Talk:Riemann curvature tensor

84: 189: 179: 158: 877:. This edit is problematic for two reasons. First of all, it changes from a well-established notation for the parallel transport (which is exactly that used by the very authoritative text of Kobayashi and Nomizu) to a misleading notation that seems to suggest that the parallel transport map depends only on the endpoint. (Parallel transport depends only on the endpoint if and only if the connection is flat.) Secondly, the formatting changes violate the 303:

curvature play, look at the basic work of Lichnerowics. As an ,algebraiker' I like to write the curvature structure in the following triple form, generalizing the concept of Lie triples (the book on Symmetric Spaces of Otmar Loos is a nice generalization of Lie theory): =R(x,y)z. This concept generalizes the notion of a Lie triple to that one of a curvature triple, where only the Jacobiidentity is missing, but a reference to the bilinear form <,: -->

74: 53: 2146: 1368:) that might benefit from a more thorough general discussion of Bianchi identities. This article, however, is about the Riemann tensor (associated to the Levi-Civita connection), not about connection forms and more general sorts of Bianchi identities. It seems that we should keep the notation as familiar as possible without getting side-tracked with discussion of tensor-valued forms and exterior covariant derivatives. 2883: 3182: 22: 1788: 902:

interested in this subject, but I'm not an expert. I think the articles should be written with non-experts at least somewhat in mind. My idea, perhaps naive, is that if I want to find out about GR, I should be able to go to the main article, read everything it references, etc., right down to the bottom of the hierarchy, and eventually understand GR.

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outline of the idea, putting it step by step into a full scientific article. This cannot be done with a scientific journal. Writing an scientific article is time consuming. In this way everybody can see how - and immediately comment. The (my)problem is time - since I work hard on my webpage, mentioned above. Hannes Tilgner

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Second, the tau symbol is used incorrectly (per the definition) except in the first equation that uses it. The subsequent two equations are incompatible with the definition. Therefore, we need a new definition or something else has to change. In the definition, the subscript of tau is the index to

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First, the definition of the tau mapping can be a little mysterious if the tau of your font doesn't look like the Greek tau you get in the math-brackets of the equations that use tau. This was a problem for me in particular because I use the Times New Roman font, which I think is a standard font, at

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is added in such a way, that R(x,y) is an element of the pseudoorthogonal Lie algebra. Note that the complete work of Ricci, Einstein and Weyl can be summarized as a decomposition of the space of curvature structures of Levi type (for Lie algebras). All this shows, that we do not yet understand this

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Thus at each event in our four dimensional space-time: A scalar (just a real number) is a rank 0 tensor; it has 4 = 1 component. A vector is a rank 1 tensor; it has 4 = 4 components. A square matrix is a rank 2 tensor; it has 4 = 16 components. There are tensors of higher rank such as the one which

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Yes, I'm considering the following: Instead of writing a full publication in some mathematical journal (I have done that too often, it didn't pay out), use the Knowledge for publication. By the rules of the scientific world, everything written down here, is published. Actually you can start with an

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The definition of Riemann curvature tensor uses a notation seldom if ever seen in physics articles, say, in Physical Review D. You might want to add a short paragraph saying "In standard notation of general relativity, the Riemann tensor takes the form" then show the equation as four terms each of

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I agree overall with the edit. I don't think there is much to be gained by assuming commutativity, and the risk of confusion is very real. But if we do decide to omit the assumption, then we should justify where the extra term comes from, possibly by adding some version of your derivation in the

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which is not defined or referenced anywhere in the article (where T is apparently a (0,2) tensor, and w a (0,1) tensor - or maybe w is a scalar field? this is also never specified in the article). Please add a definition or external reference to clarify what this bracket operation is supposed to

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I made a fix to this section a while back, but it has been reverted, perhaps for good reason. However, the reverter not only deleted the section I added to this page, but also did not make any record of what he did or why. Nor did he eliminate the original problems, which I will now attempt to

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Many visitors to this page will be students trying to acquaint themselves with the mathematics needed for problems involving general relativity. As such as student (self-study), this page appears to me to be admirably clear but situated on a plane a bit too high for my reach. It needs concrete

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Okay. My main interest is in having the text rendered more understandable. I was unaware of the discouragement of inline LaTeX formulas. If the original mystifying (to me) notation was consistent with Kobayashi and Nomizu, there should be a reference somewhere to what that notation is. I'm

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where you find more references. In the references there (look also at that one in Lecture Notes in Mathematics) you find a decomposition of the space of all curvature structures in terms of Lie and Jordan algebras. And you find how elegantly electrodynamics and gravitational waves fit into the

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My definition modified the original definition to use a curve-designator similar to the use of xt by the original author (I guess), who, I admit, uses it ambiguously as a point and as a curve-designator. Since the side of a parallelogram is a curve designator, that change rendered the uses

