# Torsion tensor: Geometric interpretation

• Terilien

#### Terilien

Can someone pleas explain to me the geometric interpretation of Torsion? Why it is true is more important.

Lol i said manifolds instead of torsion!

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I'm sorry but his notes don't talk about the geometric intepretation. He just mentions the torsion tensor and says he'll assume the christoffel symbols are symmetric(thus no torsion).

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Can someone pleas explain to me the geometric interpretation of Manifolds? Why it is true is more important.

Differential equations are used to formulate many physicsl laws. A diifferentiable manifold allows calculus to be used in a fairly general setting.

Sorry guys, I was absentmninded while reading this and said manifolds instead of torsion.

I'm sorry but his notes don't talk about the geometric intepretation. He just mentions the torsion tensor and says he'll assume the christoffel symbols are symmetric(thus no torsion).
Ohanian states
A geometry with an asymmetric Christoffel symbol is said to have torsion.
Is this consistent with what you've learned so far? I don't know the geometrical significance of what this means though. Sorry.

Note: There is another use of the term torsion other than what you're asking about which finds use in differential geometry. It refers to the rate of "twist" of a curve.

Pete

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Ohanian states
Is this consistent with what you've learned so far? I don't know the geometrical significance of what this means though. Sorry.

Pete

For one thing, Ricci need not be symmetric.
Look at http://theory.uchicago.edu/~sjensen/teaching/tutorials/GRtorsion.pdf [Broken]
as mentioned in the earlier thread I referenced above.

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I'm fine now. Actually i figured it out last night after examining the changes in geodesics ,when the chrisotffel symbols become antisymmetric.

I think the basic idea is that a space with zero torsion satisfies commutativity in displacements. In other words infinitesimal displacements commute and the parallelogram law of vector addition is well-defined. This is not the case in a space with non-zero torsion. Clearly one wants to have a very good reason for introducing such a thing at the geometrical level. It's sort of like considering functions of several variables where the mixed partial derivatives don't commute and are discontinuous.

Maybe one of you can explain to me what advantage considering non-zero torsion has at the geometrical level? Perhaps one can introduce the torsion concept in the connections of Lie Groups which are functions of a manifold while leaving the underlying geometry torsion-free.

Why would we want to talk about a geometry where displacements don't commute?

Yours, John

I think the basic idea is that a space with zero torsion satisfies commutativity in displacements. In other words infinitesimal displacements commute and the parallelogram law of vector addition is well-defined. This is not the case in a space with non-zero torsion. Clearly one wants to have a very good reason for introducing such a thing at the geometrical level. It's sort of like considering functions of several variables where the mixed partial derivatives don't commute and are discontinuous.

Maybe one of you can explain to me what advantage considering non-zero torsion has at the geometrical level? Perhaps one can introduce the torsion concept in the connections of Lie Groups which are functions of a manifold while leaving the underlying geometry torsion-free.

Why would we want to talk about a geometry where displacements don't commute?

Yours, John
Non-zero torsion means that local translational invariance is broken. Using the usual gauging process for a global->local symmetry change it's possible to put gravity into the torsion ( rather than into the curvature as in GR ) and get a perfectly good theory of gravity.

Thanks Mentz114

Non-zero torsion means that local translational invariance is broken. Using the usual gauging process for a global->local symmetry change it's possible to put gravity into the torsion ( rather than into the curvature as in GR ) and get a perfectly good theory of gravity.

Is the operation of geometric addition of infinitesimal coordinate segments commutative in spaces with intrinsic torsion?

It looks, to me, like addition of these segments in different order leads to different end-points displaced by an amount delta^a = 2 X^a_b,c d1^b d2^c, where X^a_b,c is the torsion tensor and d1 and d2 are infinitesimal coordinate displacements. I'm thinking of the torsion, X, as the anti-symmetric part of the connection, i.e, X^a_b,c = Gamma^a_b,c - Gamma^a_c,b.

I am very curious as to how the gauge theory for torsion deals with non-commutative invariant geometric addition of infinitesimal vectors. You end up at different end-points depending on how you order the additions.

It looks like we can give up flat space and introduce curvature and incorporate gravity OR give up commutative invariant geometric addition of vectors and incorporate GR. But giving up commutativity of vector addition seems a more bitter pill to accept. Addition is usually commutative in most forms of algebra... addition is the prototype for Abelian groups. But here it looks like we are going to give up Abelian addition.

Please let me know where I can read more about how the torsion gauge theory deals with the above question.

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John R. Smith
It looks like we can give up flat space and introduce curvature and incorporate gravity OR give up commutative invariant geometric addition of vectors and incorporate GR. But giving up commutativity of vector addition seems a more bitter pill to accept.
Both these situations involve parallel transport. In curved space, transporting a vector around an infinitesimal loop will result in a rotation of the vector. In a space with torsion, two parallel transports in directions dx,dy gives a different vector than when doing it the other way dy,dx. This has the effect of destroying local translational invariance, and so local conservation of momentum is lost.

The non-commutivity of infinitesimal parallel transports is 'fixed' by introducing a new covariant derivative which restores translational invariance at the cost of introducing a gauge field. Standard gauge theory.

The gauge gravity theory that results from gauging the translation group is only equivalent to GR if the equivalence principle is added as a postulate.

This is all expressed much better in these papers and references they cite.

'On the gauge aspects of gravity', Gronwald and Hehl, arXiv:gr-qc/9602013

'Topics in Teleparallel Gravity' Aldrovandi, Pereira and Vu, arXiv:/gr-qc/0312008

'Gravitation: in search of the missing torsion' R. Aldrovandi and J. G. Pereira, arXiv:/0801.4148

I was reading "On the Gauge Aspects of Gravity" by Gronwald and Hehl and came across the usual statement that Utiyama gauged the Lorentz group SO(1,3) in 1956 and recovered GR with some additional assumptions. Utiyama was working at the Institute for Advanced Study at the time. Earlier, also at the Institute for Advanced Study, H. Weyl was able to recover GR by gauging the Lorentz group, but he is passed over for being the first to gauge the Lorentz group and come up with another derivation of GR. Check out "A Remark on the Coupling of Gravitation and Electron" Phys. Rev. 77, (699) 1950 and his earlier work Proc. N.A.S. 15, 323 (1929). These papers are interesting historically. He mentioned having problems using the General Linear Group in order to deal with components of the Dirac spinors and therefore confined his attention to local frames and the Orthogonal Group and therefore to local Lorentz invariance. I believe that his original criticism was too confining because you can extend the Dirac matrix commutation relations correctly using the general tensor calculus (sure you have to use local frames in the intemediate steps, but it works out in the end).

Weyl clearly understood that he was gauging local Lorentz invariance, but perhaps not with the generality that Utiyama considered. He also considered gauging the Lorentz group because the Dirac spinors naturally fitted with orthogonal frames, but he doesn't seem to get much credit for being the first to gauge gravitation.

Anyway there is a curious statement in the 1950 Weyl paper that may relate to the question of torsion: (on page 701 in italics) "Thus by the influence of matter [jrs: the presence of Dirac wavefunctions] a slight discordance between the affine connection and metric is created".

I'm not sure if the "discordance" he talked about was an additional symmetric term in the connection or possibly an anti-symmetric torsion tensor. Weyl liked generalizing to the non-Riemannian case.

I will study his papers again to check out the symmetries of his proposed connection.

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