Talk:Exact solutions in general relativity

Rename
I've renamed the page 'exact solutions' to 'Exact solutions of Einstein's field equations' as was requested and agreed upon by several editors. --- Mpatel 11:29, 12 Jun 2005 (UTC)

Segre and Petrov classifications
For anyone interested in such things, I've created a page on the Segre classification of rank two symmetric tensors and on the Petrov classification of the Weyl tensor. ---Mpatel 17:37, 14 Jun 2005 (UTC)

Major Rewrite
I've just added much new discussion, (too many?) more examples, citations, and I've reorganized everything.

I left the material on the Ricci decomposition (into Weyl tensor, and the fourth-rank thingies built form Ricci scalar and trace-free Ricci tensor respectively), but think it should be moved to an new article on decompositions of the Riemann tensor, which would also include the even more useful Bel decomposition.

I would like to remove the citation to Adler et al., since I think this is superceded by the new citations, which are specific to exact solutions.

The section on "difficulties" omits some issues. Once articles on the full suite of fluid solutions and so forth are available, possibly examples should move to these articles as appropriate. Or maybe not?

There is an urgent need for a local structure versus global structure article. Also articles on energy conditions, congruences, closed timelike curves, completely solvable, inverse scattering transform.


 * Good reorganisation.


 * I suppose it could be argued that there are too many solutions listed - although I have no objection.


 * Agree with the comment about moving the Riemann tensor decomposition to another article, regarding which, I think EMS and myself agreed that a discussion of the Riemann tensor decomposition (and it's physical interpretation - 'Riem = Weyl + Ricci') would be good; it might also link in nicely to the Segre and Petrov classifications.


 * I was gonna mention the 'fluid' issue (including more general fluids), but I think you implicitly covered that in your comments above. I wrote down the energy-momentum tensor for a general fluid some weeks ago (see that article). Maybe an article (or a section in an article) on fluid solutions would be good: the fluid solutions could be mentioned there.


 * I changed some words like 'horrendously' and 'damn' to less POV (point of view) words. Although I agree completely with what you said, I think such words are perhaps too emotive in a wikipedia article. :)


 * There is actually a short article on closed timelike curves, although it's called closed timelike curve. I was gonna start an article on the energy conditions, but given the fact that different authors mean different things when they talk about the strong, weak and dominant energy conditions, I eventually chickened out of creating such an article. Maybe an authority like Wald should be used regarding such terminology (or even better, whoever first used the terms, 'strong energy condition', etc..., and then say that some authors have varying terminology). ---Mpatel 17:05, 23 Jun 2005 (UTC)

I said d--n?!! (Checks diff) Oh, I ""did"" say that. Good catch, thanks.

Yes, my plan was to write (or get someone else to write) an article on "fluid solution" (or possibly even better, "perfect fluid solution") and "electrovacuum solution" to go with existing articles on vacuum solution, null dust solution. Possibly some of the introductory remarks in Monochromatic electromagnetic plane wave could be moved to the new article on electrovacuum solutions.

I'll remove the the Adler et al. reference since no big objections. Before making the promised infobox for exact solutions (still mulling over design), I think a good warmup would be to make an infobox for an article on relativistic units (c=1, G=1), following the very nice appendix in the textbook of Wald.

Actually, I think it makes sense to follow the names of the energy conditions as used in some recent and widely used textbooks. ---CH

A request I forgot to make yesterday: the Types of Exact Solution section of the current version of this article contains some old material on the energy of the gravitational field. I would like to remove this, but to incorporate a similar comment in the article on the Einstein field equation, which I would like to rewrite (or get someone else to do that). ---CH

Another afterthought: re too many examples: actually, we shouldn't neglect the most basic stellar models, static spherically symmetric spacetimes in which a perfect fluid interior region is matched to the Schwarzchild vacuum exterior region. But the exact solutions article could refer to that article, which could contain a list of the most important examples, such as the Schwarzschild fluid. See also my comment on the Schwarzschild coordinates article. There are several nice papers on the arXiv which should be consulted by anyone wishing to follow up on these suggestions. The requested article on matching an interior region to an exterior region should cite the book by Eric Poisson. ---CH

Yet another afterthought: nor should we ignore minimally coupled massless scalar field solutions such as the Janis/Newman/Winacour scalar field, which models a static spherically symmetric gravitational field with an accompanying scalar field. At present, these are not important for explaining observations, but that just might change. Also, scalar fields are a good source of toy models for exploring more realistic fields. For example, they have frequently been used recently in studying gravitational collapse. Come to that, there should really be an article on the phenomenon discovered by Matthew Choptuik in numerical simulations of the formation of a black hole. --CH

Tetrad Formalism
Hi, MP,

I see you moved some material about tetrads to this article. I want to argue that in the long run, this material should be much expanded and elaborated, but should have another home.

