Talk:Christoffel symbols

affine connection vs. metric connection

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 *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 Ehresmann connection for example), and, indeed, this is a central critical idea for connections on principal fiber bundles, where there is no metric. (I mean, you can slap one on there if you happen to have a Killing form handy, but PFB's and associated bundles don't normally involve metrics in any way). By contrast, you cannot do Riemannian geometry 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 Einstein-Hilbert action 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 distinction would be a pretty hefty task. 67.198.37.16 (talk) 04:58, 1 May 2016 (UTC)

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 you.

Maybe what I'm confused about is: Should this article be distinct from Affine connection and Levi-Civita connection? Is there really so much to say about the Christoffel symbols themselves? Is there really so much to say, if we don't assume that a metric is present? (I'm honestly asking.) Mgnbar (talk) 18:00, 1 May 2016 (UTC)

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:

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.

And now pick it apart: First, some true bits:

In other words, when a surface or other manifold is endowed with a sense of differential geometry — parallel transport, covariant derivatives, geodesics, etc.

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:

the Christoffel symbols are a concrete representation of that geometry in coordinates on that surface or manifold

its true only if "that geometry" refers to (pseudo-)Riemannian geometry; its false if it refers to differential geometry. You can't coordinatize on the surface of the manifold, because e.g. on fiber bundles, there are no suitable coordinates. e.g. for a gauge connection, 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:

Frequently, but not always, the connection in question is the Levi-Civita connection of a metric tensor.

Its true if "the connection in question" 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:

• Christofell symbols apply only to (psuedo-)Riemannian geometry
• The connection in Riemannian geometry is (sometimes) called the Riemannian connection and it is the affine connection of the frame bundle.
• The Ch.. symbols are the coordinate version of the Riemannian connection,
• 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.
• 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).
• 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 structure group of a frame bundle is necessarily SO(m,n) because the metric forces it to be so (lets ignore conformal geometry for now) i.e. the metric is SO(m,n) invariant.

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. 67.198.37.16 (talk) 20:07, 1 May 2016 (UTC)

Draft first paragraph

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, (Spivak 1999) and (Choquet-Bruhat & DeWitt-Morette 1977) -/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)

-- The above statement, "the Christoffel symbols are an array of numbers describing a metric connection." is very misleading because the existence of Christoffel symbols does not guarantee the existence of a metric connection. Christoffel symbols are well-defined in terms of the affine behavior absent any type of metric connection or coordinate representation. See, for instance, Chandrasekhar (1983). I propose new wording as follows: "the Christoffel symbols 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." — Preceding unsigned comment added by 129.93.33.196 (talk) 12:40, May 1, 2019 (UTC)

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)

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 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. Justintruth (talk) 08:39, 12 July 2020 (UTC)

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)

Applications in classical (no relativistic) mechanics

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)

Christoffel symbols as tensors

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)

Perhaps a better way of explaining this is to say that $\Gamma ^{c}{}_{ab}$ does not represent the components of a type-${\tbinom {1}{2}}$ tensor (with three tensor indices), but does represent the components of a type-${\tbinom {1}{1}}$ 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)

The interpretation you mention is actually the one given in Schutz. Johnny Assay (talk) 17:10, 26 June 2019 (UTC)

References

Errors in equations?

Firstly, the equation

$\mathbf {e} _{i}={\frac {\partial }{\partial x^{i}}}=\partial _{i},\quad i=1,\,2,\,\dots ,\,n$

doesn't make much sense, in my opinion. $\mathbf {e} _{i}$ should be a vector, but ${\frac {\partial }{\partial x^{i}}}=\partial _{i}$ is a differentiation operator. If we use this definition, the definition

$g_{ij}=\mathbf {e} _{i}\cdot \mathbf {e} _{j}$

also makes little sense, because then each component in g will be a second order differential operator, while we know that the components of g are real numbers. If we instead use the definition

$\mathbf {e} _{i}={\frac {\partial {\vec {y}}}{\partial x^{i}}},\quad i=1,\,2,\,\dots ,\,n$,

where ${\vec {y}}$ is the position of an imagined Euclidean space in which the manifold exists, then $\mathbf {e} _{i}$ will be a vector, and the definition $g_{ij}=\mathbf {e} _{i}\cdot \mathbf {e} _{j}$ suddenly makes sense. I therefore wonder whether the definition of $\mathbf {e} _{i}$ is incorrectly written?

