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this idea from Wald’s
General Relativity where he first introduces the notion of a Christoffel symbol using metric spaces then generalizes the concept to connections which are “more or less arbitrary”. I believe he has some criteria the possible connections must meet. I do not have Wald with me. Perhaps someone else that has it nearby could check? If I am right then the first sentence should say that Christoffel symbols “are an array of numbers that can be used to represent an affine connection. If a metric is also introduced then a connection can be calculated from it.” If I am right the wiki is incorrect. The references currently quoted could also be confused as I think it was some time before mathematicians realized the independence of connections from a metric.
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I believe that a
Christoffel symbol can be used to define a connection where no metric has been defined. If a metric is defined then a connection can be calculated from it, but it is not necessary that a metric be defined in order to define a connection and express it with a Christoffel symbol. I got
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When I was a student, I was greatly confused about the difference between an affine connection and a metric connection. It took a while to un-confuse, and this article seems to blithly confuse the two as if they're the same thing. But there is a difference: an affine connection is what you can define
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The emphatic assertion that
Christoffel symbols are not tensors is dependent on what you view the essential qualities of tensors to be. If you view a tensor as "something that transforms like a tensor under coordinate transformations", then it doesn't obey this rule. But if you view a tensor field
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This paragraph is related to no-relativistic mechanics, so to force in spatial curvilinear coordinates. The general expression of force contravariant components in non-inertial frames is present, where
Christoffel symbols (related only to space and not spacetime) forcedly can not vanish in Euclidean
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In mathematics and physics, the
Christoffel symbols are an array of numbers describing an affine connection. In other words, when a surface or other manifold is endowed with a sense of differential geometry — parallel transport, covariant derivatives, geodesics, etc. — the Christoffel symbols are a
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In mathematics and physics, the
Christoffel symbols are an array of numbers describing an affine connection. In other words, when a surface or other manifold is endowed with a sense of differential geometry — parallel transport, covariant derivatives, geodesics, etc. — the Christoffel symbols are a
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The one-to-one relationship between tangent vectors and directional derivatives is such a basic part of the theory of tangent spaces to manifolds that it is reasonable to assume standard notation like this. However, in print it is common to "bold" the partial ∂'s to make it clear that this is the
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which you cannot do without a metric in your pocket. So I am saddened very much to see that this article wildly confuses the affine connection and the metric connection almost from the first sentence. Am I alone in seeing this, or does anyone else notice this, or care? Fixing this to draw this
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I think these are the points that a corrected first paragraph would say. Let me see if I can create such a paragraph; I guess the rest of the article is probably OK as it stands. To answer your question: yes,this article says lots of things that aren't covered, cannot be covered elsewhere.
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For the simple reason that the many young or untrained-in-the-Einstein-summation-convention will not understand it on first reading. And it is not necessary to train readers in that convention for the purpose of their being able to understand this article. Or desirable. Especially when the
543:, the concept equivalent to the Christoffel symbol is the gauge field. although conceptually similar, no one calls gauge fields "Christoffel symbols", its just not done, so the phrase "Christoffel symbol" is reserved entirely for Riemannian geometry. Thus, the next sentence is quasi-true:
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without a metric, and approx 100% of what gravitation is about involves having a metric. Now, actually having a metric really doesn't really alter the definitions of a connection, torsion, curvature, etc by much; the big difference is that having a metric allows you the write the
571:" is some Riemannian connection with torsion in it. Its false for affine connections in general: so again, in gauge theory, there is no concept of Levi-Civita. The upshot is that the rest of the article is fine, as long as we change the first paragraph to make clear these points:
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I haven't read this article in a while. The first paragraph ends with an explicit disclaimer that connections often, but not always, come from metrics. But much of the rest of the article seems to assume that a metric is present. So I find myself agreeing with
403:*without* ever having a metric, using a metric, knowing what a metric is, or any of that. One does not need a metric to define a connection, and one does not need a metric to define torsion and curvature -- these follow just fine for affine connections (see
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There. Its a good bit longer, but more accurate, and I think it opens the doors to a bigger view that is often a tripping point for students (viz the difference between affine geometry and
Riemannian geometry). I will copy this into the article shortly.
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I don't have
Chandrasekhar (1983). Is that the wording in that source? It is quite different from wording that I've seen in other sources. (And please sign your talk page posts with four tildes, like this: ~~~~.)
