Talk:Schwarzschild coordinates

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[edit] Generalize discussion to all static spherically symmetric spacetimes

Schwarzschild coordinates can be used for any static spherically symmetric spacetime. The defining characteristic of these coordinates is the fact that the radial coordinate is related to the length around circles r = r0 by the usual formula C = 2πr. Compare isotropic coordinates.

If I'm not mistaken, the Schwarzschild geometry is the geometry of a spherically symmetric spacetime - see Birkhoff's theorem (relativity). Thus, the statement that Schwarzschild coordinates can be used for any [static] spherically symmetric spacetime seems almost trivial to me. However, the defining characteristic stuff seems like a good thing to add to the article. Alfred Centauri 02:30, 25 Jun 2005 (UTC)

I think this article should be modified to reflect this, and a new companion article on isotropic coordinates added. Also, both articles should link to and from a new article on static spherically symmetric spacetimes, which should discuss matching fluid interiors to a Schwarzschild exterior region, and should mention several recently introduced techniques for producing such solutions in great profusion. ---CH

When you will be done with these new articles? ;<) Alfred Centauri 02:30, 25 Jun 2005 (UTC)

[edit] Rewrote

Page rewritten. I found it annoying that it discusses the Schwarzschild metric instead of the Schwarzschild coordinate system. We already have an article on the metric. Instead the coordinate chart is another issue, and as you can see there are plenty of interesting things to say about it.

I thought about including a section of the relationship to the metric tensors, but decided that this was threatenning to bring back into this page material on the overall Schwarzschild solution which I had just chosen to remove.

I like the idea of further generalizing this page to note the applicability of this coordinate system to any static stationary spherically symmetric spacetime. --EMS | Talk 04:31, 31 July 2005 (UTC)