2141:{\displaystyle ({\frac {d}{ds}}{\frac {d}{dt}}\tau _{tY}\tau _{sX})WZ+\tau _{tY}\tau _{sX}({\frac {d}{ds}}{\frac {d}{dt}}WZ)+\tau _{tY}{\frac {d}{ds}}\tau _{sX}({\frac {d}{dt}}WZ)+{\frac {d}{dt}}\tau _{tY}\tau _{sX}({\frac {d}{ds}}WZ)={\frac {d}{ds}}{\frac {d}{dt}}\tau _{sX}\tau _{tY}Z.} 1384:

Thanks for resolving that edit war (even if I lost it!). I agree with what you are saying but it is extremely confusing if you are looking up the Bianchi identity and Google or Bing take you to this page and you get something which claims to be the Bianchi identity but isn't.

513:, for example, that would be decidedly out of place in a general article on the curvature of Riemannian manifolds. The same sort of remark applies to the Ricci tensor, where people study things like the positivity of the eigenvalues (although in my field it's mostly the 3770:

I also rolled back the 17:17, 17 December 2021 edit by Theodornak. The Bianchi identities should indeed involve antisimmetrization, not simmetrization. The expressions with simmetrization are also zero (trivially), but that is not related to the Bianchi identities.

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The link ("second covariant derivative") in the sentence "The curvature formula can also be expressed in terms of the second covariant derivative defined as" links to an empty topic. Is the empty page in work, or does it need to link somewhere else? Thanks.

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examples with numbers, vectors, results, and definitions of all symbols. Without that connection to palpable entities, the symbols just float in some ethereal space where their meaning can be perceived only by the literati.

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A new comment :) it would be a grave mistake to mistake Riemannian curvature for the curvature of a Riemann manifold, because Riemannian curvature does not require a Riemannian metric, it only requires a connection.

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I made the calculation clearer. I tried to make it as short as possible, but it is still quite lengthy. So I do not know whether the derivation fits the Wiki style. Maybe assuming the commutativity is just fine.

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which is a product of Christoffel symbols, adding "where Gammamunualpha are the Christoffel symbols" or some such. (See Dirac's texbook "General Theory of Relativity" (John Wiley & Sons, 1975, p20).)

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Enthusiast of mathematical history will note that Riemann made a small mistake in his Habilitationschrift (sic) of 1854, in characterizing flat manifolds; see a detailed historical account in Di Scala 2001

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I don't want to get into a reversion war, but I do want the problem to be fixed. I'll take it up through complaint channels if my changes are reverted again without comment or the fixes are still needed.

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More precisely, Riemann claimed that if the sectional curvature vanish at n(n-1)/2 independent 2-planes at each point of the manifold then the manifold is flat. Several counterexamples are provided in:

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Someone who is merely trying to learn about general relativity would be confused by references to torsion which does not exist in the real world and is not a function of the metric tensor.

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I strongly disagree with the merge: you can define Riemannian curvature on any smooth manifold with a connection, but you obviously need a Riemannian metric for Riemannian manifold :)

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there seems to be evidence that it's the other way around. For the moment I left the "citation needed" tag. Might come later to check. If you know something, please share the sources.

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If you remove this article OK, but still I doubt that you catch the most general aspect of curvature in the sense of Nomizu, Kulkarni and other modern writers. Look at the reference

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there is a claim that Ricci came up with the algebraic identity while Bianchi came up with the second. I've found no evidence of this claim, which by the way is repeated in

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It may be worth adding a comment on how there are two options for sign conventions in GR, which affects, say, the formula for Riemann in terms of the covariant derivatives.

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least it is not one that I remember ever having changed for my firefox browser. So, I put the definition of the mapping on its own line using a math-bracket for clarity.

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is meant to be an overview of the various ways of understanding curvature in the Riemannian setting. The Riemann curvature tensor is one way, but it is not the only one.

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I think that it is valuable to have separate articles on the various types of curvatures (e.g., Weyl tensor). A great deal can be said about the Weyl tensor vis-a-vis

271:, I think it is bit better than this one, I plan to remove from this page everything except things directly connected with curvature tensor and link it to page above. 634:

is the subject of this article — the Riemann-Christoffel tensor is a rank 4 tensor. The Riemann-Christoffel tensor at each event is an array of 4 = 256 real numbers.

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Then the word `Bianchi identity' could link to one of the other articles you mention and the co-ordinate free definition (above) could be included in that article.

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Would anyone mind if I made the following edit. It is very little different to the current sentence, but avoids confusing anyone who is not from a GR background:

1347: 998: 1391:`On a Riemanninan manifold the Bianchi identity (often called the second Bianchi identity or differential Bianchi identity) involves the covariant derivative:' 140: 737: 851:

I'll fix this whole thing properly in a new and improved way in a few days unless I get a response, preferably one that doesn't just wipe out my edits here.

