# Talk:Spacetime symmetries

Template:Physics Ok, I created this page as another long term project. I've made a very sketchy start in writing something about symmetries in GR. I'm not sure if my description of preserving the metric is correct - in fact, now that I think about it, it's very loose. To rigorise the article, I was thinking of:

• Defining these symmetry vector fields in general and explaining why that definiton is a good one physically.
• Defining some of the more important vector fields in relativity (and mechanics, I suppose) (Killing, homothetic, conformal etc.) - obviously, their physical interpretations should be stressed.
• Maybe examples of spacetimes with certain symmetries should be stressed, along with conservation laws (e.g. Schwarzschild, Kerr and FRW are at least 3 obvious ones).

Mpatel 09:28, 10 Jun 2005 (UTC)

## Conscious start on article

Should have wrote this before I started doing it, but I'm making a conscious effort to start writing something substantial in this article. ---Mpatel 10:58, 16 Jun 2005 (UTC)

Ok, I've rigorously defined symmetries in general relativity according to Hall's definition. Examples of some symmetries have been given - still to include projective, affine and conformal symmetries. Given the level of interest in some of these symmetries, it may be better to have more detailed pages for them (certainly for Killing symmetry). Other applications of symmetries still to be mentioned (I'm struggling to find some!). ---Mpatel 13:10, 16 Jun 2005 (UTC)

## Comments on New Version

This is very nice! Three comments:

• The existing article on curvature collineations should be deleted since this concept is adequately covered here.
• I'd like to see brief explanations of the intuitive or physical significance, e.g. Killing vector fields generate self-isometries, and the Lie algebra of all Killing vector fields admitted by a Lorentzian manifold is the Lie algebra of the Lie group of self-isometries, or autometry group.
• Maybe this article should be called generalized Killing vectors? This is a fairly standard term in the literature, as you no doubt know, but my objection to using symmetries is that this should also include things like the point symmetry group of the Ernst equation. But the existing article might be getting rather long after adding the explanatory concepts. Hmmm... on the other hand, it would be nice to try to achieve a unified discussion (also to avoid exponential increase in the number of gtr articles), since generalized Killing vectors, Noether symmetries, and point symmetries of PDEs are all closely related topics.

---CH

• I agree about removing the curvature collineation article (I'll do that once I find out how to delete an article).
• Yes, there should be more explanations of the intuitive/physical significance of the symmetries. I didn't want to dwell too much on the Killing vector fields in this article as someone has already created a separate article for that - maybe some merging is in order here (but then the problem of a lengthy article arises...).
• Maybe keep the present title, as I think a unified approach would be better: mention the Noether symmetries etc., briefly, but have links shooting off to separate articles for them (I wouldn't worry too much about the number of GR articles here - after all, there are a lot of rather unsavoury and questionable articles on other topics here at WP) - there may already be a page on Noether symmetries (or something similar). As Noether symmetries are fairly general in physics, maybe the Noether symmetries of GR could be included as a subsection of a Noether symmetries article. The standard term 'generalised Killing vector(s)' could be mentioned in other GR articles (for example, in exact solutions) and a redirect to this page could be made (I suppose most people interested in such things would probably look for the standard term rather than 'symmetries in general relativity'). --MP
I've included a little Lie derivative identity to help readers see why the symmetries (which are smooth) actually form a Lie algebra (it's mentioned in a book by B. Schutz - I'll get the precise reference later). Maybe something about differentiability needs to be mentioned here (or maybe that's too technical for this article...?). --MP
Introduced another subsection on 'other types of symmetries' -- MP

## Merger of pages ?

Killing vector fields seem to be introduced in (at least) 2 places on WP: Killing vector field and projective vector fields (Killings are also mentioned briefly in spacetime symmetries). The first one is clearly a stub and the second one is more comprehensive and shows how other types of symmetry vector fields arise (a suggestion has been made to increase the 'Killing content' in the third one). Proposal: merge the contents of Killing vector field into projective vector fields and keep the discussion of Killings to a minumum in spacetime symmetries (just mention the most important features for GR). How does that sound ? ---Mpatel 28 June 2005 17:21 (UTC)

Hi, MP, I say go for it. I think we agree that it is important to try to distribute material evenly and in a way which reflects its true position in the overall scheme of things (e.g., Killing vectors and such like belong to (semi)-Riemannian geometry, so the articles should reflect this--- someone primarily interested in say Killing vectors on S^3 in relation to quaternions as pure math should not be distracted by material which is only relevant to gtr applications).

