extend the geometry, or not, but the transform itself is just a coordinate transform.

Ok, let's heuristically investigate this:
For radially moving null-geodetic light ray spherical coordinates theta and phi are constant, so their differentials vanish, if we consider this in the Kruskal line element, we have a conformal scaling factor multiplying a 2 dimensional flat space, this doesn't happen in the original Schwartzschild metric, and actually leads to:considering the Kruskal spacetime in the vanishing angular components derivatives case as conformally flat spacetime for light null geodesics.
Would there be thefore in this particular case no Weyl curvature and no bending of light possible??no vacuum solution actually??

 

[PAllen]

Ok, let's heuristically investigate this:
For radially moving null-geodetic light ray spherical coordinates theta and phi are constant, so their differentials vanish, if we consider this in the Kruskal line element, we have a conformal scaling factor multiplying a 2 dimensional flat space, this doesn't happen in the original Schwartzschild metric, and actually leads to:considering the Kruskal spacetime in the vanishing angular components derivatives case as conformally flat spacetime for light null geodesics.
Would there be thefore in this particular case no Weyl curvature and no bending of light possible??no vacuum solution actually??

I don't follow much of what you suggest. What I see is that if I consider a purely radial slice, light paths have the coordinate form U=V or U=-V (plus displacements). These, of course, have zero interval along them. Further, if I compute ds along any radial timelike path, I get the same interval as using Schwarzschild coordinates. Similarly for spacelike paths. This can be said to prove they are the same geometry.

I don't follow your analysis of curvature. The U,V plane with angles constant is not flat because r is complicated function U and V. When computing metric differentials to define curvature, you get non-trivial terms from the derivatives of r by U and V.

To analyze light bending, there is no escaping using varying angles, so I don't see what you can discern without analyzing using the complete line element.

[EDIt: To clarify how wrong it is to assume that a the U,V plane metric is flat, consider 2-d Euclidean signature geometry. I define a line element:

ds^2 = f(x,y) ( dx^2 + dy^2)

Suppose f(x,y) is coordinate distance from a circle of radius 5. Then the circumference of coordinate circle of radius 5 is zero, while its radius is not zero. On the other hand, it can be seen that this metric is asymptotically Euclidean at infinity.
]

 

[TrickyDicky]

Take a look at this link (the first 3 pages,especially equation (9) and the next paragraph, and you'll understand better what I wrote in a hurry.
http://www.ast.cam.ac.uk/teaching/undergrad/partii/handouts/GR_14_09.pdf

 

[PAllen]

I found this page (don't know if anyone gave it earlier in this thread). It derives Kruskal directly from EFE, without starting from Schwarzschild. It also shows (if your read it all through) that the curvature of Kruskal does not vanish. But to see this, you must read all the way through because the first 2/3 end up deriving 'Kruskal like' coordinates for flat spacetime (e.g. they answer the question I raised in answer to Jessem, of what limiting metric would you use if you wanted to define a coordinatea criteria for AF directly in Kruskal coordinates; that is, you would ask if the Kruskal metric approaches the flat form as r(U,V)->infinity).

http://www.mathpages.com/home/kmath655/kmath655.htm

Note that this and most derivations of this geometry don't explicitly assert *any* boundary conditions at infinity. This is because spherical symmetry + vacuum solultion imply asymptotic flatness as noted here:

http://staff.science.uva.nl/~jpschaar/report/node5.html

 

[TrickyDicky]

Wow, that's sync, at the same minute!

 

[TrickyDicky]

I found this page

The link?

Note that this and most derivations of this geometry don't explicitly assert *any* boundary conditions at infinity. This is because spherical symmetry + vacuum solultion imply asymptotic flatness as noted here:

http://staff.science.uva.nl/~jpschaar/report/node5.html

Sure, if the two conditions are met, there is AF.

