Talk:Torsion tensor

message: I think I'm starting to understand this. Does this accurately describe torsion?:
: Suppose you are in the day-to-day flat 3-D universe in a space ship just to the left of someone else in another spaceship, both of you facing in the same direction. If you both move forward that person remains just to your right.
:* If you were not in flat space, but you were in torsion-free space, as you both move forward, the person next to you might move further away, but if you only took a small step forward, the other person would still be just to your right.
:* In contrast, if the universe were flat but not torsion-free, as you moved forward, the other person would be just as far away, but might appear to orbit you, so that after some distance the other person would appear to be above you.
Is that an accurate description of torsion? —Ben FrantzDale 23:01, 28 April 2007 (UTC)
replies: text: Yes, I think so, the section on "interpretation" goes into this in a more precise way. 67.198.37.16 (talk) 04:15, 21 April 2016 (UTC)
thread_title: Understanding

message: The following makes no sense:
The components of the torsion tensor  T^c{}_{ab}  can be derived by setting  X=e_a, Y=e_b  and by introducing the commutator coefficients given by  \gamma^c_{ab} e_c :=[e_a,e_b] . We finally obtain a component expression of the torsion tensor,
: T^c{}_{ab} := \Gamma^c{}_{ab} - \Gamma^c{}_{ba}-\gamma^c{}_{ba}
It implies X=Y which is just silly. If I stretch my imagination a bit then γ might be meant to be simply [X, Y] which would make this the component version of the tensorial definition. Someone should fix or remove this. --MarSch 10:32, 20 May 2007 (UTC)
replies: text: I agree that the statement was a little odd.  The author clearly intended a and b to signify numerical indices of the frame (rather than abstract indices).  This is the only interpretation where you can have X ≠ Y.  Anyway, feel free to do with it as you like.  Physicists, I believe, generally use Latin subscripts such as i,j,k,... (or sometimes Greek indices) for numerical indices as opposed to the letters from the beginning of the alphabet such as a,b,etc.  I have preserved the notation, but perhaps it would be a good idea to bring it in line with prevailing conventions. Silly rabbit 11:48, 20 May 2007 (UTC)
thread_title:  non-sense 

message: I propose moving this to torsion (differential geometry), currently a redirect here.  That way the article can discuss torsion in a slightly broader context than just the torsion tensor (torsion form, intrinsic and extrinsic torsion, torsion of a prolongation, etc.) Silly rabbit 12:30, 20 May 2007 (UTC)
replies: text: Probably a good idea. --MarSch 13:34, 20 May 2007 (UTC)
replies: text: I agree to move this article to torsion (differential geometry). Then let's merge torsion of curves to the moved article. --Acepectif 03:20, 31 August 2007 (UTC)
thread_title:  Move to torsion (differential geometry)? 

message: * Torsion of almost complex structures:  The Nijenhuis tensor.
* The torsion of a hermitian connection vanishes if and only if the manifold is Kähler.
* Analytic torsion and the Weitzenbock identity.
* Torsion of Cartan connections.
* Weyl geometries and projective connections.
Silly rabbit 16:12, 8 June 2007 (UTC)
replies: text: Also torsion in supersymmetry. 67.198.37.16 (talk) 04:45, 21 April 2016 (UTC)
thread_title:  Some other things to think about 

message: My understanding is that expansion or contraction of a frame as it moves along a curve is also torsion. Is this wrong, or does the description of torsion need revision? 81.79.139.224 (talk) 07:29, 20 April 2008 (UTC)
replies: text: I think that could be true for a general affine connection, but it seems unlikely for a metric connection, which would still preserve the metric and therefore also the length of the frame elements. siℓℓy rabbit (talk) 14:06, 6 August 2008 (UTC)

text: Apparently Cartan discusses the general case of a geometrical interpretation of torsion for an affine connection.  I will try to add something about this over the next days, as soon as I decipher the Cartan paper (which could take some time).  siℓℓy rabbit (talk) 18:00, 8 August 2008 (UTC)
thread_title:  Does torsion only describe twisting?

message: This section suggests that a connection is torsion-free if and only if the affine developments of loops are also loops. But it seems to me that a torsion-free (but not curvature-free) connection can also take loops into open curves. For ecample, the affine developments of great triangles on the sphere are open (see PhysicsForums, Torsion, affine development and Levi-Civita connection).
Perhaps the statement is valid only for curvature-free connections. More precise description or citations will be appreciated.
89.135.19.87 (talk) 07:31, 22 November 2008 (UTC)
replies: text: You're right.  There is something wrong with the section.  An easy way to see it is that geodesics develop to straight lines.  But on the sphere the geodesics are great circles (and so are in particular closed).  I'll see what I can do to mend it. siℓℓy rabbit (talk) 14:14, 22 November 2008 (UTC)
thread_title:  Section Affine develompents seems incomplete 

message: :d(tr T) = 0
isn't true in general, and it isn't a consequence of any Bianchi identity!
Hep thinker (talk) 10:36, 13 July 2009 (UTC)
replies: text: I concur.  Since it is possible to define a connection with prescribed torsion, tr T is an essentially arbitrary one-form, at least locally.  In particular, it is not closed.  Sławomir Biały (talk) 15:58, 17 August 2009 (UTC)
thread_title:  Incorrect identity? 