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You know, I am beginning to have the same feeling. However, the both articles could probably use some refocusing and trimming, with summary style in mind.

2333:{\displaystyle ({\frac {d}{ds}}{\frac {d}{dt}}\tau _{tY}\tau _{sX})Z+{\frac {d}{ds}}{\frac {d}{dt}}WZ={\frac {d}{ds}}{\frac {d}{dt}}\tau _{sX}\tau _{tY}Z.} 3764: 3884: 3626:

The only places where I see the "bracket operation" used in the article, it is the inner product of two contravariant vector fields using the metric. So

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that addresses more general kinds of curvature forms (and their associated Bianchi identities to some extent), as well as a few other articles (such as

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To me it seems as though the equation after "The parallel transport maps are related to the covariant derivative by" needs a minus sign to conform to

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The t parameter is often used for time in physics. So, as it reads now, it looks like we have tau sometimes indexed by time and sometimes by a path.

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Agree. The notation is quite different from that used in general relativity, hence may be confusing even to physicists familiar with the subject.

3313: 211: 83: 3879: 1405:? Various formulations of the identity (in local coordinates, for instance) seem to fit to that article. (I guess maybe it should be called 873:

I don't see any indication that you have ever added a section to the page. Rather the only edit I see you ever having made to this page is

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It seems that you want to include some basic identites with curvature plus Pseudo-Riemannian case (is it?) I think it is a good idea.

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Perhaps the best way to resolve it is to qualify in this article that this is the Bianchi identity for the Levi-Cevita connection:

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Returning to the case of a torsion free connection on a tangent bundle (such as the Levi-Civita connection), we may regard

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a point along the curve of transport. After the first equation, however, it is a designator of a side of a parallelogram.

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is an equivalent method. There are many other inequivalent methods. I think having a separate overview article (written in

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But please write Lie ... with a capital L, because Lie refers to the norwegian mathematician Sophus Lie. Hannes Tilgner

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If you know of an existing article which would be an appropriate target for this, please feel free to change the link.

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One may consider removing the commuting assumption, since the general case admits the quadrilateral construction. --

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is regarded as an alternating map on pairs of vector fields, taking values in endomorphisms of vector fields. Thus

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as a trilinear map from triples of vector fields to vector fields. In this case the Bianchi identiy takes the form:

479:. However, as Riemann curvature tensor is only one of the ways to define curvature on Riemannian manifold, I think 858:

I have added my name to the WikiProject Physics table, in case its absence had anything to do with the reversion.

3302:{\displaystyle =(\nabla _{X}\nabla _{Y}-\nabla _{Y}\nabla _{X}-(\nabla _{\nabla _{X}Y}-\nabla _{\nabla _{Y}X}))Z} 3515: 3469: 3429: 1369: 882: 3776: 3756: 480: 476: 454: 616: 3610: 3845: 3817: 692: 596: 552: 344: 1036: 3496: 3455: 3418: 1502: 1414: 575: 531: 1557: 305:

curvature space completely. Especially the gravitational wave aspect needs clarification. Hannes Tilgner

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I agree to merge those two articles. But how about the opposite way? Currently, it is proposed to merge

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There was a calculation mistake in the previous edit. I give the detailed reasoning as follows. Let

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I removed the following sentence; it makes sense to experts, but is not sufficiently well-explained:

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first. As it says, "A tensor is an object which extends the notion of scalar, vector, and matrix.".

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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I don't think a merge is a really good idea. The articles do cover much of the same material but

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cover much the same sort of material. I propose merging them. Does anyone have any objections?

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http://en.wikipedia.org/Parallel_transport#Recovering_the_connection_from_the_parallel_transport

1319:{\displaystyle \nabla _{u}(R(v,w))+\nabla _{v}(R(w,u))+\nabla _{w}(R(u,v))-R(,w)-R(,u)-R(,v)=0.} 492:

Actually, I don't think we need to have each article for every kind of curvatures. For example,

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In the section symmetries and identities a bracket operation of the following syntax appears:

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is an alternating map on triples of vector fields, so the Bianchi identity may be written out:

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Hey, I just reverted an edit. It was a subtle point (but perhaps worth writing down) but

3562: 817:{\displaystyle \delta V^{\mu }=R_{\nu \sigma \tau }^{\mu }dx^{\nu }dx^{\sigma }V^{\tau }} 3709: 3537: 1428: 1357: 635: 497: 484: 89: 1332: 983: 3858: 3849: 3821: 3800: 3780: 3737: 3717: 3620: 3545: 3523: 3500: 3479: 3459: 3439: 3422: 1565: 1506: 1436: 1418: 1379: 916: 892: 867: 643: 620: 600: 579: 564: 535: 500: 487: 465: 443: 348: 333: 316: 293: 277: 556: 3698:{\displaystyle \langle x,y\rangle =g_{\alpha \beta }\,x^{\alpha }\,y^{\beta }\,.} 626:

If you do not know that, this article is much too advanced for you. Try reading

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The curvature tensor was invented by Riemann in his 1854 Habilitationvortrag "

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Point taken about merging the opposite way. Obviously this is a "slow merge".

