Schwartzschild exterior and interior solutions

[PAllen]

One hint pointing to the two line elements describing diffeent geometries is that the Schwartzschild line element defines a static spacetime(with all metric components independent of coordinate t and invariant for time reversal) while the KS line element is not that of a static spacetime. That's odd because if the two line elements are different coordinates descriptions of the same geometry, how can it be static and non-static at the same time?
Any help understanding this would be welcome.

This might mean nothing. You can call any coordinate 't'. In the interior Schwarzschild solution, coordinate r is the timelike coordinate. Dirac used SR coordinates u and v that were both mixtures of timelike and spacelike character. I don't know how to specify a test of static character for arbitrary coordinates, or a coordinate independent test for static character. If I can think of one, I'll post; hopefully someone more knowledgeble will post instead.

 

[TrickyDicky]

This might mean nothing. You can call any coordinate 't'. In the interior Schwarzschild solution, coordinate r is the timelike coordinate. Dirac used SR coordinates u and v that were both mixtures of timelike and spacelike character. I don't know how to specify a test of static character for arbitrary coordinates, or a coordinate independent test for static character. If I can think of one, I'll post; hopefully someone more knowledgeble will post instead.

Look at the second condition for staticity.The KS line element and actually any metric describing a BH is not time-symmetric, by definition.

 

[atyy]

One hint pointing to the two line elements describing diffeent geometries is that the Schwartzschild line element defines a static spacetime(with all metric components independent of coordinate t and invariant for time reversal) while the KS line element is not that of a static spacetime. That's odd because if the two line elements are different coordinates descriptions of the same geometry, how can it be static and non-static at the same time?
Any help understanding this would be welcome.

The Schwarzschld coordinates only describe a static spacetime if the coordinate radius is greater than the Schwarzschild radius. It has to be joined to another solution that describes static matter with coordinate radius greater than the Schwarzschild radius for the entire spacetime to be static.

 

[PAllen]

Look at the second condition for staticity.The KS line element and actually any metric describing a BH is not time-symmetric, by definition.

Why do you say an BH solution is non-time symmetric? I would think an eternal solution is time symmetric. Some of the references you gave earlier distinguished black hole solutions that could result from collapsing matter from eternal solutions with a wormhole. My guess would be the complete Kruskal geometry is eternal, time symmetric, and static.

 

[George Jones]

One hint pointing to the two line elements describing diffeent geometries is that the Schwartzschild line element defines a static spacetime(with all metric components independent of coordinate t and invariant for time reversal) while the KS line element is not that of a static spacetime. That's odd because if the two line elements are different coordinates descriptions of the same geometry, how can it be static and non-static at the same time?
Any help understanding this would be welcome.

Why do you say an BH solution is non-time symmetric? I would think an eternal solution is time symmetric. Some of the references you gave earlier distinguished black hole solutions that could result from collapsing matter from eternal solutions with a wormhole. My guess would be the complete Kruskal geometry is eternal, time symmetric, and static.

Kruskal-Szekeres spacetime is an extension of (external) Schwarzschild spacetime, and, as such, K-S spacetime is static everywhere Schwarzschild is, i.e., outside the event horizon, as atty noted. A region of spacetime is static if there is a hypersurface-orthogonal (gives non-rotating) timelike Killing vector field (gives stationary) in the region.

Below the event horizon, no timelike Killing field exists, so K-S is not even stationary there, let alone static.

 

[TrickyDicky]

Why do you say an BH solution is non-time symmetric? I would think an eternal solution is time symmetric. Some of the references you gave earlier distinguished black hole solutions that could result from collapsing matter from eternal solutions with a wormhole. My guess would be the complete Kruskal geometry is eternal, time symmetric, and static.

I see what you mean, I was thinking of a BH formed by gravitational collapse which obviously had a beginning and therefore is not time symmetric, but you are right, the Kruskal manifold describes a more abstract scenario with wormholes and possibly can be considered a static spacetime.
However the problem still remains IMO that the Kruskal manifold is not asymptotically flat in the coordinate-dependent way the Schwartzschild line element is.