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My first-post was a late-night rant. This morning, the main issue seems to be that the first paragraph is misleading. So, let me restate why it's misleading, and how to fix it. Let me quote:
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concrete representation of that geometry in coordinates on that surface or manifold. Frequently, but not always, the connection in question is the Levi-Civita connection of a metric tensor.
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concrete representation of that geometry in coordinates on that surface or manifold. Frequently, but not always, the connection in question is the Levi-Civita connection of a metric tensor.
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All of the concepts from differential geometry, such as parallel transport, covariant derivatives, geodesics, etc. carry over to
Riemannian geometry just fine, with very little/no change.
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yes, all of these things: parallel transport, covariant derivatives, geodesics, etc. are definable just fine without having a metric. Next sentence is true -- half-true but misleading:
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as a multilinear map from a product of vectors & dual vectors to the real numbers, then it's a perfectly fine tensor. This latter perspective is the one taken by Wald (1984):
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are an imposed algebraic structure on a tensor that exist whenever the affine map between the cotangent bundle and tangent bundle of a tensor field satisfies a homomorphism."
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or refer to a text book (e.g. T. Frankel, 1997, "The
Geometry of Physics" p.24, sectiion 1.3b). It would not be reasonable to replicate this detail in this article.
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introduction of a few sigmas will enable many readers to not need any special convention. Just say what you mean, don't make Knowledge some kind of knowledge test.
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in terms of coordinates on the manifold. Additional concepts, such as parallel transport, geodesics, etc. can then be expressed in terms of Christoffel symbols.
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In other words, when a surface or other manifold is endowed with a sense of differential geometry — parallel transport, covariant derivatives, geodesics, etc.
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The existance of a metric allows the Ch. symbols, and the various other quantities to be written in terms of the metric (as described in article).
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The metric connects the tangent and cotangent spaces of the base manifold in such a way that the one can be transformed into the other, i.e. the
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are contravariant indices, while on the right hand side, they are covariant indices. Does this make sense, or should the equation be rewritten?
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Sorry, no. Standard textbooks don't use the sum symbol; it would be inappropriate to use it here, in violation of standard conventions.
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We just need people who are willing to use the sigma summation symbol ∑, and much of this article and others will be a lot clearer.
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Please fix that section. I would do it myself except that I could not resist the urge to just delete the entire section as garbage.
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refers to one specific fixed coordinate system. As far as I know, this second interpretation isn't the usual one. --
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the Christoffel symbols are a concrete representation of that geometry in coordinates on that surface or manifold
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Because this is an encyclopedia, not a physics textbook, the Einstein summation convention should not be used.
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in the primed coordinates by the tensor transformation law, since we change tensors as well as coordinates.
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Note that, as defined here, a Christoffel symbol is a tensor field associated with the derivative operator
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Frequently, but not always, the connection in question is the Levi-Civita connection of a metric tensor.
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in physics than outside physics, and I disagree that it's a violation of standard conventions to use
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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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of a frame bundle is necessarily SO(m,n) because the metric forces it to be so (lets ignore
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to denote sum in this context. For example, defines Christoffel symbols with a number of
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Eh? Its not "garbage", its 100% textbook-standard notation. See the FredV comment below.
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can be defined without any reference to a metric, and many additional concepts follow:
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Here's the short summary, since the tangent-space article is a bit garbled: Give a
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is the position of an imagined Euclidean space in which the manifold exists, then
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which is the same as saying that when performing a matrix multiplication between
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A similar (but not quite identical) perspective can be found in Schutz (2009).
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for example), and, indeed, this is a central critical idea for connections on
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In general, there are an infinite number of metric connections for a given
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Yes, one can write something that looks like Christoffel symbols for both
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to work almost exclusively with the Levi-Civita connection, by working in
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is a differentiation operator. If we use this definition, the definition
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The Ch.. symbols are the coordinate version of the Riemannian connection,
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don't normally involve metrics in any way). By contrast, you cannot do
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Maybe what I'm confused about is: Should this article be distinct from
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in the unprimed coordinates will not be related to the components of
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to mean sum here. For accessibility purposes, I also support adding
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The interpretation you mention is actually the one given in Schutz.
