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know) from stuff which follows easily via general category theory abstract nonsense from "bundling" vector spaces into tangent bundle, jet spaces, tensor bundles, whatever. In other words, rigorously split discussion of vectors, linear operators, multilinear operators (tensors), frames, etc., from smooth sections of the corresponding bundles, which are vector fields, tensor fields, frame fields, etc. Note that "frames" are simply the
Lorentzian inner product take on the euclidean frames used to form Stiefel manifolds in mathematics, etc. Hope this makes sense---
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geometry (e.g. Grassmannians use bivectors and even p-vectors) as well as to an article on EM fields. Maybe the material on the EM field in this article should also be moved, to form part of an article on "Relativistic formalisms for electromagnetism"? Someone we need to make it easy for students to effortlessly learn all the most useful stuff. For example, isotropy groups of null versus non-null are not even mentioned in this article! But it is essential to understand why null EM fields have one more symmetry!
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442:(2) Second, it would be a lot easier to understand if canonical forms (or canonical examples) were given, and the theorem said (or were applied so as to say) that every bivector can be brought into one of these forms. The current version is about 70% of the way there, but it doesn't actually tell you what the bivectors look like. To get there, you have to think it out in your head. Which is to be avoided!
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452:(4) Naturally, it would be nice to amplify the following section (with the E's and B's) so that it connects more strongly with the "classification theorem". The section introduces the invariants |E|^2 - |B|^2 and E⋅B but only fleetingly relates them to the principal null directions -- still without actually giving a concrete form.
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P.P.P.S. Hope these postscripts will come to an end soon, gosh. I signed you and EMS up for WikiProject GR; I am still drafting the manifesto (I will have to revisit all these talk pages and try to organize all the suggestions I have made into coherent themes, which will be time consuming). Anyway,
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P.P.S. I guess I still didn't finish my thought: a general strategy which possibly should be implemented consistently is to carefully separate all the algebraic stuff holding in tangent spaces (which often has beautiful connections with algebraic geometry, stuff which increasingly physicists need to
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Hi, MP, did you ever come back to write the "juicy stuff" for this article? I'm guessing you didn't, since the promised connection between the classification of bivectors and the classification of EM fields has not been established, and the physical meaning of electrostatic, magnetostatic has not
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Hi CH. It seems like I never did get back to writing the 'juicy stuff' - I remember you mentioned some references, but I don't have them and couldn't get a hold of them. I also got distracted by other articles. I would need a bit of time to check out some of the details of your superior plans for
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I think the first half of this article should be split off to form part of an article on bivectors (if there already is one, it must need a lot of work if it doesn't already discuss the classification!), which should be in suitable math categories and should link to relevant articles in algebraic
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After fixing up all these problems, in the next round, more can be done, once we have good articles on tangent spaces and so forth. At present, the stuff about thinking about
Minkowski space as a Lorentzian manifold versus thinking about it as real inner product space will be very confusing to
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This article could use some kind of introduction that at least explains the basics to a lay person who is trying to understand if there are different types of electromagnetic fields, and if so, how they differ, and what the practical categories "look like". This article only seems to discuss the
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I tried to improve the article, but got tired before I really finished, so I think this still needs substantial work in terms of improved exposition, giving more information in less space, etc. Hope this is helpful; I might sound a bit cranky since things seem to be going so
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students, but they need to understand this in order to understand how the equivalence principle is elegantly incorporated into the geometric formalism simply by assuming that spacetime is a
Lorentzian manifold, so that its tangent spaces are all isomorphic to E.
480:(5) A final point -- the direction of my changes has actually exacerbated a problem the article already had, namely a long discursive chatty setup of the math before the theorem is stated. There is something to be said for a crisp presentation of a theorem,
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maths of it, and leaves me with a lot of questions: Are there static and dynamic electronamagnetic fields? How does the wavelength of the proton relate to the intensity of the field relate to the number of photons? Real world examples?
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envision turning every Wiki article into a review article. Rather, good Wiki articles should guide curious readers to such reviews (or to textbooks), and should help him to understand what he reads and to tie it all
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Don't |E|^2 - |B|^2 and E⋅B come from a canonical complex structure on the bivectors (given by the Hodge star operator) as the real and imaginary parts of an invariant, complex-valued quadratic form equivalent to
439:(1) First, it would be nice to know the rank in all cases, including the case of a null bivector. Is the case labelled "non-simple" exactly the case where F is non-simple? Then this should be stated explicitly.
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this article. It seems as though your plans for this article are far more ambitious than mine. You can go ahead and improve this article in any way you see fit. ---
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P.S. I didn't finish my thought: I am trying to suggest that this material should be ported/developed into more detailed discussion in two different articles:
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I rewrote this section to be fuller and more logically complete. But still is not satisfying, to me. The basic problem I have with it is that I cannot
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Just wanted to provide notice that I am only responsible (in part) for the last version I edited; see
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I had been monitoring this article, but I am leaving the WP and am now abandoning it to its fate.
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Knowledge. If you would like to participate, please visit the project page, where you can join
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Introduced the classification problem. Now for the juicy stuff ... --
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citation to a good textbook, and perhaps more. I stress that I
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bivectors, discussing material hinted at here, including
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Good luck in your seach for information, regardless!---
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