First, let me say that I agree this material is more than important--- it is essential for working with gtr! You could not possibly find a bigger fan of tetrads than myself: I have a database containing thousands of local solutions of various kinds--almost all consist of frames (vierbeins) or NP tetrads in some local chart. I always have trouble understanding why they are not the default in the physics literature, since components evaluated wrt a frame (rather than a coordinate basis) are the physical components which would be measured by an observer.

Nonspinning inertial frames are particularly important. These have the property that all covariant derivatives wrt timelike unit vector vanish, so integral curves of the timelike unit vector field are timelike geodesics and the three spacelike unit vectors do not rotate as you move along one of these integral curves. So this is as close as you can get in Lorentzian manifolds to the usual inertial frames from special relativity.

So one of my biggest goals in rewriting the GR pages here would be to help graduate students and other users of gtr to learn how to perform and interpret computations using frames. For example, the Bel decomposition I keep mentioning is always taken wrt a frame, and it can always be interpreted immediately in terms of the tidal forces and spin-spin forces which would actually be measured by the corresponding observer!

BTW, I haven't yet tried to create the planned infobox for exact solutions, but my intention was always that the "recommended version" display a frame or NP tetrad, from which one can recover the usual line element. ONe virtue of encouraging people, when given a line element (say from the Exact Solutions book) to devise a frame from which one can recover that line element (often but not always straightforward) is that this encourages users to think about observers from the outset, which is a major step toward geometrical/physical interpretation, at least locally. As we now, global interpretation is what we really want, but this is even harder to achieve--- but clearly having some local interpretation is a step in the right direction.

In a related vein, I think another important goal should be to encourage users of gtr to get into the habit of never writing down a line element without specifying the intended range of the coordinates. I can point to many instances where failure to do this has led to serious error in the literature, yet physicists resist this easy and valuable good habit.

Anyway, I am still working through where I think this material should go, but while this article should certainly link to it, I don't think it should be the ultimate "home" of this stuff. Right now I plan to write articles on frame field, Bel decomposition, NP tetrad (or NP formalism), and link out from this article and many others. Or perhaps better would be an article pulling together the two most common decompositions of the Riemann tensor (Bel decomposition and the Ricci decomposition into the Weyl tensor, a fourth rank tensor built from the traceless Ricci tensor, and a fourth-rank tensor built from the Ricci scalar), and an article pulling together coordinate bases, frames (vierbeins, dreibeins), and NP tetrads. Some experimentation may be necessary to see which works best. When I start on this task, if you have no objection, I'd like to move/amplify the material on frames which you moved to this article to the appropriate new article. --CH


 * Hi CH. I have no objections to your task of moving the tetrad section to a more comprehensive article. I moved it to this article as, for now, it seems better here than where it was before. You're right, thinking long-term, it should have a separate page. I always found it a struggle relating tetrads and metrics, as none of the textbooks really covered it adequately [Daring suggestion: have you thought about writing a book on GR ? - maybe you've already done that ? Or maybe you have a set of online notes or something along those lines ?]. ---Mpatel 29 June 2005 09:48 (UTC)

Hi, MP. I have been collecting notes for years for a book on gtr, and even have had an inquiry from a fine math/physics publisher. (Haven't spoken with the series editor for a while, so don't know if they're still interested. I hope so, because they actually pay trained people to study manuscripts in this particular series to look for errors--- some publishers don't have as rigorous a review process.  This is especially important in my case because I don't have graduate students I can put to work, heh.)

One of my concerns about writing for Wiki is that there might be some conflict between my book project and Wiki, since I am told I need to be wary about putting figures into Wiki pages which I also want to put in my book draft. Rather frustrating-- that's the hold up on the more elaborate figures I promised in the pp-wave article and such like. I do have another very slowly moving project which might lead to some separate expository papers, which might eventually turn up on a revised version of RWWW. Of course, if you read the tech news, you are told that books are heading rapidly for extinction. Unfortunately, it's not at all clear to me that Wiki will still be useful in ten or twenty years (concerned about wikispam and suchlike--- yes, I can see it's not a big problem in the GR pages at the moment, but I can see that this has the potential for spoiling our fun, much like spam has spoiled much of the fun of email.) So at this point, my book project still takes priority. Anyway, thanks much for the encouragement! I seem to need this to fuel my engine--CH (talk) 1 July 2005 03:39 (UTC)

