Talk:Petrov classification

Original Version
Made a start in describing one classification of the Weyl tensor (based on pnd's). May need to mention who Petrov was and describe other classifications briefly. Oh, and need to define a pnd ! ---Mpatel 14:30, 15 Jun 2005 (UTC)

Major Rewrite
Hi, MP, I have added much more material, including citations to the original papers of Petrov and Pirani and to books. I also reorganized the article into several sections and added a figure (converted to PNG from a quick xfig diagram I hastily drew; someone else might eventually want to make a prettier figure but this will do for now.) I see I haven't been entirely consistent about using double versus triple quotes (italic versus bold type), so you may want to change that. I didn't try to mention the complex linear operator approach to understanding Petrov types, nor did I mention the strikingingly simple characteristic forms taken by the Weyl spinors, with respect to an "adapted" Newman-Penrose tetrad, in the case of type N, type III, or type D. So eventually someone might want to add that, despite the already considerable size of this article, since it is rather useful to know.

I replaced "pnd" with "principle null direction", since I have never seen this abbreviation and didn't think it falls trippingly over the tongue.

There are (at least) two types of Penrose diagram which are important in gtr, the one I drew in the figure for this diagram, and various conformal diagrams exhibiting the causal structure of various conformal compactifications, e.g. for de Sitter spacetime or FRW dust or Kerr vacuum.

Biographical material about Petrov would be welcome, but I hope you will put it in a short biographical article which you create under A. Z. Petrov; see the link to a nonexistent article which I put in this article. Yes, there is still a lot of red ink in this version, but contrary to first impressions, I don't think I am only linearly increasing the number of dead links.

Someone should create a category of classification theorem, and add the category to appropriate mathematical articles, e.g. Thurston classification theorem, as well as this one. ---Chris Hillman


 * Hi Chris. I like the major rewrite - excellent ! Good intro. to the problem. You listed some examples of spacetimes which admit a constant Petrov type (III, N, D and O) - do you know of any spacetimes that are everywhere type I or everywhere type II ? If so, it might be good to complete the set of examples. Potentially more interesting would be examples that change their Petrov type (smoothly, continuously ?) - I don't know much about these, but I think it would be interesting if examples of these could be found. ---Mpatel 16:13, 23 Jun 2005 (UTC)

Yes, I know examples of all of those. A generic Weyl vacuum will be Type I, so choose any axisymmetric harmonic function and construct the corresponding Weyl vacuum. You might already know the recipe: let $$u(z,r)$$ solve the Laplace equation


 * $$u_{zz} + u_{rr} + \frac{u_r}{r} = 0$$

This is the consistency condition (called integrability condition) for the first-order system


 * $$v_z = 2 r \, u_z \, u_r$$
 * $$v_r = r \, \left( u_r^2 - u_z^2 \right)$$

And, from $$u(z,r)$$ you can find $$v(z,r)$$ by quadrature. Now the line element, written in the Weyl canonical chart, is


 * $$ ds^2 = -\exp(2u) \, dt^2 + \exp(-2u) \left( \exp(2v) \left( dz^2+dr^2 \right) + r^2 \,d\phi^2 \right) $$

Viola! An exact static axisymmetric vacuum. Moreover, every such solution can be put in this form. It's not quite true that these guys have a one-one correspondence with Newtonian axisymmetric static fields, though, since some nonzero $$\phi$$ give locally flat spacetimes. There is a fascinating discussion of this "correspondence" in the article by Bonnor, and I know some stuff he doesn't mention--- but this is really a topic for a proper review article.

Solutions which change type on some locus are not hard to find either. In fact, if you play with Ernst vacuums you can probably find some which are type I in most places but type II or D on a special locus.

I felt Petrov II is less interesting in the context of this article, because this is also pretty common and can be viewed as a mismash of features from III, N, and D. What I really would like is a discussion of simple III, N, D examples in the context of an article on the Bel decomposition of the Riemann tensor, since this vividly shows the distinction between the nature of the gravitational field in these three basic types. We'll see. ---CH

Bel Criteria and explicit forms of Weyl tensor
Hi, CH. No doubt you've come across the useful Bel Criteria for determining Petrov types of algebraically special Weyl tensors. I was thinking of briefly mentioning the Bel criteria in this article (or maybe another one ?) as I'm sure you know it's a quick way of determining the Petrov type once the Weyl tensor components have been calculated. Possibly more controversially, I was thinking of including explicit forms for each Petrov type (in some given suitable tetrad) - maybe I'm dreaming here; the article may become too technical. Both these would also have links going to a potential tetrad page. I await comments. ---Mpatel 1 July 2005 14:31 (UTC)

Bel Decomposition versus Weyl Spinor characterizations versus Bel Characterizations of Petrov Types
Hi, MP, Yes, I think the three key characterizations of the curvature tensor for the various types (listed in arguably decreasing order of abstraction) are


 * Weyl spinors wrt an adapted NP tetrad,
 * the Bel characterizations of the Weyl tensor I think you have in mind,
 * characterizations in terms of the Bel decomposition of the Riemann tensor-- confusingly, a distinct Bel characterization!

