Talk:Mathematics of general relativity/Archive 1

From Wikipedia, the free encyclopedia

Archive This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page.

Contents

Need for significant changes

This is not a good discussion of the math of GR. As-is, its contents would be better merged into the Einstein field equation (EFE) page.

NEEDED:

  1. Properly define a tensor in GR, which is a represntation of a quantity that is correctly transformed by the Lorentz transformations
    • Position vectors, scalar derivatives, etc. are tensors because they transform properly
    • Christoffel symbols do not transform properly and so are not tensors in GR even though they are expressed using the tensor calculus syntax.
  2. Describe metric tensors and how they
    • describe the shape of spacetime, and
    • are used to lower and (with its inverse) raise tensor indexes.
  3. Describe, position, relativistic velocity (uμ), and acceleration vectors.
    • Invariants from vectors through inner multiplication of a vector and its corresponsing form. (Example: ds2 = xμxμ).
  4. Describe important rank-2 tensors:
    • The EM field tensor
    • The stress-energy tensor
  5. Breifly describe curvature tensors.
  6. Breifly describe the EFE, leaning heavily on there already being an EFE article.
    • Describe strategies for solving and approximating solutions for the EFE.

I will work on this when I can get to it, but encourage others to tackle these in the meantime. This can and should be a good, comprehensive article. --EMS | Talk 15:35, 19 May 2005 (UTC)


User EMS, please try to be more polite when criticising someone else's work. I agree that a lot more needs to be done to improve the article.

My apologies for insulting you, but I have found in productive even when I cannot work on a page to explain my objections and suggest improvements.

In response to your suggestions:

  • Firstly, I think if all the above suggestions were included, the article would be too long (even taking into account some of the brief descriptions). A lot of what is mentioned in the suggestions is already on other pages linked to GR.
If this is about the math of GR, then it should be about that. If needed, it should reference articles about sub-topics, such as tensors in GR. But one way or another the points in the outline given above need to be touched on. If they are not present in the actual article, they should only be a link or two away. To the extent that other articles cover points in the outline, those should be linked to and allowed to provide the details.
  • The definition of a tensor in GR given in point 1 appears to be incorrect: a tensor in GR is a representation of a (usually physical) quantity that transforms according to non-singular coordinate transformations (not just Lorentz ones), each transformation being determined by two observers' reference frames.
It is a little more involved than that. Consecutive transformations on each index of a tensor should yield the correct overall transformation. That is not the case for Christoffel symbols and that page ducuments why.
  • If invariants are to be mentioned, then an explanation of why they are important (they're observer independent for example) should be given, as it's not necessarily obvious to non-specialists.
Agreed.
  • Regarding the descriptions of the Riemann and Ricci tensors, it may be better to describe the Riemann tensor as being made up of two bits: the Weyl and Ricci tensors; then possibly mention the physical significance of each bit. Of course, the geometrical significance of the Riemann tensor should be given too.
That sounds good. Dealing with those tensors are where this gets tricky, because articles on those topics already exist. The goal for this article is to tie it together, explaining why they are so important to GR.
  • The detailed strategies for solving and approximating solutions for the EFE should clearly be in the Einstein field equation article, not here, as its more technical than just describing the maths of GR. A brief description of what is involved may be appropriate in the maths of GR article.

Mpatel 14:07, 3 Jun 2005 (UTC)

EFE solutions is a topic that this page needs to touch on, but perhaps only touch on. The EFE article itself of course can say more. Beyond that, this is an area where each method could use an article of its own.
It will take some work, some experimentation, and some strategy to put together a good article here. My main gripe btw is not about what currently is in this page, but about the only math being presented here is the EFE or about the EFE. This article should provide a good overview of the math, and be a roadmap to a wealth of related articles.
--EMS | Talk 21:12, 3 Jun 2005 (UTC)


I see what you mean about this article being centred around the EFE; if I remember correctly, this article was part of the GR article, then I created this article (and made minor edits) hoping that people would include more maths (but nobody seems to have done so - yet). I've made a start in defining tensors - I hope the definition is sufficient for this article - and mentioned some examples of tensors in relativity. Perhaps a little section on tensors might be better.

Suggestion: is it worth having a (sub-)section for the Riemann tensor, given how important it is ? I mean, for example, about the Weyl/Ricci splitting and the physics of that, not to mention that the Ricci tensor is a contraction of the Riemann tensor - this might lead nicely into the EFE (once the energy-momentum tensor has been discussed).