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I notice the assumption. Hence it was not a mistake. Thanks for reminding.

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that is of interest). Again, much can be said which is not relevant here.

329:. This is a good place for 'survey articles'; but not for new results. 102: 1578: 1682:{\displaystyle W=\tau _{sX}^{-1}\tau _{tY}^{-1}\tau _{sX}\tau _{tY}.} 926:

The following subject of a recent edit war is a good addition to the

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The vector at x0 should be subtracted, not be subtracted from.

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Curvature_of_Riemannian_manifolds#Symmetries_and_identities

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Ueber die Hypothesen, welche der Geometrie zu Grunde liegen

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On a Riemannian manifold one has the covariant derivative

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Parallel transport and covariant derivative: sign error?

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Symmetries and identities - undefined bracket operation

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and so the original formula was correct as written by

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What is difference between a vector & a tensor?

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article should better be a part of this article. --

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Terrific. 3647: 3635: 1588:Calculation in the Geometric meaning section 1401:More or less off-topic, but why is there no 3875:C-Class physics articles of Mid-importance 3835: 3807: 3751:About the authorship of Bianchi identities 689:after circling an infinitesimal rectangle 152: 47: 3684: 3672: 3657: 3633: 3565: 3373: 3360: 3350: 3337: 3327: 3315: 3279: 3274: 3256: 3251: 3235: 3225: 3212: 3202: 3190: 3156: 3151: 3127: 3108: 3104: 3085: 3068: 3049: 3045: 3026: 3008: 2989: 2980: 2961: 2946: 2927: 2918: 2899: 2891: 2857: 2852: 2828: 2809: 2805: 2786: 2771: 2752: 2743: 2724: 2698: 2679: 2675: 2656: 2641: 2622: 2613: 2594: 2583: 2549: 2544: 2528: 2515: 2496: 2481: 2466: 2453: 2434: 2419: 2395: 2390: 2368: 2353: 2351: 2315: 2302: 2283: 2268: 2244: 2229: 2211: 2198: 2179: 2164: 2159: 2123: 2110: 2091: 2076: 2049: 2037: 2024: 2005: 1978: 1966: 1947: 1938: 1907: 1892: 1880: 1867: 1842: 1829: 1810: 1795: 1790: 1754: 1741: 1719: 1706: 1700: 1667: 1654: 1641: 1633: 1620: 1612: 1600: 1468: 1462: 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Mathematics 187: 177: 156: 82: 72: 51: 20: 3870:Mid-importance physics articles 970:{\displaystyle d^{\nabla }R=0.