You must get annoyed quite easily! ;<)
Seriously though, the original stub mentions the Schwarzschild metric only in the context of its line element as expressed in the Schwarzschild coordinates. In the opening line of this rewritten article, the Schwarzschild coordinate system is connected to the Schwarzschild metric in a way that is IMHO, misleading. It should go without saying that the Schwarzschild metric, like any metric, is independent of the coordinate system used to express the line element of the metric. Thus, the Schwarzschild coordinates refer to the coordinate system in which the Schwarzschild metric line element takes the form as given in the original stub. This line element of the Schwarschild metric certainly takes a different form when it is expressed in another coordinate system such as the Eddington-Finkelstein, Kruskal, Free Fall etc. etc. coordinate systems. For this reason, I am re-inserting the original opening sentence and line element expression. Alfred Centauri 20:12, 14 August 2005 (UTC)
Be careful here. The Schwarzschild spacetime (or Schwarzschild topology) is what is independent of the coordinate system. However, a specific metric cannot exist without a coordinate system to map it against. However, other than this semantic dispute I agree with what you are trying to say.
I really do not ike the Schwarzschild metric itself being here:
  1. This line item is available in the Schwarzschild solution article, and to greatest extent possible I would like to keep it isolated there.
  2. As noted in the introduction, this coordinate system can be used to map any static stationary spherically symmetric spacetime. (It can even be used to map other spacetimes, but the metric will not be anywhere as elegant.)
So I see the metric as being a distraction and a triviality. --EMS | Talk 04:54, 17 August 2005 (UTC)
I also am confused here. By Schwarzschild topology I assume you must mean the topology of the maximal extension considered as a topological manifold. In contrast, the standard interpretation of Schwarzschild spacetime would be maximal extension considered as a Lorentzian manifold. Intermediate between these two levels of structure is the smooth manifold structure.
The generalization I had in mind would be a discussion of the Schwarzschild chart (only valid for the exterior region of a black hole, of course) for a general static spherically symmetric spacetime. (Someone somewhere mentioned Birkhoff's theorem, but that only applies to electrovacuums, not perfect fluids, etc.) Here, the defining characteristic of Schwarzschild chart is that
ds^2 = -f(r)^2 dt^2 + g(r)^2 dr^2 + r^2 \, (d\theta^2 + \sin(\theta)^2 d\phi^2)
and in particular, the radial coordinate has a geometric interpretation: the coordinate spheres t=t0, r=r0 are geometric spheres (constant positive curvature) with surface area given in terms of r by the usual formula.---CH (talk) 00:55, 8 October 2005 (UTC)
On the contrary, it is my understanding that the metric tensor exists as a geometric object independent of the existence of any coordinate system. It follows that the Schwarzschild geometry exists independent of the coordinate system used to map the spacetime. Again, it is my understanding that it is the components of the metric (the terms of the line element) that depend on the existence of a coordinate system in order to be expressed. If my understanding is incorrect, I would be grateful for a reference that would set me straight.
Yes, including the Schwarzschild metric line element is redundant but not to the point of distraction IMHO. It seems reasonable to me that an article on a particular coordinate system used within the context of GR should include a mention of the metric(s) for which it is best suited along with the line element of that metric expressed in that coordinate system. However, I will not re-insert the line-element should you choose to remove it. Alfred Centauri 13:48, 17 August 2005 (UTC)
Follow-up: From the Wikipedia article Metric tensor (general relativity)...
"In general relativity, the terms metric and line element are often used interchangeably."
It occurs to me that the source of our 'semantic dispute' is that I am using the term 'metric' to refer to the metric tensor and you are using the term 'metric' to refer to the line element. Is this the case? Alfred Centauri 14:31, 17 August 2005 (UTC)
I am sorry that I lost track of this page while I was on vacation back in August. I really, REALLY dislike the Schwarzschild metric being here, and especially dislike its being at the start of the article. The Schwarzschild "chart" (as Chris would rather it be called) is a coordinate scheme and is not tied to any one metric. A link to the Schwarzschild Solution is fine, since that is a metric where this chart is used. However, the statement of the Schwarzschild Solution belongs there, not here. Coordinates are not metrics. Instead the are the rules for mapping a spacetime.
This is not a simple semantic dispute. We need to distinguish between a spacetime topology (which exists independently of any coordinate chart or metric), coordinate charts (which are independent of any spacetime topology or metric), and a metric tensor (which is a description of a spacetime topology as mapped using a chart). Look at the name of this article: It is about the Schwarzschild coordinates, not the metric tensor which is the Schwarzschild solution. --EMS | Talk 04:42, 6 October 2005 (UTC)
It seems reasonable to me that an article about a particular coordinate system would also include an example line-element of some metric tensor expressed in that coordinate system. The logical choice, in my opinion, for such an example would be the metric tensor of the Schwarzschild solution.
Quoting myself from an earlier comment: "it is my understanding that the metric tensor exists as a geometric object independent of the existence of any coordinate system". So yes, I do understand the distinction between coordinates and the metric tensor. However, I'm not sure that you do. Your statement that the metric tensor is "a description of a topology as mapped by a chart" seems wrong to me on two counts. (1) It is my understanding that the metric tensor describes the geometry of a spacetime, not the topology. (2) It is also my understanding that the metric tensor (geometry) of a spacetime is independent of any mapping. It is instead the components of the metric tensor that are coordinate dependent. Alfred Centauri 16:16, 7 October 2005 (UTC)

[edit] Presenting the Schwarzschild Solution here

Alfred Centauri wrote:

It is my understanding that the metric tensor describes the geometry of a spacetime, not the topology.

In this case, geometry = topology.

Alfred Centauri also wrote:

It is also my understanding that the metric tensor (geometry) of a spacetime is independent of any mapping. It is instead the components of the metric tensor that are coordinate dependent.

Then tell me how you can have a metric tensor without having its components. You cannot have (or express) a metric without a coordinate system. This article is not about the metric. It is about a coordinate system.