 

[TrickyDicky]

Kruskal-Szekeres spacetime is an extension of (external) Schwarzschild spacetime, and, as such, K-S spacetime is static everywhere Schwarzschild is, i.e., outside the event horizon, as atty noted. A region of spacetime is static if there is a hypersurface-orthogonal (gives non-rotating) timelike Killing vector field (gives stationary) in the region.

Below the event horizon, no timelike Killing field exists, so K-S is not even stationary there, let alone static.

Thanks, I see that now.

 

[George Jones]

as I undertand from the same wikipedia page:subsection "A coordinate-dependent definition" historically asymptotic flatness was coordinate-dependent and therefore it only allowed unimodular transformations of the coordinates (equations that only holds if g=1). This was the case at the time Scwartzschild derived his solution, see 't Hooft comment at the bottom of page 49 in [URL]http://www.phys.uu.nl/~thooft/lectures/genrel_2010.pdf[/url[/QUOTE]

The passage from 't Hooft:

In his original paper, using a slightly di®erent notation, Karl Schwarzschild replaced (r - (2M)3)^(1/3) by a new coordinate r that vanishes at the horizon, since he insisted that what he saw as a singularity should be at the origin, claiming that only this way the solution becomes "eindeutig" (unique), so that you can calculate phenomena such as the perihelion movement (see Chapter 12) unambiguously. The substitution had to be of this form as he was using the equation that only holds if g = 1 . He did not know that one may choose the coordinates freely, nor that the singularity is not a true singularity at all. This was 1916. The fact that he was the first to get the analytic form, justifies the name Schwarzschild
solution.

't Hooft is saying that in spite of two errors made by Schwarzschild, his priority "justifies the name Schwarzschild solution." One of the errors identified by 't Hooft is that Schwarzschild "allowed only unimodular transfomations". 't Hooft really means this to be taken as a an error. The reason that Schwarzschild restricted himself to these transformations was not that "historically asymptotic flatness was coordinate-dependent and therefore it only allowed unimodular transformations". Schwarzschild only had in his hands a preliminary version of Einstein's theory of gravity that allowed only unimodular transformations. when Schwarzschild formulated his solution, he was unaware of Einstein's final version of GR that allowed general coordinate transformations.

 

[TrickyDicky]

The passage from 't Hooft:


't Hooft is saying that in spite of two errors made by Schwarzschild, his priority "justifies the name Schwarzschild solution." One of the errors identified by 't Hooft is that Schwarzschild "allowed only unimodular transfomations". 't Hooft really means this to be taken as a an error. The reason that Schwarzschild restricted himself to these transformations was not that "historically asymptotic flatness was coordinate-dependent and therefore it only allowed unimodular transformations". Schwarzschild only had in his hands a preliminary version of Einstein's theory of gravity that allowed only unimodular transformations. when Schwarzschild formulated his solution, he was unaware of Einstein's final version of GR that allowed general coordinate transformations.

You are right. I interpret this to mean that Schwartzschild, due to the fact that he had a preliminary version of Einstein's equations restricted his coordinate transformations to the unimodular ones for his vacuum solution, and that given that we have the final version that stresses that the equations allow any coordinate transformation, we can actually extend the notion of asymptotically flatness to build the K-S line element.
But what I'm saying is that this might perfectly be the case in general, but specifically for the vacuum solution the restriction to unimodular coordinate transformations might be demanded by the boundary condition at infinity of this particular set-up of an isolated object.
I say this because Einstein himself,(who certainly was well aware of the general covariance of his equations) in his "Cosmological considerations" from 1917, also
admitted this boundary condition at infinity requiring unimodular transformations for the "problem of the planets" as he calls it in page 182 of the english translation, although he rejected such boundary condition at infinity for a cosmological solution.

In any case, it's easy to see that whatever the reason, be it due to Schwartzschild "error" or not, the original Schwartschild manifold obeys a different boundary condition than the Kruskal manifold, I'm not sure if this is enough for them to be different geometries.