} 879:manual of style for mathematics 660:The transformation of a vector 425:{\displaystyle \neq \nabla _{}} 240:This article has been rated as 135:This article has been rated as 3621:21:00, 28 November 2014‎ (UTC) 3391: 3386: 3374: 3320: 3293: 3290: 3244: 3195: 3152: 3147: 3141: 3023: 2896: 2853: 2848: 2845: 2721: 2715: 2591: 2588: 2545: 2540: 2416: 2391: 2220: 2161: 2070: 2046: 1999: 1975: 1928: 1889: 1851: 1792: 1307: 1298: 1286: 1283: 1274: 1265: 1253: 1250: 1241: 1232: 1220: 1217: 1208: 1205: 1193: 1187: 1171: 1168: 1156: 1150: 1134: 1131: 1119: 1113: 1077: 1059: 1056: 1040: 417: 405: 394: 368: 327:Knowledge:No original research 1: 3850:22:39, 9 September 2024 (UTC) 3822:22:47, 9 September 2024 (UTC) 3738:17:16, 31 December 2019 (UTC) 3718:22:16, 28 November 2014 (UTC) 3546:11:03, 23 November 2014 (UTC) 3524:19:02, 22 November 2014 (UTC) 1366:exterior covariant derivative 1356:There is already the article 917:00:54, 15 February 2010 (UTC) 893:16:36, 10 February 2010 (UTC) 601:17:00, 16 November 2010 (UTC) 580:14:03, 23 February 2008 (UTC) 565:06:03, 23 February 2008 (UTC) 536:02:02, 23 February 2008 (UTC) 349:16:58, 16 November 2010 (UTC) 214:and see a list of open tasks. 115:Knowledge:WikiProject Physics 109:and see a list of open tasks. 3880:C-Class mathematics articles 1480:{\displaystyle \nabla _{u}R} 1023:{\displaystyle d^{\nabla }R} 868:13:21, 6 February 2010 (UTC) 644:15:07, 10 October 2008 (UTC) 621:13:06, 10 October 2008 (UTC) 118:Template:WikiProject Physics 3585:{\displaystyle <T,w: --> 3501:19:00, 9 October 2014 (UTC) 3480:17:23, 9 October 2014 (UTC) 3460:17:01, 9 October 2014 (UTC) 3440:11:35, 9 October 2014 (UTC) 3423:09:33, 9 October 2014 (UTC) 501:08:46, 24 August 2007 (UTC) 488:08:43, 24 August 2007 (UTC) 3901: 3801:00:43, 30 April 2024 (UTC) 3781:05:32, 28 March 2022 (UTC) 1566:08:14, 18 March 2012 (UTC) 1507:15:04, 6 August 2011 (UTC) 1437:12:04, 6 August 2011 (UTC) 1419:15:57, 5 August 2011 (UTC) 1380:01:11, 5 August 2011 (UTC) 1362:connection (vector bundle) 938:may be written concisely: 682:{\displaystyle \,V^{\mu }} 444:21:47, 14 April 2007 (UTC) 141:project's importance scale 239: 172: 134: 67: 46: 3865:C-Class physics articles 481:Riemann curvature tensor 477:Riemann curvature tensor 466:20:15, 21 May 2007 (UTC) 455:Riemann curvature tensor 334:08:02, 17 May 2004 (UTC) 317:05:40, 15 May 2004 (UTC) 294:05:40, 15 May 2004 (UTC) 278:04:09, 13 May 2004 (UTC) 246:project's priority scale 3829:Notation for physicists 203:WikiProject Mathematics 3765:this mathoverflow link 3699: 3587: 3404: 3303: 3178: 2879: 2571: 2334: 2142: 1773: 1683: 1481: 1351: 1343: 1320: 1087: 1024: 994: 971: 818: 728: 683: 426: 325:Please have a look at 28:This article is rated 3700: 3588: 3516:BornRightTheFirstTime 3405: 3304: 3179: 2880: 2572: 2335: 2143: 1774: 1684: 1482: 1344: 1321: 1088: 1025: 995: 972: 819: 729: 684: 427: 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Thank you! 3121: 3098: 3062: 3039: 3002: 2974: 2940: 2912: 2822: 2799: 2765: 2737: 2692: 2669: 2635: 2607: 2509: 2494: 2447: 2432: 2381: 2366: 2296: 2281: 2257: 2242: 2192: 2177: 2104: 2089: 2062: 2018: 1991: 1960: 1920: 1905: 1823: 1808: 1569: 1552:comment added by 1497:comment added by 1342:{\displaystyle R} 993:{\displaystyle R} 591:comment added by 555:) is helpful. -- 260: 259: 256: 255: 252: 251: 151: 150: 147: 146: 3892: 3743:Metric signature 3704: 3702: 3701: 3696: 3689: 3688: 3677: 3676: 3665: 3664: 3618: 3599: 3593: 3590: 3589: 3583: 3472: 3432: 3409: 3407: 3406: 3401: 3390: 3389: 3365: 3364: 3355: 3354: 3342: 3341: 3332: 3331: 3308: 3306: 3305: 3300: 