I see no need to express the Schwarzschild metric here. Instead there is a reference to the Schwarzschild solution article. As a matter of encapsulation, that is where that equation belongs. I also worry about presentation: Having the Schwarzschild solution in the introduction leads people to think that the metric is the point of the coordinate system. Instead, the coordinate system is something independent and which has important features if its own. Certainly the connection to the Schwarzschild solution is important, but this coordinate system is also used in the internal Schwarzschild solution and the Reissner-Nordström solution, and can be used with any spherically symmetric spacetime. That overall use is what is important, and not just the first way that this coordinate system was used. --EMS | Talk 20:18, 7 October 2005 (UTC)

Geometry <> topology. "Einstein's field equations fix the local geometry of spacetime but they do not fix its topology" (MTW, figure 31.5)
The metric tensor is a function of two vectors - it "must be a rule which gives the same real number independently of the reference frame in which the vectors' components are calculated" (Shutz, p61). The components of the metric tensor in a particular coordinate system are the values of the function when its arguments are the basis vectors. Thus, the Schwarzschild geometry exists in the absence of a coordinate system. The rule to determine the interval in this geometry is the metric tensor. To express the components of the metric tensor on some basis, we need a coordinate system such as the Schwarzschild coordinates. This is why I wrote (in the original article) that "Schwarzschild coordinates refers to the coordinate system for which the line element of the Schwarzschild metric in geometrized units is given by:" Alfred Centauri 22:17, 7 October 2005 (UTC)
The Schwarzschild metric is itself the genesis of the Schwarzschild coordinate system, and most definitely is used by it. There is no question about that. However, this coordinate system has become over the years much more than that. Overall, you are now saying much the same thing as I am. I just want the emphasis to be on the coordinate system, and not on how it is used. Note that in your own writings you acknowledge that the Schwarzschild metric cannot exist without the coordinate system as well as the geometry it describes.
I also scratch my head about the MTW reference above. The local geometry of spacetime is always asymptotically that of a Minkowski spacetime, and that is indeed demanded by the Einstein field equations. (This is analogous to a surface of the Earth locally being asymptotically flat.) However, what we are talking about is what MTW calls the topology, or perhaps in my view the global geometry. *Sigh*. There seems to be some semantic ambiguity about these terms, which is surprising to me.
One other thing for you to realize: There is a Wikiproject for GR in the works. See User:Hillman/Wikiproject GTR draft. One thing that may happen because of this is that the Schwarzschild metric article may be renamed the "Schwarzschild vacuum", making that formally a description of the geometry. Also, the inappropriate presense of the Schwarzschild metric in the introduction is one of the reasons that Chris Hillman placed the cleanup and attention tags on this article. --EMS | Talk 23:21, 7 October 2005 (UTC)

[edit] Role of metric in this article

I agree that the emphasis should be on the coordinate system. I suppose our disagreement is to what extend the Schwarzschild geometry 'pollutes' this article. My desire in creating this article was to make the distinction between the coordinate system itself and the geometry so it appears that our goals are the same. When I use the term metric, I mean the geometry - a coordinate independent concept. I believe that you use the term metric in the same way that I use the term line element which is most definitely coordinate dependent. Alfred Centauri 23:46, 7 October 2005 (UTC)

I don't see it so much as pollution as a distraction. Certainly if the metric was not presented elsewhere then it would be incumbent on us to present it here. However, the actual metric is given elsewhere, giving us the ability to refer to its desciption and move on. Since this article ostensively is about the coordinate system, I am happy to not express the metric here. In any case, the metric itself would not belong in the introduction. Instead it is better to give it in a section describing how the metric is used (if it is presented here at all). For isotropic Schwarzschild coordinates, the presentation of the metric may be needed. Either that or a companion article created describing the metric and its relationship to the Schwarzschild metric. --EMS | Talk 01:02, 8 October 2005 (UTC)
I find it difficult to follow all the twists and turns above, but part of the confusion might be due to the fact that there are several levels of mathematical structure in which one can discuss coordinate charts. Lowest is topological manifold, in which the allowed transformations between two charts are local homeomorphisms on the overlap of the two domains. Here, continuity is defined by pullback to a model, Rn with the standard euclidean topology. Next is smooth manifold, in which the allowed transformations are local diffeomorphisms on the overlap of the two domains. Here, smoothness is defined by pullback to a model, Rn with the standard smooth structure. (Recall that a remarkable theorem says there is an unexpected multiplicity of possiblities when n=4.) Next is Lorentzian manifold, in which we must have an appropriate metric tensor (signature -+++), and the allowed transformations are local diffeomorphisms which preserve the metric.
The notion of Schwarzschild coordinate system which I have in mind is defined at the level of Lorentzian manifold and is strictly speaking independent of physics or any field equation. (See the metric Ansatz above.) In gtr and related metric theories, however, the geometric interpretation of the radial coordinate is of course associated with a physical interpretation.
Another way to say this is that I am talking about a class of spacetimes with certain symmetries, and a class of 'polar spherical' type coordinate charts which are defined on spacetimes with these symmetries.---CH (talk) 01:08, 8 October 2005 (UTC)