 

[JesseM]

Presumably if you have any worldline (timelike, spacelike or lightlike) defined in terms of Schwarzschild coordinates, you can then use the coordinate transformation between Schwarzschild and Kruskal-Szekeres coordinates to find the description of the same worldline in KS coordinates. Then if you use the Schwarzschild line element to integrate ds along the path in Schwarzschild coordinates, and use the KS line element to integrate ds along the same path in KS coordinates (between a pair of points which also map to one another by the coordinate transformation), you should get the same answer. (isn't the KS line element derived by doing a coordinate transformation on the Schwarzschild line element, ensuring that this will be the case?) As I understand it, "the geometry" is defined entirely in terms of path lengths along arbitrary paths, so this is all that is required for them to both be describing the same geometry.

It all seems to depend on whether this particular coordinate transformation between the Schwarzschild line element and the KS line element is valid in the context of the boundary conditions of the vacuum solution of the Einstein field equations, I know that according to standard textbooks it is.
But as I explained in my previous post, there might be reasons that lead us to think that it is not such an assured fact: an ad hoc change of the definition of asymptotic flatness to allow black holes seems to have been made thru the introduction of "conformal compactification", it is not clear to me that the original Schwartzschild manifold admits such conformal compactification since it would mean the central mass of the vacuum solution acts as a test particle (it doesn't curve the manifold) and can be then considered a minkowskian point. It makes one wonder: how can it be a gravitational source in empty space then? and originate planet precession, or bending of light.

I don't really understand how your comment relates to mine. Are you saying that the two might not be equivalent geometrically even if my statement is correct that any possible worldline expressed in one coordinate system, when mapped to the other, will have the same "length" when calculated with the line element of each system? As I said, I thought that this was basically the definition of geometric equivalence. Alternatively, are you suggesting that it might be possible to find some examples of worldlines which do not have the same length when calculated with the line element of each coordinate system? It seems rather implausible that such examples would exist and yet no physicists or mathematicians would have noticed them after all these years.

 

[TrickyDicky]

Are you saying that the two might not be equivalent geometrically even if my statement is correct that any possible worldline expressed in one coordinate system, when mapped to the other, will have the same "length" when calculated with the line element of each system?

No, that's not what I'm saying.
My point is that this specific coordinate transformation might not be allowed for this specific solution(vacuum) of the Einstein equations with a specific boundary condition at infinity.

 

[TrickyDicky]

Alternatively, are you suggesting that it might be possible to find some examples of worldlines which do not have the same length when calculated with the line element of each coordinate system?

Actually all those that describe an infalling particle going thru an event horizon are not allowed with the boundary condition at infinity of coordinate-dependent asymptotical flatness of the original Schwartzschild line element.

 

[JesseM]

No, that's not what I'm saying.
My point is that this specific coordinate transformation might not be allowed for this specific solution(vacuum) of the Einstein equations with a specific boundary condition at infinity.

What do you mean by "allowed"? According to what set of rules? AFAIK you can use any coordinate transformation that respects some basic rules like continuity and not assigning multiple coordinates to the same point in spacetime. Usually when physicists say you are not "allowed" to do something they mean that some procedure will give the wrong answer when if you try to use it to calculate some physical quantity (as in, 'you are not allowed to use the inertial formula for time dilation in a non-inertial frame'), are you saying something like that will happen here?

Alternatively, are you suggesting that it might be possible to find some examples of worldlines which do not have the same length when calculated with the line element of each coordinate system?

Actually all those that describe an infalling particle going thru an event horizon are not allowed with the boundary condition at infinity of coordinate-dependent asymptotical flatness of the original Schwartzschild line element.