Tetrads and an explicit exact solution
Hi, CH (I assume you've prob. just come from the Petrov classification talk page :)). Once an article on tetrads is created, I was wondering whether it would be appropriate to illustrate important features of the tetrad formalism (physical and mathematical) by stating an exact solution and highlighting (over and above the things you mentioned above) things like how to work out the Petrov type (nice link with Bel criteria here), then perhaps writing out the Weyl tensor in terms of basis bivectors (maybe do same for Riemann) - of course, a 'do-able' (by the casual reader) solution should be chosen, maybe a conformally flat example or a type N spacetime. Plane waves or FRW models may be a good example. Comments ? ---Mpatel 2 July 2005 11:59 (UTC)


 * Hi, MP (yep, I come from Petrovland), how about this? Choose an example (most readers here will probably be more interested in FRW than pp waves), and some computations and discussions to that article in a new section, then add pointer in this article to this example.  If this seems to work out well with FRW, you can repeat with further examples.---CH  (talk) 2 July 2005 20:39 (UTC)

Musings on Style of GTR Articles
Hi again, MP,

On a related topic, I've been reading more essays on Wiki style and can see that many wikipedians feel the encyclopedia should stop well short of turning into a textbook. I think we both feel that as long as we start simply, we can wind up getting fairly technical, particularly if the gtr pages are ultimately well organized and written according to a coherent vision. Anyway, it's important as a practical point to try to keep individual articles as short as possible. (BTW, I lost all my work on the Lorentz group article the other day when the Wikiserver got overloaded (apparently) and dropped my connection. And right now, in case of edit conflicts, large portions of an article could accidentally be deleted due to bugs in the software.  So at least right now, the whole Wiki editing process seems a bit unstable, which is one reason to try to keep individual articles short.  [Wow, I just almost lost this entire post as I was about to send it--- the Wiki server logged me out again.])

I am still struggling with the idea of infoboxes for articles on exact solutions. The examples I found are all "vertical", but we want something more "horizontal", suitable for line elements, lists of vector fields, and such like. For added flexibility, it might be best to have something like an "equationbox" for holding some of the most important items for purposes of physical/geometrical interpretation/analysis: Would there be some way to number the boxes? Sometimes one might want to discuss two different frames, and when computing say three-dimensional Riemann, to specify which basis is employed.
 * a frame field (vierbein, orthonormal basis of vectors) and its dual coframe field (orthonormal basis of one-forms),
 * a line element (the template would force the writer to insert the ranges of the coordinate variables, a simple but I think vitally important innovation),
 * geodesic equations (in some chart), perhaps with first integrals,
 * numbered generators of the Lie algebra of Killing vectors, perhaps with important subalgebras like isotropy and holonomy identified,
 * timelike vector field defining timelike congruence, together with acceleration vector and expansion and vorticity tensors,
 * box outlining computation of curature the Cartan way (coframe -> connection one-forms -> curvature two-forms),
 * Bel decomposition of Riemann or Weyl curvature computed wrt a frame (displayed as three or two three by three matrices),
 * irrotational timelike vector with three dimensional Riemann tensor of the orthogonal spatial hyperslice (computed wrt a frame field),
 * NP tetrad and list of Ricci and Weyl scalars,
 * null vector field defining a null geodesic congruence, together with its optical scalars.

The hope would be that such equationboxes would help present important technical information in a uniform and readable manner, hopefully without making articles on exact solutions grow too long or appear visually overwhelming. Somehow I'd like do two things at once: help techie readers to go straight to the good stuff (e.g. a frame field) and skip the explanatory text, and also to help nontechies skip easily over the equations. Of course, it would be essential to have articles explaining how all this machinery works and how to interpret the information held in these equationboxes!---CH (talk) 2 July 2005 20:39 (UTC)

Hi CH, I'm not sure what exactly you mean by equation boxes. Some template with all the important formulas? I believe WP policy is to keep all information, while trying to make it as accessible to everyone as possible. Some people interpret this to mean to cut out technical stuff, which is a pity. If you're going to build a lot of pages about GR, then I concur that it is important to define their structure. For tree-structure you can use the templates subarticleof and seesubarticle. BTW it is not policy to keep articles as short as possible. Articles should rather have a minimum length, otherwise they are just stubs, and articles should be merged to avoid excessive duplication. --MarSch 3 July 2005 11:09 (UTC)

New Name
Because after several weeks I just could not internalize the tongue-twisting "Exact solutions to Einstein's field equations": This article now shares the same name as new subcategory, in parallel with "Coordinate charts in general relativity" and other new subcategories I introduced earlier today.
 * Einstein's or Einstein?
 * equations or equation?
 * solutions of or solutions to?

