Talk:Einstein's field equation

Contents

1 Notation change? 2 Cleaning up needed ? 3 edits made 4 Exact solutions discussion 5 teleparallelism 6 nonlinearity of EFE 7 vacuum solutions

Notation change?

I propose that the notation should be changed in the article to reflect more standard usage in the field. In particular, spacetime indices should be greek letters, and the Einstein tensor should have the letter G, instead of E

Comments? Lethe 22:13, Jun 16, 2004 (UTC)

I think that many people nowadays use Latin letters for spacetime indices (although a lot of people still use Greek). I think that for consistency's sake, as well as keeping up to date with most research papers, it's better to use Latin letters for spacetime indices.

Totally agree with the comment about the Einstein tensor (I made the new change).

I think that the article is inappropiately using Wald's abstract index notation, where tensors are denoted by G_{ab} (with subindices a,b,c,d...) , and the subindices do not indicate the component of the tensor but rather the rank of the rank of the tensor. G_{ab} represents the tensor itself, and not its components. Tensor components are still indicated by Greek letters, G_{\mu\nu}. Many research papers use Wald notation. So I propose:

• Either to use the abstract index notation adequately (and explaining it),

• Or to work in components, as traditional, and thus change the indices from ab to \mu\nu (or ij).

--Daniel Arteaga 12:31, 28 Mar 2005 (UTC)

That last comment has clarified a few things. I would prefer to use Wald's abstract index notation. Let's try this approach with proper explanation.

Cleaning up needed ?

I think this article could do with some cleaning up.

(1) The notation change has been mentioned.

(2) Is the section on 'tensor geometry' needed ? (should it be in another article ?).

(3) The section on solutions of the field equations should surely come after the section on the field equations themselves.

(4) Perhaps a qualitative discussion of the initial value problem should be included.

That's all I can think of just now.

edits made

moved discussion of exact solutions to page on exact solutions. Mpatel 17:32, 6 Jun 2005 (UTC)

Exact solutions discussion

Lethe -

In dicussing the solutions of the EFE, the new text is much more correct than the older one you reverted to. Also, the exact solutions are of secondary importance here. Mpatel has moved them to a new page, with an updated explanation of what an exact solution is. I for one approve of this organization, but just as importantly I approve of what he is saying in the editted pages.

At the least, if you think that the exact solutions belong in the EFE page then kindly make your case for that here first. Then if we decide that it is proper one of us will merge the pages back together. In the meantime, I repeat that I highly approve of the totality of Mpatel's edits: What is written about EFE solution, both in general and of the exact kind, is a wast improvement over the previous text. At the least, that older text should be kept in the history were it belongs.

--EMS | Talk 18:44, 6 Jun 2005 (UTC)

Apologies for not fully explaining (here) why I moved that chunk of text on exact solutions. I was doing a lot of chopping and changing and must have forgotten to explain it here. I was also worried that some1 might revert my edits, but I'm glad that at least 1 person approves of them. Mpatel 11:11, 8 Jun 2005 (UTC)

Firstly, let me say that I mostly reverted because I thought I saw a fairly large deletion, without edit summary, and just assumed it was a mistake. If I had known that it was intentional, I would not have reverted. So I apologize for that misunderstanding.

But now that I bring my attention to it, I don't really agree with the change. Here are my reasons:

1. exact solutions of einsteins equations are probably the most important results of the theory, and at least some of them (and why not a list of all of them?) should be here
2. the text in question is actually quite short. it's just a list with a very short summary and separate link to each solution. if the list of solutions is nothing more than a list, why should it warrant its own article?
3. the new article has a wrong name. exact solutions?? That's far too specific! Do you envision eventually including exact solutions to Navier-Stokes, Yang-Mills, the Klein-Gordon equation, and Maxwell's equations, all in one place? What about the rest of the diff eqs? I suppose this objection could be easily laid to rest by renaming the article exact solutions to the Einstein field equations or some such.
4. the link is not prominently displayed enough. as I mentioned earlier, this is a very important part of the theory. It should have its own section, and if there is so much to write on the subject, then a subpage should be made, and linked prominently from the top of the summary in the main page, as is always done in featured articles.

Basically, I think I could be happy with the way things are, if we still had the list of exact solutions, with a 5 word commentary on each solution, along with two sentences about the general principles of exact solutions, and a link to a (properly renamed) subarticle. the subarticle would then have detailed information about the methods of finding exact solutions. maybe a synopsis of the solution for each guy in the list. well... I'm dreaming.

