Talk:Mathematics of general relativity/Archive 1

Archive 1

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 (u^μ), and acceleration vectors.
   • Invariants from vectors through inner multiplication of a vector and its corresponsing form. (Example: $ds^{2}=x^{\mu }x_{\mu }$).
4. Describe important rank-2 tensors:
   • The EM field tensor
   • The stress-energy tensor
5. Breifly describe curvature tensors.
   • Describe the connections and their use in creating the geodesic equations of motion in GR.
   • Describe the Reimann and Ricci 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 ${\ddot {x}}^{a}=\Gamma _{bc}^{a}x^{b}x^{c}$; I think the standard way of writing the geodesic equation is ${\ddot {x}}^{a}+\Gamma _{bc}^{a}x^{b}x^{c}=0$. 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 $a^{a}$ for the four-acceleration; I experimented with the notation some time ago and preferred $A^{a}$ for the four-acceleration.

• Might want to redirect electromagnetic field tensor to Faraday tensor.

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 $a^{a}$ 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 $A^{a}$ 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 $A^{a}$ was the standard notation (at least here in Scotland). Anyway, W. Rindler in 'Introduction to Special Relativity' (2nd edn.) Clarendon Press, Oxford (1991) uses $A^{a}$ 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 A^a 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, A^a 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 $\phi ^{a}$ 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 $T_{ab}$ (just like you stated $F_{ab}$ immediately afterwards) because at the moment it looks as though the general definition of the energy-momentum tensor is $\rho _{0}u_{a}u_{b}$. 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 $g_{ab}$...', 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 T_ba 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)

Assessment comment

The comment(s) below were originally left at Talk:Mathematics of general relativity/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Comment(s)

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Hello, I would to comment the statement "connection coefficients are not the components of a tensor". I think this a common mistake, or at least an approximation, to explain that Christoffel symbol are not the components of a tensor. The connexion coefficient can be defined as the difference between the unique metric compatible connexion and the partial derivative in a defined coordinate system. This is therefore linear and as a result it is a tensor. The confusion comes from the fact that if you take a different coordinate system, then the partial derivative is a different operator, and therefore the Christoffel tensor obtained is also different. This why some people say the Christoffel symbol does not transform as a tensor, actually it is because it is a different tensor, because constructed from a different partial derivative operator. This is why I would like to suggest we modify the statement, saying it is indeed a tensor because it is linear, but that it is defined respectively to a base, and therefore constructing it for a different base will lead to a different tensor. Nymbus.jp (talk) 01:53, 15 January 2009 (UTC)

Last edited at 01:53, 15 January 2009 (UTC). Substituted at 15:21, 1 May 2016 (UTC)

My turn

Chris has suggested that this article needs a HUGE cleanup - and he's right. I'd like to have a shot at this if nobody minds. I'll have a bit of time at the weekend. ---Mpatel (talk) 16:58, 15 September 2005 (UTC)

Excellent! ---CH (talk) 01:20, 16 September 2005 (UTC)

I can't say that I object. My edits were intended to get the page moving in the right direction, but it has stagnated since then. So I say to go for it. --EMS | Talk 04:11, 16 September 2005 (UTC)

P.S.: To Chris -

There is one thing that is bugging me: It is all fine and dandy to place the cleanup tag on this article and say that it needs a huge amount of work. However, it would be much better to place in this discussion page a more specific description of what you think is wrong and how this page can be improved. I have found that making explicit points does a lot to propel actual improvements. For example: When I first came to Wikipedia the GR page was terrible even in comparison to the state that you found it in. I made a number of comments about it initially, and within a few weeks, they had been addressed by others. --EMS | Talk 04:40, 16 September 2005 (UTC)

I put the tag there (before MPatel's revision). I agree that proper comments would have been much preferrable, but I was trying to get a lot done that day.---CH (talk) 01:42, 18 September 2005 (UTC)

Should we include an outline of Einstein's derivation of GR theory as described in the paper? Pepebuslo 23:13, 12 September 2007 (UTC)

The title is wrong - this is not mathematics, but a very old-fashioned writing on physics. Stay basis-free - with the notion of Singer & Thorpe, Kulkarni, Nomizu and followers - and some skilled mathematicians will jump into. In this not basis-free notion everything possible already has been found out by Einstein himself (and by Lichnerowics for gravitational waves), up to that moment where the index notation couldn't help any longer. If you don't do that, there will be no progress in GR beyond speculations like strings and higher dimensional spaces. — Preceding unsigned comment added by 130.133.134.62 (talk) 18:15, 8 June 2012 (UTC)

Mpatel's major revision

I've just made some major revisions to the article. I intend to keep as much of EMS' work as possible, as it's all essential. Just need to work around it. A few things:

• I've included some physical reasons for choosing manifolds as the 'basis' for the maths of GR.