This article should probably mention the first two, and I've been mulling over trying to include the third in an article on decompositions of the Riemann tensor, which would interlink with this article.

Probably most gtrians know the Bel decomposition as the array of Riemann components in a six by six matrix

\left[ \begin{matrix} E & B \\ B & L \end{matrix} \right] $$ where $$B,E,L$$ are certain three by three matrices suggesting the name of Lluis Bel.

That is, we treat Riemann as a linear operator on the six dimensional space of bivectors
 * $$Y^{ab} -> {R^{ab}}_{cd} Y^{cd} $$

In the Bel decomposition, taken with respect to a timelike vector field $$X$$ (not neccessarily geodesic or irrotational), these three matrices are associated with three spatial tensors (referring to the spatial hyperplanes of our observers):


 * the tidal tensor $$E[X]_{ab} = R_{ambn} X^m X^n$$, which is symmetric and controls tidal forces,


 * the magnetogravitic tensor $$B[X]_{ab} = \left(R^*\right)_{ambn} X^m X^n$$, which is traceless and controls spin-spin forces according to Papapetrou/Dixon equations,


 * $$L[X]_{ab} = \left({}^*R^*\right)_{ambn} X^m X^n$$, the "spatial components", which is symmetric.

To be precise, I should project the tensors defined above on the spatial hyperplanes to obtain explicitly three dimensional tensors. In general, they have 6, 8, 6 linearly independent components, making up the 20 Riemann components. In a vacuum, they are symmetric traceless and $$D_{ab} = -E_{ab}$$, so we have the 5+5 Weyl components.

Then as you know from Hall, in a vacuum region, if we take the decomposition wrt a carefully adapted frame (vierbein), the components fall into certain striking patterns:


 * type N shows transverse tidal forces,



E[X]_{ab} = \left[ \begin{matrix} 0 & 0 & 0 \\ 0 & -a & 0 \\ 0 & 0 & a \end{matrix} \right] B[X]_{ab} = \left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & b \\ 0 & b & 0 \end{matrix} \right] $$


 * type III shows shearing tidal forces,



E[X]_{ab} = \left[ \begin{matrix} 0 & a & 0 \\ a & 0 & 0 \\ 0 & 0 & 0 \end{matrix} \right] B[X]_{ab} = \left[ \begin{matrix} 0 & 0 & b \\ 0 & 0 & 0 \\ b & 0 & 0 \end{matrix} \right] $$


 * type D shows "Coulomb" tidal forces,



E[X]_{ab} = \left[ \begin{matrix} -2a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{matrix} \right] B[X]_{ab} = \left[ \begin{matrix} -2b & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & b \end{matrix} \right] $$

(In the last case, the Boyer/Lindquist frame in the Kerr vacuum would not be adapted in this sense.)

Types I and II should probably be thought of as combinations of these three types, according to the peeling theorem.

Ideally, we would eventually bring out the fact that these three characterizations really are talking about the same features of curvature in type N, III, D regions, etc., just expressed in different formalism. It's particularly important to make sure that readers understand that we can "rotate" tetrads to "align" them with principal null directions (or however you think of it) until eventually we achieve one of the canonical forms, and also that they understand how NP tetrads are related to vierbeins or frames and so forth.

Anyway, by all means, please go ahead and see if you can clearly and concisely discuss the Bel characterizations (and Weyl spinor characterizations?) here; if it gets too long, you can put this material in another article (e.g. the Weyl spinor characterizations could naturally go in an article on Spinor formalism for Weyl tensor or something like that).