Again, "allowed" according to what rules? Suppose I have the worldline of an infalling particle in Kruskal-Szekeres coordinates and I use the KS line element to calculate the proper time between two endpoints on that worldline, which might lie on either side of the event horizon. I can then map all points outside the event horizon into exterior Schwarzschild coordinates and use the exterior line element to calculate the proper time from the first endpoint to arbitrarily close to the event horizon (considering the limit as Schwarzschild coordinate time goes to infinity), and likewise for all points between crossing the event horizon and the second endpoint, and if I add up the proper times along these two segments I should get the same answer that I got when I used KS coordinates with the KS line element. So this should not be an example of "worldlines which do not have the same length when calculated with the line element of each coordinate system", I'm not sure if you were saying it was when you responded to that comment with "Actually..."

 

[TrickyDicky]

What do you mean by "allowed"? According to what set of rules? AFAIK you can use any coordinate transformation that respects some basic rules like continuity and not assigning multiple coordinates to the same point in spacetime. Usually when physicists say you are not "allowed" to do something they mean that some procedure will give the wrong answer when if you try to use it to calculate some physical quantity (as in, 'you are not allowed to use the inertial formula for time dilation in a non-inertial frame'), are you saying something like that will happen here?

Allowed according to the rules of differential equations and the restraints set by exact solutions satisfying the boundary conditions applied. So in this context if the boundary condition restricts the coordinate transfrmations to unimodular transformations, the solution must follow that restriction and the transformation from Scwartzschild line element to Kruskal would nt be allowed. It can be argued if that boundary condition is well posed in this particular problem, that seemed to be the understanding the understanding of Schwartzschild and Einstein but it's not the current textbook understanding as I can see.
I respect that and am not saying that one is right and the other wrong.

 

[PAllen]

I know I am no expert on this topic, but it seems to me that there is unnecessary confusion. Whether the condition for AF is stated in terms of preferred coordinates or independent of coordinates, it is a criterion of the geometry. The KS geometry contains the Schwarzshchild geometry (exterior plus vacuum interior) as a subset (black hole region, one of the exterior regions; left out is white hole region, other exterior region). If we take the Schwarzshchild subset of KS, transform to Schwarzshchild coordinates, apply the coordinate AF condition, we *must* conclude that this KS subset is coordinate AF. It is simply impossible for a coordinate transformation (that also transforms the metric) to change any geometric or topological property.

In particular, the both KS coordinates mix r and t Schwarzshchild coordinates, and you cannot pretend, e.g. V, shoud be treated as time in some meaningless application of coordinate AF condition. The coordinate AF condition presupposes you transform to a coordinate system meeting 'maximally Minkowsiki' character. Whatever you conclude in these coordinates (about AF character) is true of the geometry, irrespective of other coordinates you may use.

 

[TrickyDicky]

I know I am no expert on this topic, but it seems to me that there is unnecessary confusion. Whether the condition for AF is stated in terms of preferred coordinates or independent of coordinates, it is a criterion of the geometry. The KS geometry contains the Schwarzshchild geometry (exterior plus vacuum interior) as a subset (black hole region, one of the exterior regions; left out is white hole region, other exterior region). If we take the Schwarzshchild subset of KS, transform to Schwarzshchild coordinates, apply the coordinate AF condition, we *must* conclude that this KS subset is coordinate AF. It is simply impossible for a coordinate transformation (that also transforms the metric) to change any geometric or topological property.

Is there no problem in the way the Schwartzschild geometry is artificially glued to the rest of the KS geoemetry?
I found this interesting comment about it: http://williewong.wordpress.com/2009/10/26/conformal-compactification-of-space-time/

 

[PAllen]

Is there no problem in the way the Schwartzschild geometry is artificially glued to the rest of the KS geoemetry?
I found this interesting comment about it: http://williewong.wordpress.com/2009/10/26/conformal-compactification-of-space-time/

What do you mean artificially glued? The derivations of KS I've read start with Schwartzschild, do a perfectly ordinary coordinate transform, notice that an extension then suggests itself. An analogy:

Start with x=sqrt(y) over reals. You have curve +,+ quadrant. Transform to y=x^2, notice that it extends smoothly to +,- quadrant.

What is 'artificial gluing'?

 

[JesseM]

Is there no problem in the way the Schwartzschild geometry is artificially glued to the rest of the KS geoemetry?