• Still need stuff on tangent spaces and more on tensor and vector fields.

• I'll leave the high-powered maths of fibre bundles, spinors etc. for others (Chris ?) to deal with, but I'll create the sections and write a few words to get started.

• I think some more on the topology of manifolds should be included.

• Some info. on (covariant and Lie) derivatives + explanation of why partial derivatives are inadequate. For Lie derivatives, the intimate connection with vector fields needs discussion; then this needs to be linked with local 1-parameter diffeo.'s.

• Tonnes of references and reliable external links (both these should be easy).

I think I underestimated the magnitude of this major revision task! Comments welcome. ---Mpatel (talk) 15:09, 17 September 2005 (UTC)

A few more changes have been made. The article is still a little 'choppy' and is by no means complete, but before I try to get it to flow smoothly, I'd like some feedback on my edits. Just want to make sure we're all on the same track. ---Mpatel (talk) 11:28, 18 September 2005 (UTC)

I'll have another major burst of activity in this article on Friday, Saturday, Sunday and Monday (long weekend). In the meanwhile, feel free to make any changes you see fit (by 'you', I mean probably Chris and Ed). ---Mpatel (talk) 09:44, 19 September 2005 (UTC)

You (MP and EMS) have probably noticed that I've been working recently on some mathematical background stubs (e.g. kinematic decomposition of timelike unit vector fields). I plan to edit Energy conditions and some other articles on MP in near future, also to try to create new stubs on the Riemann tensor in gtr and on the meaning of the EFE. Regarding the latter, some of this might eventually be merged with existing articles, but in light of the recent problems with wikiservice (MUUUUCH better in last few days!), I am thinking there might be a good point to having more short articles viz. fewer but very long articles. Short articles are easier to create and organize, and we are less likely to encounter inadvertent edit conflicts. Does anyone know what the accepted wisdom is vis a vis whether many short articles or few long articles are best for wikiservers, especially in the case of articles containing a good deal of LaTeX style pseudocode?---CH (talk) 00:45, 20 September 2005 (UTC)

First of all, presentation should be the first consideration here. Wikiserver considerations should come afterwards. However, I would think that shorter articles would be better for the servers too. --EMS | Talk 16:16, 20 September 2005 (UTC)

I've managed to find some time to work on the article. If there are any suggestions for more technical info. (e.g., Chris, I know you're quite into the work on differential equations, so I can mention links to more detailed articles if you want - tell me which ones), then let me know. I've also made a few red links on solving geodesics and computing Christoffel symbols. Some juicy stuff on Christoffel symbols on the way ! ---Mpatel (talk) 16:59, 20 September 2005 (UTC)

To do list

I just added a few random items off the top of my head. From the rapidly growing length of this list (which is not yet comprehensive), I draw the conclusion that this article must focus on trying to orient the befuddled newbie by describing in nontechnical terms the 'big picture', and sending him/her to specialized articles for more detail.

Instead of offering a long long list of references in this article, it might be best to link to the graduate textbook and special topics sections of General relativity resources, and mention that we intend that all specialized articles will eventually contain appropriate references.---CH (talk) 23:46, 21 September 2005 (UTC)

Possibly useful eprint

Hi, just notice this new eprint discussing influence of gtr on differential geometry.---CH (talk) 23:36, 28 September 2005 (UTC)

Concern ...

Let me express a bit of concern about the intent and direction of this article. It seems to me to be turning into a book, rather than an article. Simply listing every topic from topology and geometry to string theory already would make this a very long article. Then, trying to provide a formal, one paragraph description of each topic will only make it longer. (And I note that this article is already almost too long for a WP article).

I'd like to get you to ask yourself "what am I really trying to accomplish with this article?", and then review if this is really the best way of accomplishing that.

In particular, the reader who already knows about metrics won't want to read the first half of this article. And the reader who doesn't know about metrics will promptly zoom off elsewhere. So I'm concerned that you may not be acheiving your goals here.

I'm also concerned that some topics I'd consider as "central" aren't even mentioned: e.g. the Lagrangian variational formulation, (which is the "natural" way to get einsteins eqns; it also opens the door to electrodynamics, kaluza klein, hopf fibrations, and string theory.) So to even mention these central topics would grow the article even further.