I have returned from my Wikivacation and am beginning by adding some of the missing articles on exact solutions. Since I seem to have become artificially hung up on designing an "equationbox" in advance of writing articles, I have decided to write some articles first and use the experience to design the equation boxes. Again, the idea is to have templates for "standard information" such as:


 * a line element with range of coordinate variables
 * definition of a frame (four orthonormal vectors, one timelike and three spacelike),
 * list of generators of the Lie algebra of Killing vectors,
 * presentation of a timelike congruence and its expansion and vorticity,
 * presentation of null congruence and its optical scalars,
 * presentation of Riemann curvature in Bel decomposition (wrt a frame).

---CH (talk) 08:52, 27 July 2005 (UTC)

Tetrads in general relativity
Created the article tetrads in general relativity. There does not appear to be a sufficiently good article on tetrads in WP (the closest one is Cartan connection applications, but it's yucky), so the new article should start pretty much from the basics. After that, the idea was to give an example or more of the use of tetrads (see sections 3 and 4 of this page). --Mpatel 14:41, 28 July 2005 (UTC)
 * I think that the best generalization of tetrads is "frame fields" ("tetrads" are very often "triads"). As such Frame fields in general relativity is a good name. Maybe this name can be shortened to just Frame field because I don't see this term used in a topic other than relativity. Lantonov (talk) 09:59, 4 August 2018 (UTC)

Local/global spacetime structure
Hi CH. I've changed the link 'local structure' to 'local spacetime structure' (same for global). I think this is a little clearer and will avoid potential ambiguities with any other articles that have a similar local/global structure demarcation like local/global structure of ..., hmmm let's see, ... the countryside. Anyway, I think you get my point and hope you agree that this change is ok. These new links are also to be found at spacetime, which also needs a little tidying up.--Mpatel 16:00, 31 July 2005 (UTC)


 * Yes, this sounds like a good idea to me too. By the way, I am planning to move the fluids to "fluid solution" article, and so forth, once I get these written.  I forgot to say I plan to very shortly write an article on "static spherically symmetric perfect fluid solutions", since these are very important, and there have recently been some interesting developments which render them MUCH more tractable!---CH  (talk) 02:18, 1 August 2005 (UTC)

Ricci decompositions article ?
Hi Chris. Where's the new article on 'Ricci decompositions' ? ---Mpatel (talk) 07:17, August 11, 2005 (UTC)


 * Try using "Go" rather than "Search" in the sidebar at the left of your browser window. And it's called "Ricci decomposition".  I just tested this and "Go" works for me. I think the server is having a hard time keeping the searchable database up to date.---CH  (talk) 09:09, 11 August 2005 (UTC)

Future plans, redux
The present state of this article suggests where I plan to expand on the themes introduced very sketchily here. Solution methods and especially symmetries and solution generating methods (Baecklund transformation like methods) deserve their own articles, and I hope to write them. Ditto for existence and linear/nonlinear stability. But I'll probably first try to write articles bluing most of the red links referring to articles on particular exact solutions which have yet to be written. That will probably require writing articles on various coordinate charts. I think there is a steep curve here: I am going slower than I would like because so much of the background is missing. I wanted to write the article on Robinson-Trautman spacetimes today, but got bogged down in writing the articles on electrovacuum solution, etc., rewriting this one, writing biographies related to these, etc. But as more and more of these background articles get written, at least as stubs, the other articles will probably come faster and faster.---CH  (talk) 09:14, 11 August 2005 (UTC)

Theme of this article
I think the theme of this article should be the interplay between specific exact solutions like de Sitter lamdavacuum, or families of solutions like Robinson-Trautman, Ernst vacuums, with work on the solution space itself, particularly questions of existence, uniqueness, and stability. I haven't yet been able to bring this out at all, but hope to do so in future rewrites. A major goal should be to put the RT vacuums, de Sitter lambdavacuum, T3 Gowdy vacuums, etc., in perspective by explaining background for general results and then in individual articles on these solutions, briefly explaining major results such as the recent proof of cosmic censorship for the above mentioned Gowdy vacuums, spikes, and so forth. Similarly, I am still trying to get to the RT vacuum article, in which I hope to explain the connection with Calabi flow and thus with Calabi-Yau manifolds; the goal is to make it appear natural that Richard Schoen's resolution of the Yamabe problem used ideas from tools developed in the context of gtr, and going the other way, David Maxwell has applied the Yamabe number to existence of solutions in gtr. And of course RT vacuums are much "in the news" lately regarding new analysis of outgoing radiation and asymptotic approach to de Sitter.

