Christoffel's symbols: symmetry in the two (lower) indices

In summary, the Christoffel symbols are always symmetric in the two lower indices in a (pseudo-)Riemannian space because of the definition of a Riemannian space as one with a torsion-free affine connection that is compatible with the fundamental bilinear form. However, in a general affine manifold with a nonsymmetric connection, the antisymmetric parts of the Christoffel symbols define the torsion of the manifold. This can be seen through the action of a covariant derivative commutator on a scalar field. When a symmetric fundamental 2-tensor is introduced, the antisymmetric parts of the connection appear in the formula for the symmetric part of the Christoffel symbols. This relationship between the Christoffel symbols and tors
  • #1

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TL;DR Summary
How must we link the Christoffel's symbols and the concept of torsion, accordingly to the fact that the Christoffel's symbols are always symmetric in the two lower indices?
I have a technical problem.
1. Accordingly to historical E.B. Christoffel’s work (I think year 1869), (Christoffel’s) symbols are symmetric in the two (today writing) lower indices.
2. These symbols have been introduced when studying the preservation of differential forms of degree two. The results which have been obtained at this time in that document have proved to be useful “a posteriori”; for example: in general relativity (GR).
3. When one looks precisely at the formalism of these symbols (second kind; see a recent article - 2007, e.g.: [01; §2 p. 217, (2)]) and exchange the positions of lower indices but impose the equality, one then automatically obtains a condition on the metric.

This fact induces questions for which some help is needed:
a. Although it is hard to believe such restriction: “Does that condition “de facto” restrict the domain of definition of GR?
b. Why do a lot of (many) scientific articles desperately introduce a concept of torsion based on a presumed possible anti-symmetry of these lower indices (which is in total contradiction with 1. and 2.)? Very roughly, something like: T(^a, _b, _c) = Gamma(^a, _b, _c) - Gamma (^a, _c, _b) which should always vanish.

c. Did I misunderstood something essential concerning these difficult topics (preservation of the element of length, connections, metrics)?

I am referring to:
[01] Annales de la Fondation Louis de Broglie, Volume 32 n. 2-3, 2007 : On a completely anti-symmetric Cartan torsion tensor.

Thanks for help.
 
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  • #2
The Christoffel symbols in a Riemannian space by definition are symmetric, ##\Gamma_{ij}^k=\Gamma_{ji}^k## by definition, because a Riemannian space by definition is torsion free. Additionally you also assume that the metric is compatible to the affine connection, i.e., that the covariant derivatives of the metric vanish, ##\nabla_{i} g_{kl}=0##. Both together determine the Christoffel symbols completely, leading to the relation with the metric:
$$\Gamma_{ij}^k=\frac{1}{2} g^{kl} (\partial_i g_{jl} + \partial_j g_{il}-\partial_l g_{ij}).$$
For a general differentiable manifold (not necessarily one with a metric) you can introduce an arbitrary affine connection with Chirstoffel symbols that are not symmetric. To have the usual properties of the corresponding covariant derivatives, the Christoffel symbols do not transform as tensor components of course, but the antisymmetriced piece does, and thus this defines a covariant specification of the resulting affine manifold, which is called torsion.

Given that there are particles with spin (particularly spin 1/2), it turns out that the space-time description in general relativity has to be extended from a pseudo-Riemannian (torsion-free) manifold to a space-time manifold with a (pseudo-)metric and with torsion (Einstein-Cartan spacetime).
 
  • #3
Well, first of all: thank you for the answer.
Now, I would not appear to be some one who is splitting hairs in four ... (you are an expert and I am not) but I can until now not be happy with this: "... are symmetric per definition".

My doubt lies on the following arguments.

I) In the original work: (i) Christoffel's symbols of the first kind -and of the second kind as well- have roughly the same definition than the actual one (except that Christoffel did not introduce the Gamma symbol but only brackets and parenthesis) and (ii) the symmetry of these symbols is not related to any concept of torsion but, a convention. I mean: Christoffel admits without demonstration that the newly introduced symbols (by himself) are symmetric. My misunderstanding may come from the fact that I am not a native German citizen.