FWIW, one of the nicest articles I've seen linking GR to topology is Raoul Bott, On some recent interactions between mathematics and physics, Canad. Math. Bull 28 (1985) p 129-165. This was an invited lecture to the Canadian math soc. and is a good whirlwind tour of the subject. (and despite being a whirlwind, i.e. giving only very breif descriptions of each point, it is still much much much longer than this article). linas 14:35, 29 September 2005 (UTC)

Linas, your concerns are noted and those of us involved in the GTR project have thought about and discussed these issues.

At the start of this article, there are 4 tags - read them.

I am in the (long) process of rewriting this article. As for the article becoming very long, I am constantly thinking about that and am trying out things to see how the article will look - the article is by no means complete; hence all the tags.

As for the Lie derivative stuff, the reason I want more details of the Lie derivative on that page is because I won't have to include more stuff on Lie derivatives in this article !!! I certainly don't want that nasty tensor expression here.

As for your argument about the readers who know/don't know about metrics, well this article will serve the needs of both readers (another thing we've included in the GTR project).

As for this article turning into a 'textbook', well that's tough. If you know anything about the maths of GR, you'll know that there's a huge amount of it. The purpose of this article is to tell people which areas of mathematics are used in GR and WHY. At the moment, it's been agreed that I (as a project member) take on this task for the time being and see what people think of my edits. I've just started revising the article, and as indicated above, am nowhere near satisifed with what's there at present.

Giving a review of this technical area will inevitably involve links to more technical areas. Like I said above, if you want less technicality in this article, then you need more technicality in other articles, which is why I feel more technical details should be given in the Lie derivative article. ---Mpatel (talk) 15:56, 29 September 2005 (UTC)

I must admit that the gargantuan list at the beginning of the article is more that a little daunting. Based on my reading of other comments, I realize that this article will not alone cover all of the listed topics. However, that list does give people that impression that it will. My suggestion is to move that grand list to Chris' Wikiproject GTR page, and condense the to-do list for this page to the items that this page needs to cover directly. --EMS | Talk 20:41, 29 September 2005 (UTC)

Linas, we have already recognized the issues you raised, so you can be sure we will bear them in mind as we develop th e article. Give us a change, please. You say you like the long review article (which I will try to obtain, BTW). So you must know that long is not neccessarily bad, but in any case it remains to be seen how long this article will turn out to be.---CH (talk) 15:53, 1 October 2005 (UTC)

Covariant derivatives and affine connections

I've decided to combine the two sections on (affine) connections and covariant derivatives, as they are intimately connected (no pun intended). I'd like some feedback on this please. ---Mpatel (talk) 15:23, 25 September 2005 (UTC)

Will eliminate most of the current work in the 'covariant derivatives' subsection, as it's adequately covered in covariant derivative. Will replace it with information on the use of the covariant derivative in GR.

Lie derivatives

Again, some feedback on this would be appreciated. Maybe we can get some of the hardcore maths buffs here - maybe User: Tosha - to help us write something on Lie dragging in the Lie derivative page. ---Mpatel (talk) 11:42, 1 October 2005 (UTC)

Tensors in GR

I've tried to include some intuition about the linearity property of tensors. ---Mpatel (talk) 11:42, 1 October 2005 (UTC)

The first paragraph of the metric tensor subsection needs to be reworked. ---Mpatel (talk) 09:48, 8 October 2005 (UTC)

I've tried to improve the intro. of this section a little. I'm still working on making this section better, especially the 2 subsections, as well as reducing some of the technicalities on tensors. ---Mpatel (talk) 14:49, 14 October 2005 (UTC)

I believe there may be an error in this section. Since I'm only learning, I'm not sure, so I let those who know best to decide. This section says:

• At $\scriptstyle \,p,$ these two vector spaces may be used to construct type $\scriptstyle \,(r,\,s)$ tensors, which are real-valued multilinear maps acting on the direct sum of $\scriptstyle \,r$ copies of the cotangent space with $\scriptstyle \,s$ copies of the tangent space.

I think it should rather say:

• At $\scriptstyle \,p,$ these two vector spaces may be used to construct type $\scriptstyle \,(r,\,s)$ tensors, which are real-valued multilinear maps acting on the direct sum of $\scriptstyle \,r$ copies of the tangent space with $\scriptstyle \,s$ copies of the cotangent space.

