Talk:Christoffel symbols

message: Is currently:
:In mathematics and physics, the Christoffel symbols are an array of numbers describing an affine connection.[1] 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 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.
Proposed:
:In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. -ref- See, for instance,  and  -/ref- The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distances to be measured on that surface.  In differential geometry, an affine connection can be defined without any reference to a metric, and many additional concepts follow: parallel transport, covariant derivatives, geodesics, etc. do not require the concept of a metric. However, when a metric is available, these concepts can be directly tied to the "shape" of the manifold itself; that shape is determined by how the tangent space to the cotangent space are attached with a metric tensor. Abstractly, one would say that the manifold is has an associated frame bundle, and an invariant metric implies that the structure group of the frame bundle is the orthogonal group SO(m,n). As a result, such a manifold is necessarily a (pseudo-)Riemannian manifold. The Christoffel symbols provide a concrete representation of the connection of (pseudo-)Riemannian geometry in terms of coordinates on the manifold. Additional concepts, such as parallel transport, geodesics, etc. can then be expressed in terms of Christoffel symbols.
:In general, there are an infinite number of metric connections for a given metric tensor; however, there is one, unique connection, the Levi-Civita connection, that is free of any torsion. It is very common in physics and general relativity to work almost exclusively with the Levi-Civita connection, by working in coordinate frames (holonomic coordinates) where the torsion vanishes.
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. 67.198.37.16 (talk) 20:16, 1 May 2016 (UTC)
replies: text: 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: ~~~~.) Mgnbar (talk) 11:41, 2 May 2019 (UTC)

text: Yes, one can write something that looks like Christoffel symbols for both affine connections and for spin connections.  However, this article, as currently written, defines the Christoffel symbols in terms of the metric. It would need some major restructuring to wedge in some other definition, and then go through some pains to demonstrate that this earlier, more primitive definition turns out to be exactly the same as the conventional formulation built from the metric tensor. 67.198.37.16 (talk) 03:39, 14 November 2023 (UTC)
thread_title: Draft first paragraph

message: JRSpriggs 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 metrics. Co-scienza (talk) 15:08, 8 June 2019 (UTC)
thread_title:  Applications in classical (no relativistic) mechanics 

message: 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 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):Note that, as defined here, a Christoffel symbol is a tensor field associated with the derivative operator \nabla_a and the coordinate system used to define \partial_a.  However, if we change coordinates, we also change our ordinary derivative operator from \partial_a to \partial'_a and thus we change our tensor \Gamma^c {}_{ab} to a new tensor {\Gamma'}^c {}_{ab}.  Hence the coordinate components of \Gamma^c {}_{ab} in the unprimed coordinates will not be related to the components of {\Gamma'}^c {}_{ab} in the primed coordinates by the tensor transformation law, since we change tensors as well as coordinates.A similar (but not quite identical) perspective can be found in Schutz (2009). Johnny Assay (talk) 19:39, 25 June 2019 (UTC)
replies: text: Perhaps a better way of explaining this is to say that \Gamma^c {}_{ab} does not represent the components of a type-\tbinom 12 tensor (with three tensor indices), but does represent the components of a type-\tbinom 11 tensor when b and c are tensor indexes, but a refers to one specific fixed coordinate system. As far as I know, this second interpretation isn't the usual one. -- Dr Greg  talk  21:27, 25 June 2019 (UTC)
replies: text: The interpretation you mention is actually the one given in Schutz. Johnny Assay (talk) 17:10, 26 June 2019 (UTC)
thread_title:  Christoffel symbols as tensors 

message: Because this is an encyclopedia, not a physics textbook, the Einstein summation convention should not be used.
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 introduction of a few sigmas will enable many readers to not need any special convention. Just say what you mean, don't make Wikipedia some kind of knowledge test.
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. 
Special:Contributions/2601:200:c082:2ea0:e1a3:7465:1ef3:b131 22:04, 26 July 2023‎
:Sorry, no. Standard textbooks don't use the sum symbol; it would be inappropriate to use it here, in violation of standard conventions. 67.198.37.16 (talk) 23:14, 12 November 2023 (UTC)
thread_title:  Einstein summation convention