In the same vein, another thing I think needs fixing is that there are several articles dealing with warp drives. These are Lorentzian manifolds, but they should not be called "solutions" of the Einstein field equation, because they are nothing like fluids, scalar fields, EM fields, Lambda term type exotic matter, or other "recognized" and more or less legitimate models of "states of matter" or of nongravitational fields.

Anyway, I entirely agree that your present article is a nice summary of various kinds of spacetime symmetries, and that writing individual articles on the most important types and putting in some intuititive explanation and examples would be appropriate in the individual pages. I do think you can probably concoct one sentence intuitive interpretations for each type which can go in the present article, however.--CH

P.S. How do you obtain the time stamp for your signature? I have been waiting for that information to come to me, as it were, rather than go hunting for it, since it's not a big priority, but after some weeks it hasn't arrived spontaneously.--CH

## Suggested Name Change for this Article

Oops, MP, just noticed that I seem to have forgotten to emphasize that in line with my comments just above, I think this article should be renamed Spacetime symmetries, and the article on Lorentzian manifolds should link "comfortably" to it. Ultimately, I would also like to see the connections with Noether symmetries brought out. As you know, that connection works and is valuable for sysems of PDEs in general, e.g. Korteweg-de Vries equation or wave equation, and has need have nothing to do with general relativity or even physics.

For example, in the Ernst vacuums, the field equations reduce to the equation of an axisymmetric hyper-Ernst potential, and the system of two coupled nonlinear real valued partial differential equations (the real and imaginary parts of the hyper-Ernst equation) has a bunch of point symmetries: euclidean similarity transformations of the three independent variables (coordinate changes referring to a flat background, if you like) and ${\displaystyle H^{2}}$ isometries of the two dependent variables. Some of these are Noether symmetries, i.e. variational symmetries for the association hyper-Ernst Lagrangian. In the case of an equation with time dependence like the KdV, such variational symmetries lead via Noether's theorem to conserved quantities like mass, energy, momentum, and various moments of a KdV soliton. So Noether symmetries can refer to internal symmetries of a physical law, not just to symmetries of the arena for physics (a particular spacetime). Another example: Wick rotation of electromagnetic fields.

--CH

## Name change

I've changed the name of this article, as suggested by CH. --Mpatel 15:00, 21 July 2005 (UTC)

## Matter collineations

I've included some more (interesting ?) info. on matter collineations. The perfect fluid example can be proven fairly easily by anyone who knows how to contract tensors. I'll get references for the electromagnetic 'results'; I think Hall's book has references for this. --Mpatel 15:21, 21 July 2005 (UTC)

## Ricci and Weyl collineations ?

I wasn't sure if it was worth mentioning Ricci and Weyl collineations (${\displaystyle {\mathcal {L}}_{X}R_{ab}=0}$ and ${\displaystyle {\mathcal {L}}_{X}C^{a}{}_{bcd}=0}$). Apart from the obvious preserving property of these tensors, I'm not sure if I can give much physical content. Not too sure about Ricci collineations (they're equivalent to matter collineations iff the Ricci scalar is zero or the vector field is Killing) - any obvious physics in there ? The Weyl tensor represents the gravitational field in regions where there is no matter, so preserving the Weyl tensor along an integral curve would mean ... ? Hmmm ... help ! --Mpatel 15:21, 21 July 2005 (UTC)

## Conformals - if you dare ...

Anyone feel brave enough to create the article on conformal vector fields ? There is huge potential for creating such a great article, as gtrians (to borrow a phrase from CH) will know - links to pp-waves etc... --Mpatel 10:26, 28 July 2005 (UTC)

## Talking to myself ?

It seems as though I'm talking to myself on this page...--Mpatel 10:29, 28 July 2005 (UTC)

If I can only fix that thing I mentioned in email and take care of some other matters, I really do intend to try to read all this at greater length, write about physical interpretation and mathematical utility of some of these more exotic symmetries, etc. Sorry for all these damnable delays! ---CH 01:18, 23 January 2006 (UTC)

## removed section

I've removed the section on 'other types of symmetries' as this article is solely about spacetime symmetries in the way that's defined in the article; any other type of symmetry, however closely related to this one, should not go in here: symmetries of differential equations, for example. Rather, it should have a separate article - dump it into the 'See also' section.

P.S. - I know I created this article, but other people are welcome to contribute (and talk !). ---Mpatel (talk) 17:21, August 10, 2005 (UTC)

## Content of article

The article is fairly mathematical. I realise that it needs a more intuitive description of spacetime symmetries. For example, there should be mention of Lorentz and general covariance and a brief description of what these mean. These ideas should then be related to (some of) the listed symmetries. Also, see the article symmetry in physics. MP (talk) 14:02, 22 January 2006 (UTC)