3289: 3288: 3284: 3283: 3266: 3265: 3261: 3260: 3240: 3239: 3230: 3229: 3217: 3216: 3207: 3206: 3183: 3181: 3180: 3175: 3173: 3172: 3155: 3140: 3139: 3135: 3134: 3122: 3120: 3109: 3099: 3097: 3086: 3081: 3080: 3076: 3075: 3063: 3061: 3050: 3040: 3038: 3027: 3016: 3015: 3003: 3001: 2990: 2988: 2987: 2975: 2973: 2962: 2954: 2953: 2941: 2939: 2928: 2926: 2925: 2913: 2911: 2900: 2884: 2882: 2881: 2876: 2874: 2873: 2856: 2841: 2840: 2836: 2835: 2823: 2821: 2810: 2800: 2798: 2787: 2779: 2778: 2766: 2764: 2753: 2751: 2750: 2738: 2736: 2725: 2711: 2710: 2706: 2705: 2693: 2691: 2680: 2670: 2668: 2657: 2649: 2648: 2636: 2634: 2623: 2621: 2620: 2608: 2606: 2595: 2576: 2574: 2573: 2568: 2566: 2565: 2548: 2536: 2535: 2523: 2522: 2510: 2508: 2497: 2495: 2493: 2482: 2474: 2473: 2461: 2460: 2448: 2446: 2435: 2433: 2431: 2420: 2412: 2411: 2394: 2382: 2380: 2369: 2367: 2365: 2354: 2339: 2337: 2336: 2331: 2323: 2322: 2310: 2309: 2297: 2295: 2284: 2282: 2280: 2269: 2258: 2256: 2245: 2243: 2241: 2230: 2219: 2218: 2206: 2205: 2193: 2191: 2180: 2178: 2176: 2165: 2147: 2145: 2144: 2139: 2131: 2130: 2118: 2117: 2105: 2103: 2092: 2090: 2088: 2077: 2063: 2061: 2050: 2045: 2044: 2032: 2031: 2019: 2017: 2006: 1992: 1990: 1979: 1974: 1973: 1961: 1959: 1948: 1946: 1945: 1921: 1919: 1908: 1906: 1904: 1893: 1888: 1887: 1875: 1874: 1850: 1849: 1837: 1836: 1824: 1822: 1811: 1809: 1807: 1796: 1778: 1776: 1775: 1770: 1762: 1761: 1749: 1748: 1727: 1726: 1714: 1713: 1688: 1686: 1685: 1680: 1675: 1674: 1662: 1661: 1648: 1640: 1627: 1619: 1568: 1546: 1509: 1489:Bianchi identity 1486: 1484: 1483: 1478: 1473: 1472: 1403:Bianchi identity 1372: 1348: 1346: 1345: 1340: 1325: 1323: 1322: 1317: 1186: 1185: 1149: 1148: 1112: 1111: 1092: 1090: 1089: 1084: 1052: 1051: 1029: 1027: 1026: 1021: 1016: 1015: 999: 997: 996: 991: 976: 974: 973: 968: 957: 956: 922:Bianchi identity 885: 823: 821: 820: 815: 813: 812: 803: 802: 790: 789: 776: 771: 753: 752: 733: 731: 730: 725: 723: 722: 710: 709: 688: 686: 685: 680: 678: 677: 653:Deleted sentence 603: 431: 429: 428: 423: 421: 420: 393: 392: 380: 379: 331:Charles Matthews 228: 227: 224: 221: 218: 197: 192: 191: 181: 174: 173: 168: 160: 153: 123: 122: 121:physics articles 119: 116: 113: 92: 87: 86: 76: 69: 68: 63: 55: 48: 31: 25: 24: 16: 3900: 3899: 3895: 3894: 3893: 3891: 3890: 3889: 3855: 3854: 3831: 3788: 3786:Examples needed 3773:JackozeeHakkiuz 3753: 3745: 3726: 3680: 3668: 3653: 3630: 3629: 3600: 3561: 3560: 3554: 3511: 3470: 3430: 3369: 3356: 3346: 3333: 3323: 3312: 3311: 3275: 3270: 3252: 3247: 3231: 3221: 3208: 3198: 3187: 3186: 3150: 3123: 3113: 3100: 3090: 3064: 3054: 3041: 3031: 3004: 2994: 2976: 2966: 2942: 2932: 2914: 2904: 2888: 2887: 2851: 2824: 2814: 2801: 2791: 2767: 2757: 2739: 2729: 2694: 2684: 2671: 2661: 2637: 2627: 2609: 2599: 2580: 2579: 2543: 2524: 2511: 2501: 2486: 2462: 2449: 2439: 2424: 2389: 2373: 2358: 2348: 2347: 2311: 2298: 2288: 2273: 2249: 2234: 2207: 2194: 2184: 2169: 2156: 2155: 2119: 2106: 2096: 2081: 2054: 2033: 2020: 2010: 1983: 1962: 1952: 1934: 1912: 1897: 1876: 1863: 1838: 1825: 1815: 1800: 1787: 1786: 1750: 1737: 1715: 1702: 1697: 1696: 1663: 1650: 1597: 1596: 1590: 