Talk:Mathematics of general relativity

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To-do list for Mathematics of general relativity: edit · history · watch · refresh
  • partial derivatives to covariant derivatives; explain what this means.
  • discussion of levels of structure and local vs. global,
  • exterior calculus, including covector and Hodge dual,
  • intuition for Spinors,
  • Cartan-Karlhede algorithm,
  • Newman-Penrose formalism, GHP formalism, ADM formulation, other initial value formulations, Cauchy evolution, characteristic evolution, Cauchy characteristic matching - relation to computing wave generation.
  • matching/junction conditions.
  • EIH approximation (geodesics and inertial motion of test particles),
  • geodesic equations and Papapetrou-Dixon equations (motion of spinning test-particles),
  • relativistic multipole moments,
  • link to article on computation of connection and curvature via Cartan's method using exterior calculus,
  • congruence (general relativity) and discussion of kinematical decomposition for timelike and null vector fields,
  • the Ricci tensor directly couples to immediate presence of mass-energy-momentum,
  • via Lanczos tensor potential for Weyl tensor, Weyl indirectly couples to mass-energy,
  • local versus global isometries, conserved currents,
  • isotropy subgroup, holonomy subgroup, new article on spacetime holonomy (groups, classification of spacetimes etc.) .
  • integration by parts in curved spacetimes,
  • nonlocality of gravitational field energy-momentum in general relativity,
  • discussion and links to article explaining triangularization of Killing equations (etc.),
  • conformal structure, e.g. Carter-Penrose diagrams,
  • list various methods used in establishing Penrose-Hawking singularity theorems,
  • list open problems in gtr having more to do with math than physics, e.g. rigorous classification of spacetime singularities,
  • links to general references, point out specialized articles contain specialized references.

Throughout, article should stress

  • intuitive meaning,
  • levels of structure,
  • degree to which a given mathematical technique/concept is special to gtr, Lorentzian manifolds, etc. (for example, triangularization via Gröbner basis methods is very widely applicable in applied mathematics, as are perturbation theory methods, appropriate bundling procedure very important for smooth manifolds).

Talk:Mathematics of general relativity/Archive 1

Contents

[edit] 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)

[edit] 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)

[edit] 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)


[edit] Possibly useful eprint

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

[edit] 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)

[edit] 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.

[edit] 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)

[edit] 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)

[edit] 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)

[edit] Lagrangians

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

[edit] 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 Γ/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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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 1.731x108m / 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)