Mpatel 16:01, 4 Jun 2005 (UTC)


I think that a short section on tensors and what they are is essential. Unlike a general treatement of tensors (like the classical treatment of tensors page, you can emphsize that face the GR tensors all are with respect to a four-dimensional pseudo-Riemannian manifold. Of course you need to hit on the tensor syntax, but if you do this right you can lean heavily on that classical-treatment page.

A section on the physical interpretations of the Riemann, Ricci, and Weyl tensors is a good idea. Note that the math is treated elsewhere, but should be briefly touched on and the approprate related articels linked to. Indeed, this kind of thing is why I am so critical of the current state of this article: It is not (yet) touching on these subjects but very much needs to.

--EMS | Talk 02:57, 6 Jun 2005 (UTC)

topology of spacetime in GR

As my knowledge of topological structures in GR is poor, perhaps some1 out there can contribute to the section on 'topological structure of spacetime'. Maybe some examples of spacetimes with (physically) interesting topologies (for example, ones which permit time travel, wormholes etc.) could be included. Mpatel 16:13, 6 Jun 2005 (UTC)

The "topological structure" of spacetime is that of a curved, four-dimensional, pseudo-Riemannian manifold. The individual topologies are given by various solutions to the EFE, being composed of a coordinate basis/system and a metric for the manifold as mapped with that coordinate system.
The details of specific EFE solutions are better off being covered in the Einstein field equations article, if not in articles for the solutions themselves.
--EMS | Talk 16:50, 6 Jun 2005 (UTC)

Revisions ready in my sandbox

A rewritten version of this page is now available at User:Ems57fcva/sandbox/mathematics_of_general_relativity. I propose to replace the current article with this one soon. So comments are being sought.

Do note that the part on the EFE is much shorter than the current text. So the tetrad stuff needs to be moved over to the EFE page. If noone does this first, I will do so before moving over my revisions.

--EMS | Talk 04:43, 9 Jun 2005 (UTC)

Good to see that you have made a conscious effort in rewriting the maths of gr page, EMS. Just a few picky comments:
  • As far as I'm aware, the term 'line item' is not standard in describing the metric, but I think 'line element' is standard (though maybe that's what you meant).
  • You've written the geodesic equation as \ddot{x}^a=\Gamma^{a}_{bc}x^bx^c ; I think the standard way of writing the geodesic equation is \ddot{x}^a+\Gamma^{a}_{bc}x^bx^c=0. I can't remember all the details, but I think your geodesic equation arises as you're using a metric with trace -2, whereas to get the geodesic equation in standard form, the metric with trace +2 is used (basically, I think your Christoffel symbol is the negative of the one I use as a result of the different metric signature). Although many people use the metric with trace -2 (but not me), I think most people would recognise the geodesic equation in it's standard form.
  • There is a page four-vector which I editted some time ago. It has notations for four-velocity, four-acceleration etc. (although there may be some clashes on other pages). It may be better to use the notation currently in four-vector, as it's the main article on four-vectors - in any case, I hated using aa for the four-acceleration; I experimented with the notation some time ago and preferred Aa for the four-acceleration.
Forgive the pickiness of the above comments. I appreciate that your proposed version is still in it's infancy and will no doubt be expanded.
Mpatel 15:41, 9 Jun 2005 (UTC)
The pickiness is quite forgiven. The business with the geodesic equation is sloppiness on my part, and will be changed. (I use the sandbox for major revisions and let them sit for a day or two because I am prone to that kind of stuff.) The other comments are also good.
I share your dislike for aa for the acceleration 4-vector, but all the same that seems to be the standard notation. I guess that it is hard to confuse the a's in it, or to confuse it with {a^a}_b (The Loretnz Transformation tensor). Even so I came close to using your Aa in that draft. In the end, I decided that we are acting here as reporters on the field: It is not our job to introduce new and novel concepts and notations, even if they are better. (However, if you can find an alternate notation for the acceleration four-vector in the literature, we can reference the article and run with it.) --EMS | Talk 18:25, 9 Jun 2005 (UTC)

Changes done

The new version of this article is now in place. The part of the tetrad formalism of EFE solutions has been moved to the EFE page. I saw nothing else in the old article that needed to be moved.