1575: 1547: 1541:A.J. Di Scala, 1515: 1492: 1464: 1459: 1458: 1370: 1331: 1330: 1177: 1140: 1103: 1098: 1097: 1043: 1035: 1034: 1007: 1002: 1001: 982: 981: 948: 943: 942: 932:in this article 924: 883: 830: 804: 794: 781: 744: 736: 735: 714: 701: 691: 690: 669: 662: 661: 655: 609: 586: 451: 400: 384: 371: 363: 362: 356: 319: 296: 280: 265: 225: 222: 219: 216: 215: 193: 186: 166: 120: 117: 114: 111: 110: 88: 81: 61: 32:on Knowledge's 29: 12: 11: 5: 3898: 3896: 3888: 3887: 3882: 3877: 3872: 3867: 3857: 3856: 3830: 3827: 3826: 3825: 3787: 3784: 3752: 3749: 3744: 3741: 3725: 3722: 3721: 3720: 3707: 3706: 3705: 3694: 3687: 3683: 3675: 3671: 3663: 3660: 3656: 3652: 3649: 3646: 3643: 3640: 3637: 3595: 3594: 3581: 3578: 3575: 3572: 3569: 3553: 3550: 3549: 3548: 3534: 3510: 3507: 3506: 3505: 3504: 3503: 3485: 3484: 3483: 3482: 3471:Sławomir Biały 3448: 3447: 3443: 3442: 3431:Sławomir Biały 3411: 3410: 3399: 3396: 3393: 3388: 3385: 3382: 3379: 3376: 3372: 3368: 3363: 3359: 3353: 3349: 3345: 3340: 3336: 3330: 3326: 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977: 966: 963: 960: 955: 951: 923: 920: 906: 905: 904: 903: 896: 895: 884:Sławomir Biały 829: 826: 811: 807: 801: 797: 793: 788: 784: 780: 775: 770: 767: 764: 760: 756: 751: 747: 743: 721: 717: 713: 708: 704: 700: 676: 672: 654: 651: 649: 647: 646: 631: 613:118.95.114.159 608: 605: 583: 582: 541: 540: 539: 538: 521: 520: 519: 518: 504: 503: 490: 450: 449:Merge proposal 447: 433: 432: 419: 416: 413: 410: 407: 403: 399: 396: 391: 387: 383: 378: 374: 370: 355: 352: 312: 311: 310: 289: 288: 287: 273: 264: 261: 258: 257: 254: 253: 250: 249: 238: 232: 231: 229: 212:the discussion 199: 198: 182: 170: 169: 161: 149: 148: 145: 144: 137:Mid-importance 133: 127: 126: 124: 107:the discussion 94: 93: 90:Physics portal 77: 65: 64: 62:Mid‑importance 56: 44: 43: 37: 26: 13: 10: 9: 6: 4: 3: 2: 3897: 3886: 3883: 3881: 3878: 3876: 3873: 3871: 3868: 3866: 3863: 3862: 3860: 3853: 3851: 3847: 3843: 3839: 3828: 3823: 3819: 3815: 3811: 3805: 3804: 3803: 3802: 3798: 3794: 3785: 3783: 3782: 3778: 3774: 3768: 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3771:-- 3736:) 3716:) 3686:β 3674:α 3662:β 3659:α 3648:⟩ 3636:⟨ 3617:) 3613:• 3544:) 3522:) 3499:) 3478:) 3458:) 3438:) 3421:) 3413:-- 3371:∇ 3367:− 3358:∇ 3348:∇ 3344:− 3335:∇ 3325:∇ 3277:∇ 3272:∇ 3268:− 3254:∇ 3249:∇ 3242:− 3233:∇ 3223:∇ 3219:− 3210:∇ 3200:∇ 3125:τ 3102:τ 3083:− 3066:τ 3043:τ 3021:− 3006:τ 2978:τ 2959:− 2944:τ 2916:τ 2826:τ 2803:τ 2769:τ 2741:τ 2719:− 2696:τ 2673:τ 2639:τ 2611:τ 2526:τ 2513:τ 2479:− 2464:τ 2451:τ 2313:τ 2300:τ 2209:τ 2196:τ 2121:τ 2108:τ 2035:τ 2022:τ 1964:τ 1936:τ 1878:τ 1865:τ 1840:τ 1827:τ 1752:τ 1739:τ 1717:τ 1704:τ 1665:τ 1652:τ 1643:− 1631:τ 1622:− 1610:τ 1564:) 1560:• 1505:) 1466:∇ 1435:) 1417:) 1397:W 1378:) 1314:0. 1278:− 1245:− 1212:− 1179:∇ 1142:∇ 1105:∇ 1049:∇ 1013:∇ 965:0. 954:∇ 934:: 915:) 891:) 866:) 824:. 810:τ 800:σ 787:ν 774:μ 769:τ 766:σ 763:ν 750:μ 742:δ 720:σ 707:ν 675:μ 642:) 619:) 599:) 578:) 563:) 534:) 402:∇ 398:≠ 386:∇ 373:∇ 347:) 3844:( 3816:( 3795:( 3775:( 3732:( 3712:( 3693:. 3682:y 3670:x 3655:g 3651:= 3645:y 3642:, 3639:x 3609:( 3586:} 3577:w 3574:, 3571:T 3540:( 3518:( 3495:( 3474:( 3454:( 3434:( 3417:( 3398:. 