Anyway. I think the move could be good in the long run, might not be so good now, but whatever, as long as it wasn't a weird deletion, I don't care. The text is still somewhere, and is linked to from this article. I didn't realize that at the time, and that is reason enough to revert my undeletion. -Lethe | Talk 12:53, Jun 8, 2005 (UTC)

I agree with you that the "exact solutions" article should be renamed. I can also agree that some coherent discussion of solutions of the EFE should be in this page. Indeed, solving the EFE in general needs to be discussed.

My big reason for being vehement on keeping the changes is that exact solutions of the EFE are not "physically realizable" solutions as the old text claimed. Instead they are a subset of the EFE whereby a metric tensor can be formulated with an Einstein tensor that is equal to the stress-energy of the spacetime in question. Some exact solutions (such as the Alcubierre metric) may not be physically realizable. (The Alcubierre metric uses negative energy, whetever that is and if it even is.) At the same time, other cases lacking exact solutions (such as two bodies orbiting each other) are quite obviously not just physically realizable but physically realized.

So overall you have some good thoughts here. All I ask is that their implementation be done in a way that moves the article forwards.

--EMS | Talk 14:56, 8 Jun 2005 (UTC)

In making the edits, I was thinking long term. The articles exact solutions (ES) and Einstein's field equation are quite short at the moment and they will no doubt be expanded in the future (for example, in ES, I was thinking of perhaps including descriptions of various techniques used in searching for exact solutions of EFE - Petrov types, Segre types etc.).

I agree that some of the exact solutions could be mentioned in the EFE article, but they should definitely be mentioned in ES - the problem/question then becomes: 'is it OK to have that same list on both pages ?' Certainly, a handful of the important ones could be mentioned in passing on the EFE page but a more complete list should be in ES. I don't think there is much point in listing all the known exact solutions (Kramer, Stephani et al have done that).

Agree about the name change for ES.

Mpatel 16:51, 8 Jun 2005 (UTC)

Mpatel - I think that you are on the right track with these thoughts. "'[I]s it OK to have the same list on [multiple] pages?'" To me it depends on the size of the list and its importance to the article(s). However, in general the answer would be "No". Some overlap is needed to relate the pages, but their covering the same ground to a large extent often is troublesome. (For example, before you fixed it the EFE and math-of-GR pages disagreed as to what an "exact solution" is. [I was about to fix that myself, but you beat me to it.]) So I approve of your idea of listing some important metrics here, while keeping the full list in the exact solutions page.

We may also want to consider having the subarticle be on solutions instead of just exact solutions. However, it should be noted that exact solutions are preferred since anything odd in a non-exact solution begs the question of "is this real, or is it an artifact of the computation that would vanish if a (more) exact solution was known?".

--EMS | Talk 02:18, 9 Jun 2005 (UTC)

teleparallelism

Considering the fact that Einstein himself wrote a paper on the teleparallel formulation of gravity, I think it should stay in the article. No? -Lethe | Talk 22:35, Jun 10, 2005 (UTC)

I have been looking at this, and it seems to be a failed attempt to create a unified field theory incorporating causitive mechanisms for both gravitation and electromagnetism. I will first direct you to the comment at the end of this report of Einstein's (http://www.lrz-muenchen.de/~aunzicker/einst.html). It seems that he was having trouble generating correct equations of motion. This is only one of the reports I found on this teleparalellism web site (http://www.lrz-muenchen.de/~aunzicker/ae1930.html). Overall it seems to be an effort that got dropped after 1930, and that probably says a lot about it. I have also found a 2002 newsgroup posting by Chris Hillman (http://www.lns.cornell.edu/spr/2002-01/msg0038083.html) on this matter which leads to me believe that the theories are distinguishable. There is also an implication that the predictions become identical in the special case of no torsion, in which case the theories have become identical!

I don't know anything about teleparallelism being used to unify field theory and gravity, but it doesn't surprise me that people would try that. I mean, they tried it with GR (cf. Kaluza-Klein), and teleparallelism looks superficially more similar to field theory than GR does. For a more conservative introduction to teleparallelism (i.e. without any unification), see gr-qc/0011087. Equation 21 of that paper is the Einstein field equation in the teleparallelism formalism. My recollection is that the two theories are identity on topologically trivial spacetimes, but not necessarily so on nontrivial spacetimes. Of course, no one knows the topology of our spacetime yet, so there's no experimental reason to prefer one over the other. -Lethe | Talk 02:36, Jun 11, 2005 (UTC)

Unification seems to have been Einstein's goal with this. I gather he at some point decided that he was barking up the wrong tree, and dropped it.