message: I think that characterizing Finsler geometry as a "non-metric situation" is non-sensical. A better contrast to Riemannian geometry would be that Finsler-manifolds are "not infinitesimally Euclidean" like Riemanian manifolds are in the sense that the norm induced into each T_xM is given by an inner product.
As matter of fact there is a generalization of the concept of torsion (more adequate for Finsler-geometry) given by the Frölicher-Nijenhuis bracket
: T(X,Y):=[J,v](X,Y)
where J is the tangent structure and  v is the vertical projection of a nonlinear connection on (TM,\pi,M). Perhapt this should be incorporated into the article.  —Preceding unsigned comment added by 82.181.89.137 (talk) 12:53, 13 August 2009 (UTC)
replies: text: Yes, it should be.   There is another formulation that makes no reference to metrics, and that is this: if one has a solder form and writes θ for it, then the torsion tensor Θ is given by Θ = D θ (with D the exterior covariant derivative).  For any given connection ω, there is a unique one-form σ on the tangent bundle TE that is vanishing in the vertical bundle, and is such that ω+σ is another connection 1-form that is torsion-free.   The resulting one-form ω+σ is nothing other than the Levi-Civita connection. One can take this as a definition: since the torsion is given by \Theta = D\theta = d\theta + \omega \wedge \theta, the vanishing of the torsion is equivalent to having d\theta = - (\omega +\sigma) \wedge \theta, and it is not hard to show that σ must vanish on the vertical bundle, and that σ must be G-invariant on each fibre (more precisely, that σ transforms in the adjoint representation of G).  Note that this defines the Levi-Civita connection without making any explicit reference to any metric tensor. Properly speaking, the metric is an example of a solder form. I am somewhat confused by exactly when and where one can write down a solder form, and thus don't quite know how to turn this into a fully general statement (and don't currently have any references that provide/clarify such details).  67.198.37.16 (talk) 21:30, 22 April 2016 (UTC)
replies: text: This means that the article is wrong when it says that "torsion" in Finsler geometry is the same as the torsion of some affine connection. Where did all this word salad come from? --Svennik (talk) 22:15, 23 August 2023 (UTC)
replies: text: I just looked. I don't see any place where the article says that it's the "same thing". 67.198.37.16 (talk) 04:12, 11 November 2023 (UTC)
replies: text: The article suggests that "torsion" in Finsler geometry is generalised by "torsion" of a smooth manifold equipped with an affine connection. At least that's what the introduction to the article suggests in paragraphs 1 and 2. While that doesn't mean it's saying they're the same thing, it fails to provide evidence that those two notions are related in anything but name. And indeed, one discussion below suggests that there's no relationship between these at all. The article is then simply wrong. Svennik (talk) 19:42, 22 November 2023 (UTC)
thread_title:  Comment on Finsler-geometry 

message: moved Torsion field to Torsion field (pseudoscience), and then converted the redirect to point to Torsion tensor. (With similar changes to the talk pages.) I presume that this was because he thought that there was something scientific about torsion which made it distinct from the pseudoscience described in Torsion field (pseudoscience). I am not convinced that that is so. What evidence do you have that any such thing as torsion exists in the real world? JRSpriggs (talk) 09:22, 29 March 2011 (UTC)
replies: text: !? The torsion tensor is a standard concept in differential geometry; its covered in all textbooks on differential or affine geometry. (also covered in newer books on general relativity, e.g. misner, thorne, wheeler, 1972) The torsion-field thing is an interesting bit of pseudoscience. Quite unrelated, except for two facts: first, that pseudoscience loves to mis-appropriate science terms, thus adding to the confusion, and, second, that anything that "spins" is like catnip to UFO enthusiasts. 67.198.37.16 (talk) 04:26, 21 April 2016 (UTC)

text: A very simple example in the real world is the torsion coefficients of non-orientable surfaces such as the Klein bottle and Boy surface - these manifolds look very pretty when made of glass. It is an open question whether the Universe as a whole has torsion. — Cheers, Steelpillow (Talk) 08:02, 21 April 2016 (UTC)
thread_title:  Pseudoscience 

message: In the (sub-)section Twisting of reference frames this passage appears:
"The case of a manifold with a (metric) connection admits an analogous interpretation. Suppose that an observer is moving along a geodesic for the connection. Such an observer is ordinarily thought of as inertial since she experiences no acceleration."
But this is the only occurrence of the word "acceleration" that appears in the article. So this passage is more confusing than helpful.  (And incidentally, the linked article on acceleration contains no reference to the word "geodesic".)
If there is a relationship between geodesics and the absence of acceleration, this article would be a good place to explain it, not merely refer to it mysteriously.Daqu (talk) 23:22, 5 October 2015 (UTC)
replies: text: Concept from general relativity: geodesics are precisely those curves given by observers in free-fall, i.e. no acceleration. 67.198.37.16 (talk) 04:01, 21 April 2016 (UTC)
replies: text: An object in free-fall does accelerate. --Svennik (talk) 22:01, 23 August 2023 (UTC)
replies: text: Um, hey, by definition, free falling reference frames do not accelerate. They are inertial frames, and they have that name for a reason. See Equivalence principle if unclear on the concept. 67.198.37.16 (talk) 04:15, 11 November 2023 (UTC)
replies: text: I never studied GR so I made a mistake. Thanks for the link. Svennik (talk) 19:45, 22 November 2023 (UTC)
thread_title:  Classic extremely bad exposition