This is because functionals from the cotangent space would apply on vectors of the tangent space. What do you think? Ferred (talk) 17:18, 12 January 2013 (UTC)

Riemann tensor

Created the page Riemann tensor (general relativity). Can dump (and organise) any relevant technical details in this new page.---Mpatel (talk) 10:25, 8 October 2005 (UTC)

• Why? What's wrong with the Curvature tensor page? I know that it covers the Riemann tensor generally in differential geometry, but why not just have a section within that page which talks about its implications to general relativity? Mike Peel 13:07, 26 January 2006 (UTC)

There is a lot that needs to be discussed about the Riemann tensor in GR, for example, the importance of certain Riemann invariants, classification of the Riemann tensor in GR, physical significance of the Riemann tensor, splitting of Riemann tensor into 'Ricci' + 'Weyl' and what each of these bits mean, to name a few... MP (talk) 14:01, 27 January 2006 (UTC)

Lagrangians

Just created a new article: Variational methods in general relativity. ---Mpatel (talk) 08:33, 8 October 2005 (UTC)

Analysis of edits

Mpatel -

I have given the article a once-over per your request. There are two items that I uncomfortable with. The first is your description of a tensor in the section Tensors in GR. I cannot see how that introduction communicates what a tensor is or how it is used. That equation looks correct, but the indexing is unwieldy, and you do not explain what T is. I would advise looking at Wald pp. 20-22 for an example of a more coherent description.

The second item is your description of the covariant derivative. You need to tell people what $\Gamma$/connections are before then.

Overall, it is good to see this article being taken well beyond where I left it. I think that you pretty much have a good overview of the subject in place. However, I very much advise being sure that you have the basics firmly in place and leave the more obtuse materials for other articles in the hierarchy. Be advised that my concerns with the GR pages are becoming less ones of making sure that the material is correct and reasonably complete and more ones of seeking to have the material be organized and accessible based on what is in the article and/or the articles that are being linked to. As you continue to edit this article please keep in mind that a correct but unintelliable article is no better than an incorrect but understandable one. (This is not to say that this article is curently unintelligable, but rater that that concern is the reason for my comments above.) --EMS | Talk 03:12, 14 October 2005 (UTC)

Spacetime as a manifold

I plan to reduce this section slightly, once the article spacetime has enough details. Also need to mention the local isometry problem in here somewhere (and the Cartan-Karlhede algorithm). ---Mpatel (talk) 15:26, 14 October 2005 (UTC)

EFE

It is absolutely essential to somewhere discuss the evolution equation which explains how mass/momentum indirectly creates long-range gravitational interaction. That is, the EFE says that the immediate presence of mass/momentum in some region causes Ricci curvature there, but we need to explain how Ricci curvature causes Weyl curvature. In particular, in the course of concentrating matter to form the Sun, we move stuff around which causes gravitational radiation and gradually curls up the vacuum surrounding the increasingly dense concentration of matter. Roughly speaking. See my little essay on the RelWWW website.---CH (talk) 18:20, 18 October 2005 (UTC)

Burnout

Ok, I've been slaving away at this article for about a month and I feel burned out. I've removed the cleanup tag as I think the article has been sufficiently 'cleansed'. The inuse tag has been removed as I plan to stay away from this article for a while (I've been looking at it for too long).

I also feel as though I've been hogging this article, so feel free to make any edits you (CH and EMS ?) think are appropriate. Please bear in mind that a lot of time has gone into rewriting the article, so I'd suggest making any major edits judiciously - or discuss these first. Of course, the article still needs a lot of content to be mentioned (see the todo list - I've just added a few more things on this list) as well as external links and references, but some of the stuff on the 'pending task' list will inevitably be mentioned elsewhere (for example, infinitesimal holonomy groups - too technical to even be mentioned in this article). Enjoy...

So, now I can start exploring some of the other GR pages... ---Mpatel (talk) 14:10, 21 October 2005 (UTC)

You've done some great work! I'll try to work up the courage to assess how to continue, and I'll ask you before I make major changes. ---CH (talk) 18:35, 21 October 2005 (UTC)

Thanks. Just a point about the 'Why tensors ?' section at the start. I felt that an explanation of why tensors are used in relativity should be given straight away. I did elaborate on this justification later ('Tensors in GR'), so I don't mind if a mini-merger is done here. ---Mpatel (talk) 08:28, 22 October 2005 (UTC)

Felber vandal

An anon using IP 141.155.124.124 added the following to this and to General relativity:

On February 11, 2006, noted physicist Dr. Franklin Felber announced a new exact solution to the Einstein field equations. His new solution takes into account the gravitational field of a mass moving close to the speed of light, which has never been done before. His research shows that a mass moving faster than 57.7% of the speed of light (approximately $1.731x10^{8}m/s$) will generate a narrow antigravity beam in front of it, thereby gravitationally repelling other masses lying in its immediate path. The greater the velocity of the mass, the stronger the force of the beam. Dr. Felber believes that his new solution can revolutionize space travel, predicting the possibility of a payload to be transported at a sufficient percentage of the speed of light by the end of the century.

