Torsion Tensor

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Torsion tensor 1

Torsion tensor

Torsion along a geodesic.

In differential geometry, the notion oftorsion is a manner of characterizing a twist or screw of a moving frame around a curve. Thetorsion of a curve, as it appears in the Frenet Serret formulas, forinstance, quantifies the twist of a curve about its tangent vector as thecurve evolves (or rather the rotation of the Frenet Serret frame aboutthe tangent vector). In the geometry of surfaces, thegeodesic torsiondescribes how a surface twists about a curve on the surface. Thecompanion notion of curvature measures how moving frames "roll"along a curve "without twisting".

More generally, on a differentiable manifold equipped with an affineconnection (that is, a connection in the tangent bundle), torsion and curvature form the two fundamental invariants of the connection. In this context, torsion gives an intrinsic characterization of how tangent spaces twist about a curve

when they are parallel transported; whereas curvature describes how the tangent spaces roll along the curve. Torsionmay be described concretely as a tensor, or as a vector-valued two-form on the manifold. If is an affine connectionon a differential manifold, then the torsion tensor is defined, in terms of vector fields XandY , by

where [ X ,Y ] is the Lie bracket of vector fields.Torsion is particularly useful in the study of the geometry of geodesics. Given a system of parametrized geodesics,one can specify a class of affine connections having those geodesics, but differing by their torsions. There is a uniqueconnection whichabsorbs the torsion, generalizing the Levi-Civita connection to other, possibly non-metricsituations (such as Finsler geometry). Absorption of torsion also plays a fundamental role in the study of

G-structures and Cartan's equivalence method. Torsion is also useful in the study of unparametrized families of geodesics, via the associated projective connection. In relativity theory, such ideas have been implemented in theform of Einstein Cartan theory.

The torsion tensorLet M be a manifold with a connection on the tangent bundle. Thetorsion tensor (sometimes called theCartan(torsion) tensor ) is a vector-valued 2-form defined on vector fields XandYby

where [ X ,Y ] is the Lie bracket of two vector fields. By the Leibniz rule,T ( fX ,Y ) = T ( X , fY ) = fT ( X ,Y ) for any smooth

function f . SoT is tensorial, despite being defined in terms of the non-tensorial covariant derivative: it gives a 2-formon tangent vectors, while the covariant derivative is only defined for vector fields.
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Torsion tensor 3

The curvature form and Bianchi identities

The curvature form is thegl(n)-valued 2-form

where, again, D denotes the exterior covariant derivative. In terms of the curvature form and torsion form, thecorresponding Bianchi identities are[2]

1.2.Moreover, one can recover the curvature and torsion tensors from the curvature and torsion forms as follows. At apointu of Fx M , one has

[3]

where againu : Rn Tx M is the function specifying the frame in the fibre, and the choice of lift of the vectors via 1 is irrelevant since the curvature and torsion forms are horizontal (they vanish on the ambiguous vertical vectors).

Torsion form in a frame

The torsion form may be expressed in terms of a connection form on the base manifold M , written in a particularframe of the tangent bundle (e1,...,en). The connection form expresses the exterior covariant derivative of these basicsections:

The solder form for the tangent bundle (relative to this frame) is the dual basis i T* Mof theei, so that i(e j) =

i j

(the Kronecker delta.) Then the torsion 2-form has components

In the rightmost expression,

are the frame-components of the torsion tensor, as given in the previous definition.

It can be easily shown that i transforms tensorially in the sense that if a different frame

for some invertible matrix-valued function (gi j), then

In other terms, is a tensor of type (1,2) (carrying one contravariant and two covariant indices).Alternatively, the solder form can be characterized in a frame-independent fashion as the T M -valued one-form on Mcorresponding to the identity endomorphism of the tangent bundle under the duality isomorphism End(T M ) T M T* M . Then the torsion two-form is a section of

given by

where D is the exterior covariant derivative. (See connection form for further details.)
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Torsion tensor 4

Irreducible decomposition

The torsion tensor can be decomposed into two irreducible parts: a trace-free part and another part which containsthe trace terms. Using the index notation, the trace ofT is given by

and the trace-free part is

where i j is the Kronecker delta.

Intrinsically, one has

The trace ofT , tr T , is an element of T* M defined as follows. For each vector fixed X T M , T defines an elementT ( X ) of Hom(T M , T M ) via

Then (trT )( X ) is defined as the trace of this endomorphism. That is,

The trace-free part ofT is then

where denotes the interior product.

Characterizations and interpretationsThroughout this section, M is assumed to be a differentiable manifold, and a covariant derivative on the tangent

bundle of Munless otherwise noted.

Twisting of reference frames

In the classical differential geometry of curves, the Frenet-Serret formulas describe how a particular moving frame(the Frenet-Serret frame)twistsalong a curve. In physical terms, the torsion corresponds to the angular momentumof an idealized top pointing along the tangent of the curve.The case of a manifold with a (metric) connection admits an analogous interpretation. Suppose that an observer ismoving along a geodesic for the connection. Such an observer is ordinarily thought of as inertial since sheexperiences no acceleration. Suppose that in addition the observer carries with herself a system of rigid straightmeasuring rods (a coordinate system). Each rod is a straight segment; a geodesic. Assume that each rod is paralleltransported along the trajectory. The fact that these rods are physicallycarried along the trajectory means that theyare Lie-dragged , or propagated so that the Lie derivative of each rod along the tangent vanishes. They may, however,experience torque (or torsional forces) analogous to the torque felt by the top in the Frenet-Serret frame. This force ismeasured by the torsion.More precisely, suppose that the observer moves along a geodesic path (t ) and carries a measuring rod along it. Therod sweeps out a surface as the observer travels along the path. There are natural coordinates (t , x) along this surface,wheret is the parameter time taken by the observer, and x is the position along the measuring rod. The condition thatthe tangent of the rod should be parallel translated along the curve is

Consequently, the torsion is given by
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If this is not zero, then the marked points on the rod (the x = constant curves) will trace out helices instead of geodesics. They will tend to rotate around the observer. Note that for this argument it was not essential that is ageodesic. Any curve would work.