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suddenly makes sense. I therefore wonder whether the definition of
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harv error: no target: CITEREFChoquet-BruhatDeWitt-Morette1977 (
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The connection in Riemannian geometry is (sometimes) called the
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The way in which I have seen the inverse be defined before is
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Special:Contributions/2601:200:c082:2ea0:e1a3:7465:1ef3:b131
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symbols. My understanding is it's more common to drop the
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Tangent space#Tangent vectors as directional derivatives
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also makes little sense, because then each component in
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SO(m,n). As a result, such a manifold is necessarily a (
700:-/ref- The metric connection is a specialization of the
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Perhaps a better way of explaining this is to say that
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That was a mistake by me which I have now corrected. —
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1498:A First Course in General Relativity
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3538:Manifolds and Differential Geometry
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3374:Einstein summation convention
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989:{\displaystyle \partial _{a}}
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823:-- The above statement, "the
803:) where the torsion vanishes.
261:This article is supported by
226:Knowledge:WikiProject Physics
220:and see a list of open tasks.
109:and see a list of open tasks.
3566:B-Class mathematics articles
676:. -ref- See, for instance, (
229:Template:WikiProject Physics
3591:B-Class relativity articles
962:{\displaystyle \nabla _{a}}
756:of the frame bundle is the
3612:
3194:pushforward (differential)
3032:{\displaystyle x=\varphi }
1915:{\displaystyle {\vec {y}}}
849:) 12:40, May 1, 2019 (UTC)
569:the connection in question
252:project's importance scale
3521:13:45, 12 June 2024 (UTC)
2269:04:55, 9 April 2021 (UTC)
2247:23:06, 8 April 2021 (UTC)
2229:22:47, 8 April 2021 (UTC)
2214:22:14, 8 April 2021 (UTC)
1461:17:10, 26 June 2019 (UTC)
1447:21:27, 25 June 2019 (UTC)
1237:19:39, 25 June 2019 (UTC)
881:08:39, 12 July 2020 (UTC)
264:the relativity task force
260:
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3511:symbols to the article.
2040:is incorrectly written?
1654:should be a vector, but
1496:Schutz, Bernard (2009).
1411:are tensor indexes, but
928:15:08, 8 June 2019 (UTC)
141:project's priority scale
3504:{\displaystyle \Sigma }
3484:{\displaystyle \Sigma }
3464:{\displaystyle \Sigma }
3444:{\displaystyle \Sigma }
3424:{\displaystyle \Sigma }
2398:07:49, 2 May 2023 (UTC)
2043:Secondly, the equation
865:11:41, 2 May 2019 (UTC)
819:20:16, 1 May 2016 (UTC)
632:20:07, 1 May 2016 (UTC)
468:18:00, 1 May 2016 (UTC)
439:04:58, 1 May 2016 (UTC)
426:Einstein-Hilbert action
409:principal fiber bundles
98:WikiProject Mathematics
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388:Lowercase sigmabot III
28:This article is rated
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1453:Johnny Assay
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1371:tensor when
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869:
837:— Preceding
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594:frame bundle
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413:Killing form
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137:Mid-priority
136:
96:
62:Mid‑priority
40:WikiProjects
3248:by writing
873:Justintruth
678:Spivak 1999
662:mathematics
112:Mathematics
103:mathematics
59:Mathematics
3560:Categories
1469:References
920:Co-scienza
655:Proposed:
169:Relativity
3197:one-forms
3013:that is,
2261:JRSpriggs
2221:Anita5192
918:metrics.
915:JRSpriggs
734:geodesics
710:manifolds
708:or other
2603:pullback
2507:gradient
2405:manifold
1435:Dr Greg
890:and for
839:unsigned
706:surfaces
383:365 days
349:Archives
789:torsion
762:pseudo-
740:to the
666:physics
300:Please
250:on the
223:Physics
214:Physics
164:Physics
139:on the
30:B-class
1893:where
857:Mgnbar
714:metric
668:, the
460:Mgnbar
36:scale.
3199:from
2426:, an
2390:FredV
1281:does
720:, an
688:and (
3542:ISBN
3517:talk
3399:talk
3364:talk
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2291:talk
2265:talk
2243:talk
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2117:and
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1391:and
1328:does
1233:talk
924:talk
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877:talk
861:talk
847:talk
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696:help
684:help
664:and
628:talk
464:talk
454:and
445:you.
435:talk
3228:to
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2764:on
2239:Kri
2206:Kri
1283:not
1023:to
704:to
660:In
304:to
242:Mid
131:Mid
3562::
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1987:⋅
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