OK. This all sounds reasonable to me. Alfred Centauri 03:56, 8 October 2005 (UTC)

[edit] Major revision (8 Oct 2005)

I have removed the tags and stub, since this is assuming a fairly complete form. Still to be done:

  • add computation of Einstein tensor and wave equation via exterior calculus,
  • write or expand stubby articles cited in this article

Hope everyone will be pleased with the revision.---CH (talk) 16:01, 8 October 2005 (UTC)

It's definitely a change. I am not sure that it is not at too high a level now, but I am not about to roll it back. At the least, you not only knew where you wanted to go but also how to get there. Things like the radial coordinate not giving distances between the spheres are important, and yet although I was aware of it I had not gotten to the point of "fitting" it in. For now, I will just study this article. In many ways this article is now an example of how to do this kind of thing. --EMS | Talk 17:03, 8 October 2005 (UTC)

[edit] Concerns

I am getting convinced that this article now exists at too high a level, although I am loathe to "dumb it down" until there is some sense of either where some of the higher-level material ought to go, or a consensus exists that it is inappropriate. For now, here are the areas that are of greatest concern to me:

  • The last paragraph of the Definition section is inaccessible to me. This needs to either be clarified or removed.
  • The Killing vector fields section is inaccessible to anyone not already familiar with Killing fields. Given its current state and the current state of Wikipedia, there is no way for a non-expect to glean that these are the infinitessimal transformations that one can do and have their view of the spacetime remain the same. This situation needs to be rectified: The material in this section is important and relevant to the following sections, and so cannot be left inaccessible.
  • The Coordinate singularities section is quite dense. I won't call it inaccessible, but it has some fairly hefty conceptual prerequisites as-is. In essense, this is a milder version of the problem with Killing vector fields.
  • For the Visualizing the static hyperslices section, I cannot figure out where you are going or why this is important. This section needs to either be clarified or removed.
  • The A metric Ansatz section is beautiful, but I wonder if it's inclusion is fair to readers given its high level matertial. Those concerns are doubled in the last half where you work out the Bel decomposition. Without the related articles in place to permit people to ferret out what a Bel decomposition is about, this is quite unfair. My temptation here is to remove the Bel decomposition material to the Bel decomposition article itself, where it may make an excellent example.

For now, I will not "fix" these. Some care is needed in editing Chris' work, and with any edits my goal is to improve on what is already here. --EMS | Talk 16:20, 9 October 2005 (UTC)