This interpretation of torsion plays a role in the theory of teleparallelism, also known as Einstein-Cartan theory, analternative formulation of relativity theory.

The torsion of a filament

In materials science, and especially elasticity theory, ideas of torsion also play an important role. One problemmodels the growth of vines, focusing on the question of how vines manage to twist around objects.[4]The vine itself is modeled as a pair of elastic filaments twisted around one another. In its energy-minimizing state, the vine naturallygrows in the shape of a helix. But the vine may also be stretched out to maximize its extent (or length). In this case,the torsion of the vine is related to the torsion of the pair of filaments (or equivalently the surface torsion of theribbon connecting the filaments), and it reflects the difference between the length-maximizing (geodesic)configuration of the vine and its energy-minimizing configuration.

Torsion and vorticity

In fluid dynamics, torsion is naturally associated to vortex lines.

Geodesics and the absorption of torsionSuppose that (t ) is a curve on M . Then is anaffinely parametrized geodesic provided that

for all timet in the domain of . (Here the dot denotes differentiation with respect tot , which associates with thetangent vector pointing along it.) Each geodesic is uniquely determined by its initial tangent vector at timet =0,.

One application of the torsion of a connection involves the geodesic spray of the connection: roughly the family of all affinely parametrized geodesics. Torsion is the ambiguity of classifying connections in terms of their geodesicsprays:

Two connections and which have the same affinely parametrized geodesics (i.e., the same geodesic spray)differ only by torsion.[5]

More precisely, if XandYare a pair of tangent vectors at p M , then let

be the difference of the two connections, calculated in terms of arbitrary extensions of XandYaway from p. By theLeibniz product rule, one sees that does not actually depend on how XandY ' are extended (so it defines a tensoron M ). LetS and Abe the symmmetric and alternating parts of :

Then

is the difference of the torsion tensors. and define the same families of affinely parametrized geodesics if and only ifS ( X ,Y ) = 0.

In other words, the symmetric part of the difference of two connections determines whether they have the sameparametrized geodesics, whereas the skew part of the difference is determined by the relative torsions of the two
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Torsion tensor 6

connections. Another consequence is: Given any affine connection , there is a unique torsion-free connection with the same family of affinely

parametrized geodesics.This is a generalization of the fundamental theorem of Riemannian geometry to general affine (possibly non-metric)connections. Picking out the unique torsion-free connection subordinate to a family of parametrized geodesics is

known asabsorption of torsion , and it is one of the stages of Cartan's equivalence method.

Notes[1] See Kobayashi Nomizu (1996) Volume 1, Proposition III.5.2.[2] Kobayashi Nomizu (1996) Volume 1, III.2.[3] Kobayashi Nomizu (1996) Volume 1, III.5.[4] Gorielyet al. (2006).[5][5] See Spivak (1999) Volume II, Addendum 1 to Chapter 6. See also Bishop and Goldberg (1980), section 5.10.

References

Bishop, R.L.; Goldberg, S.I. (1980),Tensor analysis on manifolds, Dover Cartan, . (1923), "Sur les varits connexion affine, et la thorie de la relativit gnralise (premire pa

(http:/ / www.numdam. org/ item?id=ASENS_1923_3_40__325_0), Annales Scientifiques de l'cole NormaleSuprieure 40: 325 412

Cartan, . (1924), "Sur les varits connexion affine, et la thorie de la relativit gnralise (premire pa(Suite)" (http:/ / www. numdam.org/ item?id=ASENS_1924_3_41__1_0), Annales Scientifiques de l'cole Normale Suprieure 41: 1 25

Elzanowski, M.; Epstein, M. (1985), "Geometric characterization of hyperelastic uniformity", Archive for Rational Mechanics and Analysis 88 (4): 347 357, doi: 10.1007/BF00250871 (http:/ / dx.doi. org/ 10. 1007/ BF00250871)

Goriely, A.; Robertson-Tessi, M.; Tabor, M.; Vandiver, R. (2006), Elastic growth models (http:/ / math. arizona.edu/ ~goriely/ Papers/ 2006-biomat. pdf), Springer-Verlag Unknown parameter|collection= ignored (help)

Hehl, F.W.; von der Heyde, P.; Kerlick, G.D.; Nester, J.M. (1976), "General relativity with spin and torsion:Foundations and prospects" (http:/ / rmp. aps. org/ abstract/ RMP/ v48/ i3/ p393_1), Rev. Mod. Phys.48, 393.

Kibble, T.W.B. (1961), "Lorentz invariance and the gravitational field" (http:/ / dx. doi. org/ 10. 1063/ 1.1703702), J. Math. Phys.2, 212.

Kobayashi, S.; Nomizu, K. (1963), Foundations of Differential Geometry, 1 & 2 (New ed.), Wiley-Interscience(published 1996), ISBN 0-471-15733-3

Poplawski, N.J. (2009), "Spacetime and fields", arXiv:0911.0334 (http:/ / arxiv. org/ abs/ 0911. 0334) