Schwartzschild exterior and interior solutions

[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.

 

[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 relevent). It clearly contains a black hole, invalidating claim 2.

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I remain very interested in the alleged counter-argument.