This apparently refers to an eprint I must have missed, [1]. (Felber's affiliation is given as Physics Divsion, Starmark in San Diego, which might raise a few eyebrows; a minute with Google fails to clarify whether this "Starmark" is the same as a similarly named real estate company in that area, but with more effort this can no doubt be determined.) Needless to say, this kind of "announcement" of an unpublished (and possibly controversial!) eprint is utterly unacceptable in an encyclopedia!

I haven't had a chance to look at the eprint yet, which dates from May, but I noticed right away that the claim that this alleged solution takes into account the gravitational field of a mass moving close to the speed of light, which has never been done before suggests a profound ignorance of the literature; see Aichelburg-Sexl_ultraboost and citations therein. This is not a good sign, but I'll keep an open mind until I look at the eprint. ---CH 03:34, 14 February 2006 (UTC)

I'll have to come back to this later (since I'm in the middle of other stuff), but a glance at the actual eprint suggests that a better description would be that someone (possibly Carmeli, not Felber) has allegedly noticed a previously overlooked aspect of test particle motion in a known solution, namely some ultraboost (more than one is possible, confusingly enough!) of the Kerr vacuum, not to have found a new exact solution of the EFE. Felber apparently claims to have given a solution of the geodesic equations which validates the alleged effect.

The abstract is confusingly worded, but hopefully Felber knows the difference! This should become clear once I have a chance to read the eprint more carefullly. I did notice from glancing over the first few paragraph that it features some odd turns of phrase. Mashhoon does mention Felber's eprint in his own recent eprint Beyond Gravitoelectromagnetism: Critical Speed in Gravitational Motion, but this mention is very low key--- certainly I wouldn't read it as an "endorsement". I did see this Mashoon eprint, but haven't studied it yet.

Exotic propulsion is a fringe subject which has numerous very enthusiastic fans. I would caution them against letting their hopes overwhelm their critical judgment, and remind them that in an encyclopedia, we should strive to give an accurate and balanced account of the current mainstream thought in areas of potential scientific controversy. Let me just stress this: I would caution everyone against accepting uncritically anything said by anyone about alleged anti-gravity propulsion. Newcomers often mistake coordinate effects for physical effects (in fact, even experienced physicists do this much more often than I would like), and while Felber must be discussing some kind of "ultraboost", there are various possible "limits" of the Kerr or Schwarzschild vacuums, so this quickly gets very thorny. Carmeli and Mashhoon are well known in classical gravity, but I certainly wouldn't buy any of this until I have had a chance to check some computations myself, and no-one else should either!

And just to be clear, inserting "research announcments" in these WP articles is clearly vandalism, quite apart from anyone's judgement of the scientific validity (or lack thereof) of the claims made in the announcment.---CH 04:40, 14 February 2006 (UTC)

Perspective on article

I've come back to this article again. This time, I'm more interested in the organisation and how it relates to content. For example, I've just added a handful of 'main article' links. This is to reduce the content in this article that is (possibly) repeated (or explained better) in another article. Related to this, affine connection is currently a stub and some of the work in 'maths of GR' could be transferred to that stub. MP (talk) 18:14, 17 February 2006 (UTC)

I've transferred the material in the affine connection section of this article to the affine connection article. Now I can slim the affine connection part of this article and include relevant notions for GR. MP (talk) 17:07, 1 April 2006 (UTC)

Students beware

This article concerns a topic dear to my heart, and I had been monitoring it for bad edits, but I am leaving the WP and am now abandoning this article to its fate.

I emphatically do not vouch for anything you might see in more recent versions, although I hope for the best.