We will need articles on all of the following (no particular order): Each of these should cite the original paper, the best recent paper(s) giving most relevant latest theorem(s), and the best available review, if any is known. Note that I have been using existing templates to make the appearance of the pages more uniform.
 * Dimension counting (get some handle on how tiny our families of known solutions are, compared to the entire solution space),
 * Existence and uniqueness of solutons (huge subject, so could easily develop into a set of pages),
 * "Minisuperspaces" from families of solutions, applications of dynamical systems theory to perturbations within the family, Lyapunov stability of flows (ditto),
 * Nonlinear global stability theorems,
 * Initial-value formulations,
 * ADM mass,
 * quasilocal energy-momentum,
 * gravitational energy-momentum complexes,
 * much improved stress-energy tensor article,
 * energy conditions,
 * positive energy theorem,
 * Bianchi classification, with individual articles (the one on Type IX will be most important for Mixmaster, NUT, rotating cosmologies, etc.),
 * Similarity conjecture (B. J. Carr),
 * Weyl curvature conjecture (Penrose),
 * Penrose inequalities (stronger than positive energy theorems),
 * BKL conjecture,
 * Classification of singularities (scalar, nonscalar, strong, weak; simple explicit examples given in pp wave articles),
 * Cauchy horizons,
 * Event horizons,
 * Smoothness issues in gtr (c.f. impulsive waves, nonunique extensions),
 * Mixmaster models.

The article as I have rewritten it to date already segues into (for example) a more specialized articles on fluid solutions, which should eventually segue into an article on static spherically symmetric perfect fluids, which should link to articles on the solution schemes and perhaps to some individual articles on say the Schwarzschild fluid, the Buchdahl theorem, etc. Hopefully anyone wishing to help out will understand my intent of a hierarchical descent into more and more detailed articles.

Articles yet to be written on some exact solutions which are particularly important in connection with the big picture in say the review by Friedrich, include:
 * NUT vacuum,
 * RT vacuums < RT null dusts < RT spacetimes (three increasingly specialized articles),
 * dS lambdavacuum (merge two existing articles into this),
 * T3-Gowdy vacuum < Gowdy vacuum (related to CPW and also to important stuff in Friedrich's review).

Hopefully, it will be possible to gracefully inform the reader that the higher articles slur over many important points (e.g. in my first attempt at a very rough account of basic facts about the solution space in this article, I only mentioned two groups of equations for initial value formulation, but really one should discuss four groups, each with their own difficulties and partial resolutions).

Also, I am not trying to write a review article myself; the idea is not to cite everything, just to cite the most useful reviews or, in the most detailed articles, to cite the really key papers as well as the best specialized reviews available.---CH (talk) 23:20, 11 August 2005 (UTC)

New article on 'solution generating methods (GR)'
The section on 'Constructing solutions' seems very large. How about chopping down this section to a paragraph or two and sending the rest of that section to a new article solution generating methods (general relativity) (or a similar title) ? This would also fit in nicely with trying to organise the mathematics of general relativity article, where I believe a link to an article like the one I just proposed would be better. Comments ?---Mpatel (talk) 10:16, 11 October 2005 (UTC)

Franklin Felber about to propose an antigravity 'solution'
From the talk page on EFE, someone brought this to my attention. It sounds dubious to me, but it sounds like it's not an exact solution. Will have to wait till Tuesday to find out. MP  (talk) 11:42, 12 February 2006 (UTC)


 * Look at his submitted articles, here, here, and here. This last one is quite funny, he introduces the Wiechert-Lienart potential of classical electromagnetism and formulates it in some 'language' used in electromagnetism including scalar and vector potentials.


 * I haven't closely followed his argument but I think he is fantasizing. In any case extreme caution is warranted regarding his claims. 145.97.223.119 18:42, 21 September 2006 (UTC)


 * Felber and his company, "Starmark", seem to have dropped from sight, so this this may be "overkill", but FWIW, Felber's eprints were among the worst I have seen at the arXiv, an incoherent mismash of beginner's mistakes.---CH 07:08, 7 August 2007 (UTC)

"Difficult"
I'm a newbie to higher mathematics and I wonder, when you say that something (such as solving an equation) is "difficult", what does that mean? That it's impossible? Possible only in certain circumstances? Always possible but time-consuming? Some clarification would be welcome. Meneth 14:35, 16 June 2006 (UTC)


 * By 'difficult', we mean theoretically possible, but any standard techniques to solve equations are not particularly helpful. Hence, new techniques have to be found - or the old ones can be used but would be very time-consuming. Hope that clarifies some things. Cheers. MP  (talk) 17:10, 16 June 2006 (UTC)

Students beware
I have extensively edited earlier versions of this article, which concerns a topic particularly 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.

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, although I hope for the best.