 

[PAllen]

Take a look at this link (the first 3 pages,especially equation (9) and the next paragraph, and you'll understand better what I wrote in a hurry.
http://www.ast.cam.ac.uk/teaching/undergrad/partii/handouts/GR_14_09.pdf

Ok, fine, so they define a concept of a 2 space that has curvature but also has a conformally flat metric by their definition. So what? There is not the slightest hint of a claim or argument that any of the transforms they have done change geometry. All they are saying is the metric takes this form in these coordinates. And...

Note, especially, they say things like:

"We can find lots of different coordinate transformations that preserve the conformal nature
of the 2-space defined by equation (5)."

This is just a nice metric form that can be achieved by coordinate transforms. Reading any of this to say these coordinate transforms change the geometry is a false reading of this material.

 

[PAllen]

The link?

Sorry, forgot the link, it's there now.

 

[PAllen]

As I understood the references you gave before about a type of conformal transform that changed geometry, the meaning was completely differfent: these were not just coordinate transforms, they introduced 'points at infinity' which certainly does change the geometry.

 

[TrickyDicky]

Ok, fine, so they define a concept of a 2 space that has curvature but also has a conformally flat metric by their definition. So what? There is not the slightest hint of a claim or argument that any of the transforms they have done change geometry. All they are saying is the metric takes this form in these coordinates. And...

Note, especially, they say things like:

"We can find lots of different coordinate transformations that preserve the conformal nature
of the 2-space defined by equation (5)."

This is just a nice metric form that can be achieved by coordinate transforms. Reading any of this to say these coordinate transforms change the geometry is a false reading of this material.

Let's not jump into conclusions yet (me included), do you agree that the line element in (9), when considering only null geodesics (light), in other words considering the angular components of the metric constant, is that of a conformally flat spacetime?

Conformally flat spacetimes have a vanishing Weyl curvature, any vacuum solution is incompatible with a vanishing Weyl curvature.

If so (and I insist considering a radiation only situation), how could this line element be a vacuum solution for light?

 

[PAllen]

Let's not jump into conclusions yet (me included), do you agree that the line element in (9), when considering only null geodesics (light), in other words considering the angular components of the metric constant, is that of a conformally flat spacetime?

Conformally flat spacetimes have a vanishing Weyl curvature, any vacuum solution is incompatible with a vanishing Weyl curvature.

If so (and I insist considering a radiation only situation), how could this line element be a vacuum solution for light?

I am not familiar with the formal definition and properties of conformal flatness. Your reference defines that this metric is conformally flat, noting carefully, that it still has curvature using the precisely the same argument about derivatives that I gave in my post #94.

I am not very familiar with the Weyl curvature tensor. A quick look up in Wikipedia suggests that in dimension >=4, vanishing Weyl tensor is necessary and sufficient for conformal flatness. However, we are talking about conformal flatness of a 2-d slice of a manifold. The same Wiki article (on Weyl tensor) observes that *any* 2-d manifold is conformally flat, so all your other conclusions are unfounded. That is, I conclude from all this that you can take any 2-d slice you want of any 4-d manifold, and you can find coordinates that express the metric in a conformally flat form on that slice.

Obviously, the conclusion that KS can't be a vacuum solution is wrong, as I gave you a reference to a complete derivation of it directly from the vacuum field equations.

 

[TrickyDicky]

I am not familiar with the formal definition and properties of conformal flatness.
I am not very familiar with the Weyl curvature tensor.

That says a lot.

Obviously, the conclusion that KS can't be a vacuum solution is wrong, as I gave you a reference to a complete derivation of it directly from the vacuum field equations.

Your reference doesn't even derive the conformal nature of the line element, is put by hand in the very first mathematical expression by saying the metric must have a diagonal form.

 

[George Jones]

Let's not jump into conclusions yet (me included), do you agree that the line element in (9), when considering only null geodesics (light), in other words considering the angular components of the metric constant, is that of a conformally flat spacetime?

No, (9) does not give a conformally flat spacetime. Why should the angular components be constant for null geodesics?