3395:Z 3392:) 3387:] 3384:Y 3381:, 3378:X 3375:[ 3362:X 3352:Y 3339:Y 3329:X 3321:( 3318:= 3297:Z 3294:) 3291:) 3286:X 3281:Y 3263:Y 3258:X 3245:( 3237:X 3227:Y 3214:Y 3204:X 3196:( 3193:= 3170:0 3167:= 3164:t 3161:= 3158:s 3153:| 3148:) 3145:Z 3142:) 3137:X 3132:Y 3129:t 3118:t 3115:d 3111:d 3106:s 3095:s 3092:d 3088:d 3078:Y 3073:X 3070:s 3059:s 3056:d 3052:d 3047:t 3036:t 3033:d 3029:d 3024:( 3018:Z 3013:X 3010:s 2999:s 2996:d 2992:d 2985:Y 2982:t 2971:t 2968:d 2964:d 2956:Z 2951:Y 2948:t 2937:t 2934:d 2930:d 2923:X 2920:s 2909:s 2906:d 2902:d 2897:( 2894:= 2871:0 2868:= 2865:t 2862:= 2859:s 2854:| 2849:) 2846:) 2843:Z 2838:Y 2833:X 2830:s 2819:s 2816:d 2812:d 2807:t 2796:t 2793:d 2789:d 2784:+ 2781:Z 2776:X 2773:s 2762:s 2759:d 2755:d 2748:Y 2745:t 2734:t 2731:d 2727:d 2722:( 2716:) 2713:Z 2708:X 2703:Y 2700:t 2689:t 2686:d 2682:d 2677:s 2666:s 2663:d 2659:d 2654:+ 2651:Z 2646:Y 2643:t 2632:t 2629:d 2625:d 2618:X 2615:s 2604:s 2601:d 2597:d 2592:( 2589:( 2586:= 2563:0 2560:= 2557:t 2554:= 2551:s 2546:| 2541:) 2538:Z 2533:X 2530:s 2520:Y 2517:t 2506:t 2503:d 2499:d 2491:s 2488:d 2484:d 2476:Z 2471:Y 2468:t 2458:X 2455:s 2444:t 2441:d 2437:d 2429:s 2426:d 2422:d 2417:( 2414:= 2409:0 2406:= 2403:t 2400:= 2397:s 2392:| 2387:Z 2384:W 2378:t 2375:d 2371:d 2363:s 2360:d 2356:d 2328:. 2325:Z 2320:Y 2317:t 2307:X 2304:s 2293:t 2290:d 2286:d 2278:s 2275:d 2271:d 2266:= 2263:Z 2260:W 2254:t 2251:d 2247:d 2239:s 2236:d 2232:d 2227:+ 2224:Z 2221:) 2216:X 2213:s 2203:Y 2200:t 2189:t 2186:d 2182:d 2174:s 2171:d 2167:d 2162:( 2136:. 2133:Z 2128:Y 2125:t 2115:X 2112:s 2101:t 2098:d 2094:d 2086:s 2083:d 2079:d 2074:= 2071:) 2068:Z 2065:W 2059:s 2056:d 2052:d 2047:( 2042:X 2039:s 2029:Y 2026:t 2015:t 2012:d 2008:d 2003:+ 2000:) 1997:Z 1994:W 1988:t 1985:d 1981:d 1976:( 1971:X 1968:s 1957:s 1954:d 1950:d 1943:Y 1940:t 1932:+ 1929:) 1926:Z 1923:W 1917:t 1914:d 1910:d 1902:s 1899:d 1895:d 1890:( 1885:X 1882:s 1872:Y 1869:t 1861:+ 1858:Z 1855:W 1852:) 1847:X 1844:s 1834:Y 1831:t 1820:t 1817:d 1813:d 1805:s 1802:d 1798:d 1793:( 1767:. 1764:Z 1759:Y 1756:t 1746:X 1743:s 1735:= 1732:Z 1729:W 1724:X 1721:s 1711:Y 1708:t 1677:. 1672:Y 1669:t 1659:X 1656:s 1646:1 1638:Y 1635:t 1625:1 1617:X 1614:s 1606:= 1603:W 1556:( 1501:( 1475:R 1470:u 1431:( 1413:( 1374:( 1337:R 1311:= 1308:) 1305:v 1302:, 1299:] 1296:u 1293:, 1290:w 1287:[ 1284:( 1281:R 1275:) 1272:u 1269:, 1266:] 1263:w 1260:, 1257:v 1254:[ 1251:( 1248:R 1242:) 1239:w 1236:, 1233:] 1230:v 1227:, 1224:u 1221:[ 1218:( 1215:R 1209:) 1206:) 1203:v 1200:, 1197:u 1194:( 1191:R 1188:( 1183:w 1175:+ 1172:) 1169:) 1166:u 1163:, 1160:w 1157:( 1154:R 1151:( 1146:v 1138:+ 1135:) 1132:) 1129:w 1126:, 1123:v 1120:( 1117:R 1114:( 1109:u 1081:= 1078:) 1075:w 1072:, 1069:v 1066:, 1063:u 1060:( 1057:) 1054:R 1045:d 1041:( 1018:R 1009:d 988:R 962:= 959:R 950:d 911:( 887:( 862:( 806:V 796:x 792:d 783:x 779:d 759:R 755:= 746:V 716:x 712:d 703:x 699:d 671:V 638:( 615:( 595:( 574:( 559:( 530:( 418:] 415:v 412:, 409:u 406:[ 395:] 390:v 382:, 377:u 369:[ 343:( 248:. 143:. 42::

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