Good luck in your search for information, regardless!---CH 23:27, 30 June 2006 (UTC)

Clean-up needed (Confusing text)
This article relies so heavily on advanced physics terminology that I have severe problems understanding it. I have marked it as confusing. Could someone rewrite this so that at least the introduction will make sense to the average reader? Dr. Submillimeter 21:05, 17 March 2007 (UTC)


 * Email me (see my user page) if you want to learn more about the now defunct WikiProject GTR. Our work was not without flaw, and it was left incomplete once we quit WP in despair, but we actually tried hard to write articles which would be comprehensible to a variety of audiences, taking as our model the best of John Baez's expositions. ---CH 07:02, 7 August 2007 (UTC)

Some bad edits of this article since June 2006
Correctly User:Dr. Submillimeter and User:CH !!! The articles "Exact solutions in general relativity" and Solutions of the Einstein field equations and link [[Lorentzian manifold] (here covariant written tensors there contravariant tensors) simply show drastically, that nobody here seems to know about real solutions. Trying to put some solutions failed. Sorry we're not very good in English, please help TOGETHER to enhance it!

Sorry, but our 6 by name known solutions were simply "RV" by an in his HP for destruction known user. Sorry, could it be that educated WIKI-physicists are not at all educated about the basis of GR and Big-bang? wfck84.158.250.58


 * Well, Rottenburg area Deutsche Telekom anon, aka, aka "wfck", whoever you are IRL:


 * I quit Wikipedia in June 2006 due to the ease with which opinionated but ill-informed anons like yourself can disrupt the work of knowledgeable and responsible registered editors like myself, User:Mpatel, User:Pdn, and EMS, the former members of the now defunct WikiProject GTR--- all of whom, incidentally, posted under our real names and possess verifiable qualifications to write articles about GTR.


 * General relativity is widely acknowledged as one of the crowning achievements of twentieth century physics. It is a reasonably well-defined, very well-tested and perfectly comprehensible theory, not inordinately difficult for intelligent, determined and well-prepared students to learn and to use, but it nonetheless presents unique intellectual challenges to the student, and is in some respects quite subtle. As such, GTR intrigues---but is mostly misunderstood by--- the general public.  Since the public (and teachers, journalists, policy makers and even jurists) increasingly turn to WP for information about even such specialized and technical topics as GTR, it seemed to us important to try to apply our knowledge of GTR to present an accurate, readable, and reasonably comprehensive portrait in Wikipedia of current mainstream knowledge in classical gravitation physics.  From roughly June 2005 to June 2006, we attempted to work toward this goal. Unfortunately, due to frustration over constant and steadily increasing disruptions of our work by ignoramuses and trolls, one by one we all quit editing WP, or at least physics articles; AFAIK only EMS still occasionally edits physics articles here.


 * Regarding my own qualifications to write about exact solutions of the Einstein field equation, I suspect that you are simply trolling, "wfck", but since memories are short, for benefit of current Wikipedians, let me point out that I have solved the EFE literally thousands of times (most but not all of "my" solutions turned out to be rediscoveries of previously known solutions), and I have the database to prove it. I also have a working familiarity with a large proportion of the published exact solutions discussed in the standard monograph by Kramer et al., the monograph by Griffiths, and other books which discuss exact solutions (see the citations in my own last version of this article); anyone else who is familiar with these standard sources will have no difficulty, after skimming my expository writings, in verifying that I am indeed familiar with mainstream gtr and with much current research in classical relativity.  Thus, your apparent suggestion that I might not know what I am talking about is not only manifestly absurd, it suggests that you yourself have little knowledge of this field.


 * I encourage any Wikipedians who wish to form their own opinion to compare the versions of articles I wrote, which are listed here, with standard GTR textbooks and with standard monographs on exact solutions. Those who are familiar with these sources should have no difficulty verifying that the computations and other information presented in these articles is mathematically correct, physically sensible, and in complete accordance with current mainstream opinion in gravitation physics.


 * It is sad indeed to see what has become of this article since June 2006, as exemplified by two particularly ill-informed, misleading, and inaccurate edits by two different editors.


 * "Wfck", as you yourself acknowledge, your command of the English language may be insufficient for you to contribute usefully to the English language encyclopedia. But what really matters here is that your edit of this article consists of assertions which are incorrect in any language. User:DVdm was quite right to immediately revert it on those grounds alone, but there were further reasons why your edit was disruptive and greatly marred the previous version.  Your opening sentence, "The well-known solutions are mainly linear and exponential scalar Potential-fields, homogeneous and isotropic without infinitesimal tensor properties" is as good an example of sheer nonsense as I have yet seen in the Wikipedia.  Several of the so-called "named solutions" you mentioned are not to be found in the standard monograph on exact solutions by Kramer et al., for the good reason that they exist only in your imagination.  Those you mentioned which are indeed well-known "named solutions", such as the de Sitter lambdavacuum, FRW dusts, and Schwarzschild fluid solution are discussed in Wikipedia articles which are or were listed in this category, contrary to your assertion that Wikipedia does not discuss them.  The German language textbook by Stephani happens to be outstanding in its discussion of some important exact solutions, "wfck", and even a brief glance at this book will show just how wildly inaccurate is the picture which you attempted to present.