The planned article on the Bel decomposition should also discuss the relationship of $$B,E,L$$ with normal coordinate charts, since this takes us right back to the roots of curvature.---CH (talk) 1 July 2005 22:19 (UTC)

P.S. An article on bivectors in Lorentzian spacetime would be good, if there isn't one already. A key point is to explain simple bivectors and the Plücker relation picking out the submanifold of simple bivectors in the Grassmannian. ---CH (talk) 1 July 2005 22:25 (UTC)


 * Hi, CH. I've made a start in writing down the Bel Criteria (or characterisation, if you prefer). Is it worth mentioning something about repeated pnd's ? Anyway, I didn't want to go into too much detail as regards the equivalent criteria for each alg. special type (Hall has collected them together). I'm not too hot on spinors, so I think I'll leave the Weyl spinor material for you (or someone else) to write. I may make a start on the Bel decomposition soon. I also want to discuss something with you on the exact solutions talk page. ---Mpatel 2 July 2005 11:41 (UTC)

"Characterization": I've never tried to use the Bel criteria to actually find the Petrov type; I use the flow chart in the exact solutions book. I think GRTensorII uses a similar algorithm, and if I recall correctly, D'Inverno also discusses the same flow chart, and I've seen many papers which follow this method. Is your method superior as a practical algorithm for determining the Petrov type? Staying close to things technical readers are likely to encounter in say the arXiv is probably a legitimate consideration in deciding what to say and what to leave out. I know I'd violate this rule by discussing Bel decomposition of Riemann in extenso in these pages, but in this case I think an exception is clearly warranted: this decomposition is far superior for physical insight than its competitors.

About pnds, yes, I think you should try to explain clearly and concisely how the null vectors appearing the Bel criteria are related to the pnds. In type N I know the relation is simple: the quadruple pnd is the same as the null vector in the criterion (also, the same as the wave vector). But in type III, the obvious guess is that the triple pnd is the null vector appearing in the criterion, but that's wrong, isn't it? Also, as you can see, I am having second thoughts about "PND"; if you are going to use this term again in the new section, maybe the abbreviation is worth making after all :-/ ---CH  (talk) 2 July 2005 21:06 (UTC)

Classification of e/m fields
Hi CH. I want to create an article called classification of electromagnetic fields. My reason for doing so is that there are nice analogies between the e/m and grav. fields both physically and mathematically and that the proposed article will clarify this to some extent. The approach used will have to be looked at: is the spinor approach better (it's mathematically elegant, but is it physically clearer ?), or perhaps just crunching out the classification using tensors ? Certainly, the importance of (principal) null directions should be stressed (and maybe even repeated principal null directions), as these appear to be important in the classification (both e/m and grav.). Need to think about this. --Mpatel 09:00, 27 July 2005 (UTC)


 * Great! I fervently agree that there is a need for an article discussing null versus non-null in special relativity.  Make sure to consult Gravitation and Inertia by Ciufolini & Wheeler, since this has a very nice discussion of the analogy between the two quadratic invariants of the EM field tensor and two of the quadratic invariants of the Riemann tensor.  I guess you probably know the monograph on "Exact solutions", but if not that has a very good discussion of two major approaches to studying the symmetries of symmetric or antisymmetric second rank tensors (including the EM field tensor).


 * I also agree that there is a need for a clear discussion of analogies between EM and gravitation as treated in gtr. Somewhere there should also be an explanation of the relationship between Killing vectors and electrovacuums.  These have been on my list, but I haven't gotten to them yet, so by all means have a shot at them.---CH  (talk) 23:08, 27 July 2005 (UTC)

Peeling theorems and new article
Hi Chris.

I've just tagged on a sentence mentioning the electromagnetic peeling theorem, on which there should be an article as there are nice comparisons with the gravitational case. Which brings me onto another topic - given the many similarities between gravitation and electromagnetism (mathematical and physical), perhaps there should be an article devoted to this. Not quite sure what to call it, as there are many differences between the two as well. Maybe, Comparisons between gravitation and electromagnetism (yuck !), or Similarities between GR and EM (still yuck). Hmmm... . Anyway, I think you see my point.

MP  (talk) 10:01, 21 December 2005 (UTC)

Students beware
I extensively edited an earlier version of this article and 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 to User:Mpatel and to all students searching for information, regardless!---CH 02:50, 1 July 2006 (UTC)

I do not believe the expression for the Weyl tensor in terms of the Weyl scalars is correct. I see no way that this expression reduces to the on given in Chandrasekhar's classic text on Black holes.

Pirani and independence
The current introduction claims that (emphasis by me):
 * The classification was found in 1954 by A. Z. Petrov and independently by Felix Pirani in 1957.

I don't feel comfortable editing this, but I find this claim false, judging by, where Pirani references the work of Petrov already in 1956 (meaning that he couldn't independently discover the classification in 1957). Trosos (talk) 13:48, 22 January 2023 (UTC)