What do you mean "artificially glued"? Are you familiar with the idea of coordinate "patches" which only cover a partial region of a larger spacetime, like Rindler coordinates which only cover the "Rindler wedge" of a full Minkowski spacetime? (and which have a different line element than the line element in Minkowski coordinates) Do you think the Rindler geometry is artificially glued to the rest of the Minkowski geometry? If not, what's the relevant difference?

 

[TrickyDicky]

What do you mean artificially glued? The derivations of KS I've read start with Schwartzschild, do a perfectly ordinary coordinate transform, notice that an extension then suggests itself. An analogy:

Start with x=sqrt(y) over reals. You have curve +,+ quadrant. Transform to y=x^2, notice that it extends smoothly to +,- quadrant.

What is 'artificial gluing'?

What do you mean "artificially glued"? Are you familiar with the idea of coordinate "patches" which only cover a partial region of a larger spacetime, like Rindler coordinates which only cover the "Rindler wedge" of a full Minkowski spacetime? (and which have a different line element than the line element in Minkowski coordinates) Do you think the Rindler geometry is artificially glued to the rest of the Minkowski geometry? If not, what's the relevant difference?

All those extensions rely on certain geometrical manipulation of the spacetimes, based on the notion of conformal infinity and the conformal compactification of the manifolds. However mathematically sound they may seem, I'm not sure about their physical justification. On what observations are they built upon?

 

[atyy]

All those extensions rely on certain geometrical manipulation of the spacetimes, based on the notion of conformal infinity and the conformal compactification of the manifolds. However mathematically sound they may seem, I'm not sure about their physical justification. On what observations are they built upon?

Observations are a completely different issue.

No one is saying that the maximally extended vacuum Schwazrschild solution exists to be observed. People are just saying it is a possible reality consistent with Einstein's equations.

There are other possible realities consistent with Einstein's equations, and our universe seems to be consistent with a perturbed FLRW solution.

 

[JesseM]

All those extensions rely on certain geometrical manipulation of the spacetimes, based on the notion of conformal infinity and the conformal compactification of the manifolds.

Are you including the extension from Rindler spacetime to Minkowski spacetime here? If you lived in a flat spacetime and happened to be using Rindler coordinates to do some calculations, would you be skeptical that spacetime extends beyond the Rindler horizon where these coordinates end?

However mathematically sound they may seem, I'm not sure about their physical justification. On what observations are they built upon?

I think the basic physical justification is the idea that the spacetime should be "maximally extended" as discussed in the last paragraph here, so that worldlines don't end at finite proper time unless they run into a physical singularity. Does it really make physical sense that any worldline would end at some finite time just because it takes infinite coordinate time to reach that proper time in some arbitrarily-chosen coordinate system? Right now it's 7:44 PM here, one could design a coordinate system where it takes an infinite coordinate time for my clock to reach the time of 7:50 PM, do I really need a physical justification for believing that the mere existence of such a coordinate system doesn't imply my life is actually going to end at 7:50?

 

[JesseM]

Observations are a completely different issue.

No one is saying that the maximally extended vacuum Schwazrschild solution exists to be observed. People are just saying it is a possible reality consistent with Einstein's equations.

There are other possible realities consistent with Einstein's equations, and our universe seems to be consistent with a perturbed FLRW solution.

All this is true, but although real astrophysical black holes are not described by the Schwarzschild solution (since that solution only describes a black hole which has existed eternally from the perspective of external observers), I think TrickyDicky and Mike_Fontenot are suggesting we should be skeptical about whether spacetime actually continues beyond the event horizon of real black holes, as it definitely would in the maximally extended version of whatever solution describes the spacetime outside a real astrophysical black hole like the one at the center of our galaxy.

 

[PAllen]

All those extensions rely on certain geometrical manipulation of the spacetimes, based on the notion of conformal infinity and the conformal compactification of the manifolds. However mathematically sound they may seem, I'm not sure about their physical justification. On what observations are they built upon?

Not in derivations I've read. See MTW pp. 826-41. No mention of conformal anything. My analogy to extending half a parabola to a whole parabola seems precisely equivalent to what is going on here.