It's good you are not rushing to 'fix' anything, since I probably mostly agree with your complaints. But you certainly should not remove anything. Once you know the background I have in mind, you will, I think, very much appreciate this material. You'll have to be patient since adding all the background material to appropriate articles (new or old) will be a time consuming process.
It might be helpful to say that I don't envision any of Wikipedia article on a highly technical subject (such as this one) as something which should neccessarily be instantly comprehensible to every reader. The goals should be to avoid alienating readers but rather to encourage them to follow links, learn stuff, come back, and gradually understand more and more. Right now, unfortunately, not much of this background material exists on Wikipedia. This is an unavoidable problem, since I, at least, can't write the background first; I need to have specific examples where I plan to use this background in mind, so I pretty much need to write top-down. So simply removing stuff from this article will only hinder me in preparing the background articles which I admit are sorely needed at present. Although I can see that unfortunately I alienated even one well-prepared reader (yourself), I hope this condition is only temporary!
In the mean time, if you think about the background for a full year gtr course at the level of MTW, you will appreciate that noone, however well prepared, should neccessarily expect to understand everything right away, and this article is referring to some subtle points. Understanding will come once both background articles and well integrated articles applying/illustrating those ideas are in place.
BTW, when I get a chance to write about static spherically symmetric perfect fluids, you may appreciate that this article is itself background for that forthcoming article, so I do try to sometimes write bottom-up. In any event, please be patient.---CH (talk) 22:23, 9 October 2005 (UTC)
Chris - I see what you are trying to do, and actually do appreciate much of what you have placed here. For myself, I am happy to sit back and learn what I can. However, this is not just about me. As I see it, the technical level of an article must be appropriate to the type of audience the it will likely attact and to the subject matter itself. In essense, the sooner someone would find it reasonable to look at this article, the less technical it should be. At the same time, if covering the subject to a reasonable level of detail requires the assumption that the reader has a certain level of technical knowledge, then it should be more technical. So what we have here is a balancing act.
For example, the main general relativity article is fairly but not too technical. It needs to cover a fair amount of ground, but can do so at a fairly high level. It both can and should be reasonably accessible, even to someone lacking experience in the subject. However, it does make demands on the reader to master certain concepts in order to fully understand it, and I am not opposed to that. This is not simple after all.
I am concerned that this article is one that people will come to relatively quickly in the future. The Schwarzschild chart will be prominently referenced by the Schwarzschild vacuum article (assuming that we go with your naming convention, of which I approve BTW). So it will attact people of a somewhat lower technical level than others will. I therefore see this as an article that can and should be written at a somewhat lower technical level, and help to ramp those who are interested up to a higher level of understanding.
So I agree with you that it is unreasonable to expect every reader of this article to grasp it immediately. I know that you cannot do this subject any justice unless you assume that the reader possesses a certain amount of prerequisite knowledge. However, you can run people into the technical equivalent of a brick wall by being too technical. Are you sure that you should be writing about "Killing vectors" instead of isometries of spacetime (for example)? I see this article as a chance to introduce the topic of isometries, explaining it in a directly linked article if not also doing so briefly in this article itself.
To be functional, the GR articles will need to progress from less to more technical in a coherent fashion. It is something that should be planned out and considered as this is built up. It also does no good to run far ahead of what the underlying artcile structure can support. I programming, I have found top-down vs. bottom-up to be a false distinction. You cannot properly test the bottom without driving it from the top, nor can you fully test the top without running the supporting functions at the bottom. So I always start with a minimal set of basic functions that I implement fully, and finish the project by adding in the remaining functionally. In essense I create a hourglass and then fatten it up. Something similar is needed here. --EMS | Talk 01:43, 10 October 2005 (UTC)
P.S. Actually, I do plan to move some material to the Bel decomposition article as soon as possible, augmenting the discussion by explaining how this is related to the Hodge star of the curvature two-forms and to Fermi normal coordinates. ---CH (talk) 00:57, 11 October 2005 (UTC)

[edit] Equation changes - please check

I have changed this equation:

d\sigma^3 = \sin(\theta) \, dr \wedge \sin(\theta) \, d\phi + r \, \cos(\theta) \, d\theta \wedge dr = -\left( \frac{\sin(\theta)\, d\theta}{g(r)} + \cos(\theta) \, d\phi \right) \wedge \sigma^1

to:

d\sigma^3 = \sin(\theta) \, dr \wedge \, d\phi + r \, \cos(\theta) \, d\theta \wedge d\phi = -\left( \frac{\sin(\theta)\, d\phi}{g(r)} \wedge \sigma^1 + \cos(\theta) \, d\phi \wedge \sigma^2\right)

so that it will be consistent with these equations:

d\sigma^\hat{m} = -{\omega^\hat{m}}_\hat{n} \, \wedge \sigma^\hat{n}
{\omega^1}_3 = -\frac{\sin(\theta) \, d\phi}{g}
{\omega^2}_3 = -\cos(\theta) \, d\phi