 * A slightly earlier bad edit by runs counter to current mainstream  physics, and I stand by what I wrote: speculative models of allegedly "stable wormholes" proposed by some authors rest upon dubious and speculative theoretical hypotheses which presently have no experimental support and little if any solid theoretical support.  The eprint by Krasnikov is speculative and it is seriously misleading to imply that his claims have been generally accepted.  Similarly, as I stated, "warp drive" models are Lorentzian manifolds but not "solutions of the EFE".  Of course, science is constantly evolving and this situation might change.  It might be that it will someday turn out that warp drives or stable wormholes are physically realizable; I can't say, since I possess no crystal ball.  But I can say that at present the best scientific judgement is that these possibilities are dubious.  Inaccurately presenting idiosyncratic opinions as mainstream is inappropriate in an encyclopedia, or even in, to adopt the phrase of Larry Sanger, "the largest thing often called an encyclopedia".


 * I'd like to add some general comments addressed to Wikipedia users generally.


 * It seems to me that the initiative bravely announced more than a year ago by Jimmy Wales to improve quality and eradicate nonsense from the WP has failed, having lapsed into despair and apathy among qualified editors--- at least insofar as physics articles are concerned. This is exactly what I expected, but it is still very sad to see this, since I support the putative encyclopedic goals of Wikipedia, and more generally the goals of the "open information" movement.  While I remain sympathetic with honest and well-informed Wikipedia editors who still seek to pursue these goals, I myself have despaired of doing that at Wikipedia.  I would however be very glad to be proven wrong by a new generation of Wikipedia editors possessing the fortitude to push through long-needed wikiconstitutional and policy reforms, and the skill and expertise to improve existing content and to add new good content.


 * I wish I had some workable advice for current Wikipedians trying to maintain previously "pretty good" articles by reverting obviously terrible edits. Unfortunately, WikiProject Physics, a project which formerly included a small critical mass of Ph.D. physicists, is as far as I can tell largely defunct, with most of the former members having quit WP for essentially the same reasons why I left: under existing policies, it is ridiculously time consuming to try to maintain articles, much less improve them.


 * The articles I wrote during 2005-2006 were part of an ambitious plan to provide Wikipedia readers with a well planned and mutually supporting set of comprehensive, accurate, and up-to-date articles on classical gravitation physics, at the level suitable both for a general audience and also suitable for helping more advanced students deepen their grasp of the subject. Unfortunately, I was forced by bad edits and trolling to give up my attempts to create good content in order to find time to try to maintain the work I had already done. Then I was forced to give that up, in order to find time to attempt to discuss specific policy reforms intended to ameliorate the WP "brain drain" alluded to above.  Then I was forced to give this up too, in order to find time to attempt to discuss reforms of the the policy-making process itself.  Finally I was forced to give up even this attempt to help Wikipedia grow into a reliable and reasonably stable information resource.  Many other knowledgeable former Wikipedians have reported similar experiences, and I now see little if any hope that Wikipedia will ever succeed in presenting anything which I would recognize as a genuine encyclopedia.


 * Sad to say, in my judgement Wikipedia has failed in its stated mission, but even worse, Wikipedia has succeeded in redefining "good information" from "reliable information" to "conveniently located and easily manipulated information". I fear that this will have grave future consequences for society, and this is the ultimate reason why I now wish I had never become involved with Wikipedia at all.


 * To repeat what I wrote in June 2006: good luck to students in your on-line search for reliable information about physics generally and gtr in particular! I only wish could point to some reliable on-line resources suitable for autodidacts, but at present the best advice I can offer is the tried and true: get into graduate school, master some standard textbooks, and study classic research papers as well as recent arXiv eprints. ---CH 06:59, 7 August 2007 (UTC)


 * Speaking for myself. I had a huge grin on my face when I saw your name pop up on my watch list, Chris. You have been a tremendous asset here in the past. But I suspect you might be right about Wikipedia redefining good information. Some topics, and GR is a great example, are not something you should try to pick up from an encyclopedia. For a technical subject like this, the need is for a simple comprehensible description of what it is about, guided by genuine expertise. But an article here should not attempt to be a introductory technical tutorial, I think. This is the kind of topic that an open encyclopedia like this is inclined to mangle.