As noted by Mpatel, this is not a final product, but I think that it is a much better article on the math of GR than what preceeded it. More work is needed to flesh out the linkages to more abstract tensor calculus subjects. Note that I am trying to off-load much of the details on tensor calculus to the appropriate articles on the subject. This page should be more about how GR uses the tools of tensor calculus than about the tools themselves. (As Mpatel noted this would be an awfully big article if it covered all of the details. However, to me a greater problem is that most of those details are or will be or should be covered elsewhere.) --EMS | Talk 16:26, 10 Jun 2005 (UTC)

The changes made by EMS are a great start to what should ultimately be a great article. Regarding the notation for the four-acceleration, I was convinced that Aa was the standard notation (at least here in Scotland). Anyway, W. Rindler in 'Introduction to Special Relativity' (2nd edn.) Clarendon Press, Oxford (1991) uses Aa for the four-acceleration. Maybe we can get Chris Hillman to give us more insight. --- Mpatel 17:06, 10 Jun 2005 (UTC)
The change in the four-acceleration terminology is now done. If Rindler (who is an authority on relativity) uses that syntax in his SR textbook, then my request for some coverage in the literature has been met. As for Chris Hillman: If we can resolve something like this ourselves we should do so. Chris is an excellent resource, but I would think that he would rather be dealing with issues more important than whether an "a" should be capitalized or not.
BTW - I was not comfortable with referencing the existing four-vector article in this one. The four-vector article as-is is very SR oriented. For the most part, the same laws do apply in GR, but often with the regular derivatives replaced with covariant derivatives. (That is one of the big holes in this page btw - The covariant derivative and how it is used. A related issue is how the math of GR relates to the math of SR.)
One more (albeit minor) thing: You decapitalized a "The" following a colon. My understanding of the rule on capitalization following a colon is that a full sentence should behave like one (first word capitalized), but a fragment is not to start with a capital letter. In the discussion of why the EFE has only 10 values, I considered the text following the colon to be a full sentense. Hence the capital T. --EMS | Talk 19:22, 10 Jun 2005 (UTC)
I'm glad that four-acceleration notation has been sorted - it's been bugging me for ages. Regarding the sentence after the colon, I don't think I ever recall seeing a capital letter after a colon, but I'm not too hot on the grammatical role of colons, so you're probably right. I don't have a problem with using the capital T. I'll change it back. --- Mpatel 10:56, 11 Jun 2005 (UTC)
Regarding the four-acceleration, I finally remembered why people are loathe to use Aa for: That is also the symbology for the EM four-potential! My feeleing at this point is to leave it alone. In principle we can use any symbology that we like, as long as the usage is properly defined. So until and unless someone else gives us an alternate scheme and good reason to support it, Aa is and will be the Wiki "standard" notation for four-acceleration.
In the meantime, thanks for restoring my "T". Now I can drink my "T" without worrying about my "T". :-) --EMS | Talk 15:50, 11 Jun 2005 (UTC)
I was worried about that clash of notation between the four-acceleration and the four-potential. Here on WP, I use \tilde{A}^a for the four-pot. - I know it's ugly, but guess what, Rindler solves this problem for us as well: he uses φa for the four-pot. Later, I think I'll change the four-pot. to Rindler's notation and reference it. I just spent ages responding to some user who started making claims about spacetime being a myth (see GR talk page). --- Mpatel 16:08, 11 Jun 2005 (UTC)

In the list of important tensors in relativity section, there is the energy-momentum tensor for dust. Perhaps it would be better to just write Tab (just like you stated Fab immediately afterwards) because at the moment it looks as though the general definition of the energy-momentum tensor is ρ0uaub. Or even better, a little discussion of the energy-momentum tensor and important examples could be given. Just a suggestion. In case you're interested, I wrote down the energy-momentum tensor for a viscous fluid in energy-momentum density and also the EM one - you might want to check the notation for consistency. ---- Mpatel 16:33, 11 Jun 2005 (UTC).

Thanks for pointing that out. I will fix it now. --EMS | Talk 20:45, 11 Jun 2005 (UTC)

Abstract index notation

I think we should use the abstract index notation (AIN) for tensors consistently on the GR pages. Previously, I wrote things like, '...the metric tensor components are gab...', but now I believe that we should make it a standard for the GR pages to use the AIN. I hope we can reach a quick consensus (in the positive) on this point. ---Mpatel (talk) 12:33, August 14, 2005 (UTC)

I certainly can go along with it. However
  1. This is a point that should be covered in standards for Chris' Wikiproject GTR, and
  2. It is not fair to others to ask for this until it is fully explainted in the abstract index notation article.
So this is a good idea, but I think that you are ahead of things a bit. --EMS | Talk 04:46, 19 August 2005 (UTC)

Editing formulas

I have inserted a space between the Tba and the trailing comma because it made the formula look like it was a prime insted of a. Unfortunately, now the line is broken and the comma shows up first character of the new line. Can someone who knows how to insert a non-break space please fix this? Thanks. 84.160.232.20 06:35, 10 September 2005 (UTC)