Also, shouldn't this equation

{\omega^0}_1 = \frac{f' \, dt}{g}

read

{\omega^0}_1 = -\frac{f' \, dt}{g}

? Alfred Centauri 17:48, 8 April 2006 (UTC)


About σ1,σ2, good catch. Maple indexes 1,2,3,4 but in these articles I have tried to decrement to 0,1,2,3 cause that is way more cool. But due to buggy libraries I don't always decrement consistently, arghgh. As for the other question, did you think about Lorentzian transpose versus euclidean transpose? ---CH 03:04, 10 April 2006 (UTC)


I believe the killing vectors listed to be incorrect, they are currently listed as

\partial_\phi, \; \; \sin(\theta) \, \partial_\theta + \cot(\theta) \, \cos(\phi) \partial_\phi, \; \; \cos(\theta) \, \partial_\theta - \cot(\theta) \, \sin(\phi) \partial_\phi

however I believe the correct versions should be:

\partial_\phi, \; \; \sin(\phi) \, \partial_\theta + \cot(\theta) \, \cos(\phi) \partial_\phi, \; \; \cos(\phi) \, \partial_\theta - \cot(\theta) \, \sin(\phi) \partial_\phi

Neil Butcher 81.96.79.153 15:20, 14 May 2006 (UTC)

You are right--- I must have copied them incorrectly from my notes. Thanks for the correction! ---CH 03:50, 16 May 2006 (UTC)

[edit] Students beware

I completely rewrote the Oct 2005 version of this article and had been monitoring it for bad edits, but I am leaving the WP and am now abandoning this article to its fate.

Just wanted to provide notice that I am only responsible (in part) for the last version I edited; see User:Hillman/Archive. I emphatically do not vouch for anything you might see in more recent versions. I'd like to hope for the best, but see my warning in Talk:Karl Schwarzschild#Students beware.

Good luck in your search for information, regardless!---CH 17:27, 1 July 2006 (UTC)

[edit] Hats and signatures

I would do away with the hats above indices. They indicate the coefficients are with respect to a moving frame (a not necessarily coordinate basis) but they are IMHO over-pedantic and simply look bad when typeset. Nobody of course uses them in actual calculations. Besides, if they are used with ijkl they must be used with 0123 as well for consistency. And this would be just plain unbearable!

In the "Generalizations" section the staticity assumption is removed. When this is done, two forms of the metric actually satisfy the given symmetry and signature constraints (the first one is listed in the article):

ds^2 = -f(t,r)^2 \, dt^2 + g(t,r)^2 \, dr^2 + r^2 \, \left( d\theta^2 + \sin^2(\theta) \, d\phi^2 \right),

and:

ds^2 = +f(t,r)^2 \, dt^2 - g(t,r)^2 \, dr^2 + r^2 \, \left( d\theta^2 + \sin^2(\theta) \, d\phi^2 \right),

They both must be put into the Einstein equation and then it turns out the first one is valid for \scriptstyle2m<r<\infty and the second one for 0 < r < 2m.

Finally, the first section says: "Birkhoff's theorem states that every isolated spherically symmetric vacuum or electrovacuum solution of the Einstein field equation is static" but this phrasing is true only in the exterior (Birkhoff says that f and g above are independent of t which is true both in the exterior and interior). In general one would say "Birkhoff's theorem states that every isolated spherically symmetric vacuum or electrovacuum solution of the Einstein field equation is a patch of the Schwarzschild metric", but this formulation would no longer be relevant to that section. Perhaps it's best to remove it and insert the corrected version somewhere else where it would be more a propos. JanBielawski 21:10, 13 August 2006 (UTC)

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