 * I've been looking at this particular article recently, because there are exactly solutions around; but unfortunately &mdash; as you note &mdash; some monomaniacal individuals want to insert all kinds of weirdness into "exact solutions". The last time there were actual examples of solutions here, they included some genuine examples like Schwartzchild, and some cases were merely unconventional and failed cosmological ideas that have never been made into exact solutions of the field equations at all: like Zwicky and Arp. The associated discussion, however, was peppered with unphysical gibberish. Some time I may try to write up some informal text on genuine examples of exact solutions. But I am not sure that this page even belongs in the encyclopedia at all. It ought to be within the GR article; and I am thinking that the GR article should not really be attempting to explain the full detail of solutions. &mdash;Duae Quartunciae (talk · cont)  07:19, 7 August 2007 (UTC)

Equations?
The introduction currently contains the text:

the result must satisfy the Einstein field equations (written here in geometrized units)

and then gives a single equation (one I have seen before). It then says equations (plural) again straight after. Why does it say equations but only gives one? Deamon138 (talk) 10:09, 23 June 2008 (UTC)


 * The Einstein field equations say
 * $$R_{\mu \nu} - {1 \over 2}R\,g_{\mu \nu} + \Lambda g_{\mu \nu} = {8 \pi G \over c^4} T_{\mu \nu}.$$
 * This is short-hand for a system of 16 equations. &mu; is a variable which ranges from 0 to 3. &nu; is another variable ranging from 0 to 3. So for example, if we set &mu;=0 and &nu;=2, then one of the equations is:
 * $$R_{0 2} - {1 \over 2}R\,g_{0 2} + \Lambda g_{0 2} = {8 \pi G \over c^4} T_{0 2}.$$
 * Six of the equations are essentially duplicates due to the symmetry of the tensors; leaving ten equations. Those ten equations satisfy another four dependencies due to the requirement of general covariance under coordinate transformations. The remaining six independent equations determine the time-evolution of the gravitational field. JRSpriggs (talk) 20:08, 23 June 2008 (UTC)


 * Thanks for the info, I see now: both variables can be from 0 to 3 (I'm guessing only integers are allowed) which means each variable can take 4 values. Therefore, there are 4x4=16 field equations. Cool. If only the rest of General Relativity was that easy! Deamon138 (talk) 21:43, 23 June 2008 (UTC)


 * Yes. The four values of an index correspond to the four basic directions in spacetime. Typically, 0 is taken to stand for t (time); 1 for x; 2 for y; and 3 for z. So $$R_{1 2} = R_{2 1}$$ is the curvature (see Ricci tensor) in the x-y plane. And $$T_{0 0}$$ is the time component (i.e. density) of the flux of the time component (i.e. energy) of momentum (see Stress-energy tensor). JRSpriggs (talk) 02:12, 24 June 2008 (UTC)


 * Yeah that makes some sense awesome. Come October, I should be studying Physics (and therefore Special Relativity) at University, and who knows, maybe I'll go onto General some time after too, and what you've put here has done something to qualm my thoughts that its a lofty and virtually unapproachable area of physics. Still, I find that a fair amount of the Wikipedia articles on physics to be very hard going for a layman, almost incomprehensible, and I must say, I don't follow the tensor article at all (and googling "tensor" didn't help). Sorry to be a nuisance, but do you by any chance know of any good links discussing tensors from basics, or are the external links provided on that article the best I'm going to get? If so, I'll look at those. Deamon138 (talk) 03:15, 24 June 2008 (UTC)


 * Personally, I stick with what Tensor calls the classical approach. See "The Mathematical Theory of Relativity" Cambridge University Press by Sir Arthur Stanley Eddington. See Covariant transformation. I have not looked for a good external link. Most "modern" work on tensors suffers from excessive abstraction and too much attention to "basis vectors" which obfuscates the important points. JRSpriggs (talk) 05:28, 24 June 2008 (UTC)

Suggested merge
I'd like to suggest that this article be merged with Solutions of the Einstein field equations. There doesn't seem to be enough difference in scope to warrant two articles.TimothyRias (talk) 14:18, 30 November 2010 (UTC)

All the material in the Solutions of the Einstein field equations article seems to be reproduced here. It might as well be deleted because there are two separate articles for exact and non-exact solutions already. AHusain (talk) 06:44, 8 January 2014 (UTC)

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Schwarzschild solution
is Schwarzschild solution an exact solution? Jackzhp (talk) 16:29, 3 April 2019 (UTC)


 * Yes, of course. It is the second known exact solution after Minkowski space. JRSpriggs (talk) 08:25, 4 April 2019 (UTC)