 

[JesseM]

Not in derivations I've read. See MTW pp. 826-41. No mention of conformal anything. My analogy to extending half a parabola to a whole parabola seems precisely equivalent to what is going on here.

Maybe TrickyDicky is thinking of Penrose diagrams? A point which is at an infinite distance from the horizon in Schwarzschild coordinates is only at a finite distance in a Penrose diagram, at a point on the diagram which is said to represent "conformal infinity" (see here), but this is not true of a diagram in Kruskal-Szekeres coordinates.

 

[TrickyDicky]

Observations are a completely different issue.

No one is saying that the maximally extended vacuum Schwazrschild solution exists to be observed. People are just saying it is a possible reality consistent with Einstein's equations.

There are other possible realities consistent with Einstein's equations, and our universe seems to be consistent with a perturbed FLRW solution.

You'll agree with me observations are important in physics and should be a starting point for any theoretical construction.
KS black holes are eternal, how does that agree with a Bing Bang universe with a finite past origin?

Maybe TrickyDicky is thinking of Penrose diagrams? A point which is at an infinite distance from the horizon in Schwarzschild coordinates is only at a finite distance in a Penrose diagram, at a point on the diagram which is said to represent "conformal infinity" (see here), but this is not true of a diagram in Kruskal-Szekeres coordinates.

Yes, I'm actually referring to Penrose diagrams, and yes the KS diagrams were created a bit earlier and are sort of a precursor, but nevertheless KS diagrams imply the the modified definition of asymptotic flatness to weakly asymptotic flatness based on conformal compactification, see pages 47-49 in http://arxiv.org/PS_cache/gr-qc/pdf/9707/9707012v1.pdf

BTW your link is broken

 

[JesseM]

Yes, I'm actually referring to Penrose diagrams, and yes the KS diagrams were created a bit earlier and are sort of a precursor, but nevertheless KS diagrams imply the the modified definition of asymptotic flatness to weakly asymptotic flatness based on conformal compactification, see pages 47-49 in http://arxiv.org/PS_cache/gr-qc/pdf/9707/9707012v1.pdf

P. 47-49 don't depict diagrams of Kruskal coordinates, look on p. 46 where they say they are drawing a "CP diagram for the Kruskal spacetime", where CP is defined on p. 40 as a Carter-Penrose diagram. The diagrams on p. 47 and 49 are likewise CP diagrams. And page 48 specifically distinguishes between the Kruskal spacetime M and its conformal compactification \tilde{M} (M with a tilde over it), just like they distinguish earlier on the page between Minkowski spacetime and its own conformal compactification.

BTW your link is broken

Link works fine for me, try it again. And can you please address my questions from post #51?

 

[TrickyDicky]

P. 47-49 don't depict diagrams of Kruskal coordinates, look on p. 46 where they say they are drawing a "CP diagram for the Kruskal spacetime", where CP is defined on p. 40 as a Carter-Penrose diagram. The diagrams on p. 47 and 49 are likewise CP diagrams. And page 48 specifically distinguishes between the Kruskal spacetime M and its conformal compactification \tilde{M} (M with a tilde over it), just like they distinguish earlier on the page between Minkowski spacetime and its own conformal compactification.