Good luck to User:Mpatel and to all students searching for information, regardless!---CH 02:02, 1 July 2006 (UTC)

Suggestion for recently added subsection

May I suggest that the new subsection on Solving the Einstein field equations be moved to the article solutions of the Einstein field equations, as it clearly belongs there and fits in better over there than it does here. The current article is supposed to be an overview of the maths of GR. Thanks. MP (talk) 22:16, 23 January 2007 (UTC)

Merge proposal

From the talk page of the "introduction" article (Talk:Introduction to mathematics of general relativity#The header), there seemed in June of last year to be a reasonable consensus to merge the articles, but this wasn't actioned. I've therefore started the formal procedure, which will result in the articles being merged by default in two weeks (or sooner if consensus develops here). I agree with the participants on the "introduction" talk page that the main article is, in fact, a great deal more accessible to the non-technical reader, and I don't expect that there's going to be a lot of actual content that needs merging. Tevildo (talk) 15:03, 13 January 2008 (UTC)

• Support as nominator. Tevildo (talk) 15:03, 13 January 2008 (UTC)

The introduction article to me on a cursory scan seems not to be an introduction, but rather a summary of core mathematics, which is a quite different thing entirely. Would others agree with this? LinaMishima (talk) 04:57, 28 January 2008 (UTC)

• Support. The introduction article is actually more technical than this article. It merely summarizes the math used for general relativity, not introduces it. That article doesn't have much that this article doesn't have. Ummonk (talk) 20:21, 4 February 2008 (UTC)

• Protest. If the articles are too similar, maybe an introduction could be a bit more general and less technical. General relativity is a vast issue, and the mathematics required covers (partly) a slew of undergrad maths courses. There should indeed be two mathematical introductions: one for understanding the definitions of the equations (tensors, metrics, etc) and other for more general knowledge (maybe to help understand the particular solutions?). It serves more people than making the merged article pithy and technical enough for the students of the subject, for then mere amateurs would have no access to the subject on a broader level. Both are also rather lengthy. --Sigmundur (talk) 22:06, 5 February 2008 (UTC)

• Oppose. Ummonk, no offence intended, but do you actually understand the math in either of these articles? This article requires a much deeper level of understanding of abstract algebra and differential geometry than the introductory one, which requires much less background to understand, but also less rigorous. 98.203.237.75 (talk) 04:57, 20 July 2008 (UTC)

• Oppose. For those of us without formal higher-level mathematical education, many of the mathematical articles on Wikipedia are too difficult to understand. Having the separate Intro article serves to keep much of this material still accessible to the average reader. I wish there were more mathematical articles that did not lapse into jargon that's impenetrable to the lay reader. TJRC (talk) 18:30, 3 September 2008 (UTC)
• Oppose. Both articles are very different in nature of length and depth. Should not be merged. The introduction article already has more than enough info to be fitted into the article without the general introduction. Keep both, but mention something like "For an introduction on..., please visit...." Prowikipedians (talk) 15:23, 25 November 2008 (UTC)

• 'comment. Maybe some kindhearted mathematician will write an introduction to the introduction for those of us stuck back here in precalc. I think there's quite a lot of people now who have read layman's texts on relativity (Einstein even wrote a basic-level text) but have no math background. How about an introduction to the math of fields and vectors? I don't know whether that addresses the merger, but I'll leave that up to you guys. —Preceding unsigned comment added by 206.53.64.108 (talk) 12:47, 11 June 2009 (UTC)

• Oppose. I have been attempting to learn the mathematics of general relativity and have made some slow progress. I found this article a very good mathematical introduction. It presented all the mathematical tools in a concise and direct way. Its major strength was in the discussion of "Parallel transport". That was an entirely introductory and comprehensible description around which the mathematics became understandable. Many more advanced articles do not give the same insight.Mikebernstein (talk) 21:55, 26 October 2009 (UTC)

• Support The fact that this article is easier to understand than Mathematics of general relativity is completely irrelevant to the question of whether these two should be merged or not. That is an excuse for content forking. What it does indicate is that the Mathematics of general relativity needs to be better written (i.e. that it is probably violating WP:NOTTEXTBOOK). --Mcorazao (talk) 21:36, 7 June 2010 (UTC)
  • Comment (This is still live?) It would appear from the above that there's a consensus against merging, so it might be time to remove the tags. However, I still feel I should point out that the issue is that _this_ article (Mathematics of general relativity) is _easier_ to understand than the so-called "Introduction" article. Perhaps we should just swap them over. Tevildo (talk) 01:54, 10 June 2010 (UTC)

Editor Assistance

This article is currently under discussion at Wikipedia:Editor assistance/Requests#Math of General Relativity and page swap D O N D E groovily Talk to me 16:31, 25 September 2010 (UTC)