Anyway, when you just exchange i and j in the above writing, you can not come to the logical conclusion that the value of the symbol is unchanged, except if you accept constraints on the metric; for example: that metric is symmetric per convention.

II) Except if I misunderstand the concept of torsion, torsion has nothing to do with the anti-symmetry of the Christoffel's symbols (see point b in my opening thread) but, is related to Cartan's part of the connection; and that part is anti-symmetric.

Discussion is welcome.
 
  • #4
I've just told you the usual definitions:

(pseudo-)Riemann space: A differentiable manifold with a positive definite (non-degenerate) fundamental bilinear form and the torsion-free affine connection that is compatible with the fundamental form.

An affine manifold with a general connection can have torsion, which is a tensor whose components are just the antisymmetric parts of the Christoffel symbols.
 
  • #5
Blackforest said:
Summary: How must we link the Christoffel's symbols and the concept of torsion, accordingly to the fact that the Christoffel's symbols are always symmetric in the two lower indices?
Start with a manifold equipped with a generic nonsymmetric connection ##\Gamma^\lambda_{~\mu\nu}##, which has been introduced as a device for defining a covariant derivative, e.g., $$\nabla_\mu V^\lambda ~:=~ \partial_\mu V^\lambda ~+~ \Gamma^\lambda_{~\mu\nu} \, V^\nu ~.$$ At this early stage, there is no metric, and the ##\Gamma##'s are defined only via the requirement that ##\nabla_\mu V^\lambda## must transform like a 2nd rank tensor.

Torsion is defined via the action of a covariant derivative commutator's action on a scalar field ##f##, i.e., $$[\nabla_\mu , \nabla_\nu] f(x) ~=~ 2\Gamma^\lambda_{~[\mu\nu]} \partial_\lambda f(x) ~.$$ Now introduce a fundamental symmetric 2-tensor (usually called the "metric") with components ##g_{\mu\nu}##, and assume it is constant parallel transport, which is equivalent to ##g_{\mu\nu}## being covariantly constant, i.e., $$0 ~=~ \nabla_\lambda g_{\mu\nu} ~\equiv~ \partial_\lambda g_{\mu\nu} ~-~ g_{\alpha\nu} \Gamma^\alpha_{~\lambda \mu} ~-~ g_{\mu\alpha} \Gamma^\alpha_{~\lambda \nu} ~.$$If you perform the usual computation by cyclic permutation of indices (where ##g## is symmetric, but ##\Gamma## is not), one can derive an expression for the symmetric part of ##\Gamma## which is essentially a sum of the usual Christoffel symbols (1st kind) and 2 extra terms involving the antisymmetric part of the connection ##\Gamma^\lambda_{~[\mu\nu]}##. (Try it as an exercise -- it's instructive.)

Thus, if ##\Gamma^\lambda_{~[\mu\nu]}## is assumed to vanish, one gets the usual Christoffel symbols. Otherwise the torsion ##\Gamma^\lambda_{~[\mu\nu]}## enters (both here, and in the formula for the curvature tensor).

(This is further complicated by anholonomicity if one is not working in a coordinate-adapted (holonomic) basis, where it introduces a non-tensorial contribution to ##\Gamma^\lambda_{~[\mu\nu]}##. Strictly speaking, "torsion" is better understood as the part of ##\Gamma^\lambda_{~[\mu\nu]}## which remains nonzero when one reverts to a holonomic basis.)
 
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  • #6
Super. Thank you, I think I got it. I did the proposed exercise (very instructive indeed) and this allows me to reach equation (7) in the given reference.

I beg your pardon for a small supplementary and perhaps completely stupid question (because it has an evident answer); it is related to my initial one: "Does this way of thinking and proceeding insure the preservation of the element of length?" (This preservation was Christoffel's motivation for creating his symbols; see point 2 above).
 
  • #7
Blackforest said:
"Does this way of thinking and proceeding insure the preservation of the element of length?"
Well, "covariantly-constant metric components" means the metric is the same "there" as it is "here", provided you use parallel transport to get from "here" to "there".

So,... yes.
 
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