You are missing my point. I know those are CP diagrams of Kruskal spacetime, I mean that the coordinate change of the Schwartzschild metric to get the Kruskal spacetime requires the extension of the definition of asymptotic flatness to admit weakly asymptotically simple spacetime, which is related to the conformal compactification of the Penrose diagrams. I think it's licit to ask for the physical justification of this seemingly ad hoc redefinition of asymptoticaly flat spacetime. Yes, I know it's compatible with GR and with the general covariance of the equations, but we are addressing a particular case, not a general case, i.e. the context of the unique vacuum solution of the Einstein equations, and in this context is where I think an extension of the original boundary condition, that demanded restriction to unimodular coordinate tranformations for this particular problem, must be physically justified by some very convincing observational fact, not mere speculation about wormholes, eternal blacK holes and white holes. Once again all these may very well be compatible with the GR equations and their freedom of coordinate transformations, but we are talking about the restricted case of a singular solution of the specific problem of Ric=0. Here we must make a choice about the boundary condition at infinity, either it approachesthe metric of compactified Minkowski spacetime(the conformal manifold into which Minkowski space-time is embedded with the points mentioned below not fixed by the metric) as r → ∞,in which case the coordinate transformation to obtain the Kruskal spacetime is perfectly valid) or it approaches the metric of Minkowski spacetime manifold, that with the start and the end-point of null,time-like and space-like geodesics points fixed at the boundary by the metric, as r → ∞.
I think at the very least be should acknowledge this choice when we use the KS solution, and therefore be able to sustain it on some physical consideration that makes us choose the compactified Minkowski manifold boundary instead of the Minkowski spacetime boundary.

can you please address my questions from post #51?

Rindler extension I have really not thought of in these context.
Your second questions has implicit the choice of weakly asymptotical flatnes, all I can say is that if you choose the coordinate-dependent boundary condition this problem doesn't even arise, because the "coordinate" singularity" or event horizon does not belong tothe manifold, and the spacetime is defined as an empty (no Ricci curvature sources) manifold with a determined (by the specific problem) Weyl curvature (determined by the 2GM/r parameter).

 

[PAllen]

You'll agree with me observations are important in physics and should be a starting point for any theoretical construction.
KS black holes are eternal, how does that agree with a Bing Bang universe with a finite past origin?

The Townsend reference you give discusses this. You just use part of the Kruskal solution and join it to a solution for inside horizon stellar collapse. Anyway, this has got nothing to do with AF criteria.

Yes, I'm actually referring to Penrose diagrams, and yes the KS diagrams were created a bit earlier and are sort of a precursor, but nevertheless KS diagrams imply the the modified definition of asymptotic flatness to weakly asymptotic flatness based on conformal compactification, see pages 47-49 in http://arxiv.org/PS_cache/gr-qc/pdf/9707/9707012v1.pdf

No, paper chooses to use only the coordinate free definition of AF. It does not state or imply in any way that the use of the coordinate definition would yield a different result (it simply doesn't mention the coordinate definition). I have given, I believe, a very specific argument that this position is untenable. So far, you have not provided any reference or any argument to support the idea that KS geometry is anything more than Schwazrschild exterior+interior + white hole + alternate exterior; and the either exterior part of KS would meet any definition of AF that is met by Schwazrschild.

Carter-Penrose diagrams have nothing to do with KS geometry per se. They can be used with any geometry and starting coordinates as a way to conveniently represent horizons and singularities. The Townsend paper shows them being used to elucidate the coordinate horizon in Rindler coordinates. Does this imply that use of Rindler coordinates changes the geometry of spacetime? (Jessem has asked you this a couple of times as well).

 

[TrickyDicky]

The Townsend reference you give discusses this. You just use part of the Kruskal solution and join it to a solution for inside horizon stellar collapse. Anyway, this has got nothing to do with AF criteria.

I think it has, in the context of the boundary conditions used in the solution of the Einstein equations for empty space. Only in this context. I've said it many times.

No, paper chooses to use only the coordinate free definition of AF. It does not state or imply in any way that the use of the coordinate definition would yield a different result (it simply doesn't mention the coordinate definition). I have given, I believe, a very specific argument that this position is untenable. So far, you have not provided any reference or any argument to support the idea that KS geometry is anything more than Schwazrschild exterior+interior + white hole + alternate exterior; and the either exterior part of KS would meet any definition of AF that is met by Schwazrschild.

KS extended solution does not meet the definition of AF that restricts transformation of coordinates to be unimodular. Read my previous post.

 

[PAllen]

I think it has, in the context of the boundary conditions used in the solution of the Einstein equations for empty space. Only in this context. I've said it many times.


KS extended solution does not meet the definition of AF that restricts transformation of coordinates to be unimodular. Read my previous post.

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.