I also agree that there is no known experimental reason to prefer GR to teleparallelism. However, I will point out three things:

• I myself am working on a modification of GR to remove the black hole from theory. It also agrees with extant experimental results within the margins of error for the observations. However, I will not present it in Wikipedia due to its being truly original research, and its lacking any support in the field at this time.
• There is this little thing called Occam's Razor which says that when two theories give results which are in accord with observation, preference should be given to the simpler one. GR seems to be a special case of teleparallelism, and so is the simpler theory.
• Unless I am mistaken, noone has seen if teleparallelism works in the medium strength gravitational fields of the binary pulsars. This is a serious issue as the binary pulsar observations blew Rosen's bimetric theory out of the water, and have set a lower bound on the parameter ω for the Brans-Dicke scalar-tensor theory which is so large that the difference between it and GR is at best miniscule.

Einstein embraced teleparallelism as a chance to extend GR. It seems to have failed to to live up to that promise. If one cannot distinguish between the theories, then teleparallelism is useless unless you can show that it is a better fomulation of GR. On the other hand, if there is a difference, then the statement that they give the same results is false and this is an alternate theory. Either way, it does not belong here. --EMS | Talk 05:21, 11 Jun 2005 (UTC)

My impression is that this is an alternate theory, and one that is followed by a small minority of physicists and possibly not any of any standing. If you like, we can ask Chris for his opinion. (I don't want to bother him unless it is needed and will be a good use of his talents, but this issue appears to qualify on both counts.) However, mine is that it does not belong here. If it deserves mentioning then it should be done in the GR article itself in the alternate theories section or in a subarticle on alternate theories. Indeed, the Einstein-Cartan formalism (which this seems to be related to) also should not be here as I see it, but I am not yet sure of what to do with it.

Yes, teleparallelism is an alternate theory, but a viable one. It deserves a mention on wikipedia. If you argue that it should go in the GR article and not this one, I'm willing to listen to that argument (though I think it could well go in both). Maybe I agree with your argument, and then we can move it, but at the time, it looked like you were deleting, not moving, and that's why I reverted (as with MPatel's edit earlier.) -Lethe | Talk 02:36, Jun 11, 2005 (UTC)

I was deleting it. I felt that it does not belong here, and still do. Yes. I think it should be moved to the GR article and listed as an alternate theory. Either that or the reference should be dropped completely. Once again, this article is about the EFE. If an alternate theory supports the EFE, it is redundent here. If it does not support the EFE, then it is irrelevant here. Either way, discussions of alternate theories belong elsewhere. So I won't touch that reference for now. I make no promises for later, or for anyone else.

BTW - I couldn't care less about whether teleparallelism is viable, and neither does Wikipedia according to the Wikipedia NPOV policy. In that policy it is stated that if only a very small part of a community supports a certain viepoint, its being covered in Wikipedia is inappropriate even if their viewpoint is correct. So don't show me the teleparallelism is viable. Instead show me that it has support in the community of physicists! --EMS | Talk 05:21, 11 Jun 2005 (UTC)

So for the moment I will leave the teleparallelism reference alone. It's having a defender is reason enough to do so for now. However, I ask you consider that for reasons of scope that it does not belong here, and that for reasons of its being followed by a very small minority of physicists may not even be appropriate to mention in Wikipedia as an alternate theory (but it may be worthy of mentioning in the development of general relativity article). --EMS | Talk 01:48, 11 Jun 2005 (UTC)

I would have to agree with EMS's view that teleparallelism is a theory and does not belong in the EFE article. I think that even a passing mention of teleparallelism's field equations are inappropriate in the EFE article, as 'EFE' is almost always taken to mean 'the field equations that Einstein wrote down in formulating GR', no mention of teleparallelism being mentioned - as EMS wrote above, teleparallelism was taken to be an extension of GR. --- Mpatel 10:30, 11 Jun 2005 (UTC)

I moved teleparallelism and  Einstein-Cartan theory to general relativity.  -Lethe | Talk 06:57, Jun 13, 2005 (UTC)

nonlinearity of EFE

Mentioned a few things about EFE being nonlinear in the metric. I think it's ok to mention differences between EFE and other dynamical equations like Maxwell's equations (ME) and Schrodinger's equation (SE) (yes, I know there is more than 1 type of SE), as it's a pretty important mathematical difference with physical implications - of which I have yet to explore properly. ---- Mpatel 14:20, 11 Jun 2005 (UTC)

vacuum solutions

Vacuum solutions incorporated into EFE, as the vacuum field equation page seemed more like a definition rather than an article. --- Mpatel 15:45, 13 Jun 2005 (UTC)