Secondly, the equation

$g^{ij}=\left(g^{-1}\right)_{ij}$

looks wrong to me. On the left hand side, i and j are contravariant indices, while on the right hand side, they are covariant indices. Does this make sense, or should the equation be rewritten?

The way in which I have seen the inverse be defined before is

$g^{\mu \rho }g_{\rho \nu }=\delta ^{\mu }{}_{\nu }$,

which is the same as saying that when performing a matrix multiplication between g with indices up and g with indices down, you get the identity matrix. —Kri (talk) 22:14, 8 April 2021 (UTC)

Did you really mean to write "covariant" twice?—Anita5192 (talk) 22:47, 8 April 2021 (UTC)

That was a mistake by me which I have now corrected. —Kri (talk) 23:06, 8 April 2021 (UTC)

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. JRSpriggs (talk) 04:55, 9 April 2021 (UTC)

Eh? Its not "garbage", its 100% textbook-standard notation. See the FredV comment below. 67.198.37.16 (talk) 20:49, 12 November 2023 (UTC)

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 vector not the operator, e.g.:

$\mathbf {e} _{i}={\frac {\boldsymbol {\partial }}{{\boldsymbol {\partial }}x^{i}}}={\boldsymbol {\partial }}_{i}$

For more detailed explanation see Tangent space#Tangent vectors as directional derivatives 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. FredV (talk) 07:49, 2 May 2023 (UTC)

Here's the short summary, since the tangent-space article is a bit garbled: Give a manifold $M$, an atlas (topology) consists of a collection of charts $\varphi :U\to \mathbb {R} ^{n}$. Given some arbitrary real function $f:M\to \mathbb {R}$, the chart allows a gradient to be defined:

$\partial _{\mu }f\equiv {\frac {\partial \left(f\circ \varphi ^{-1}\right)}{\partial x^{\mu }}}$

This is called a pullback because it "pulls back" the gradient on $\mathbb {R} ^{n}$ to a gradient on $M$. The pullback does not depend on the actual function $f$, it is the same no matter what $f$ is used. Thus, the standard vector basis $({\vec {e}}_{1},\cdots ,{\vec {e}}_{n})$ on $\mathbb {R} ^{n}$ pulls back to a standard ("coordinate") vector basis $(\partial _{1},\cdots ,\partial _{n})$ on $M$. There is some abuse of notation. The abuses are:

$\partial _{\mu }\equiv {\frac {\partial }{\partial x^{\mu }}}$

as well as writing $(\varphi ^{1},\ldots ,\varphi ^{n})=(x^{1},\ldots ,x^{n})$ that is, $x=\varphi$ or $x^{\mu }=\varphi ^{\mu }$. This provides a vector basis for vector fields on $M$ Common notation is

${\vec {X}}=X^{\mu }\partial _{\mu }=X^{\mu }{\frac {\partial }{\partial x^{\mu }}}$

The same abuse of notation is used to pushforward (differential) one-forms from $\mathbb {R} ^{n}$ to $M$ by writing $dx^{\mu }=d\phi ^{\mu }$ which are soldered to the basis vectors as $dx^{\mu }(\partial _{\nu })=\delta _{\nu }^{\mu }$. I'll add a version of this note to the article, so that its self contained. 67.198.37.16 (talk) 21:24, 12 November 2023 (UTC)

Einstein summation convention

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)

Actually, yes, some textbooks do use $\Sigma$ to denote sum in this context. For example, defines Christoffel symbols with a number of $\Sigma$ symbols. My understanding is it's more common to drop the $\Sigma$ in physics than outside physics, and I disagree that it's a violation of standard conventions to use $\Sigma$ to mean sum here. For accessibility purposes, I also support adding $\Sigma$ symbols to the article. 35.1.160.159 (talk) 13:45, 12 June 2024 (UTC)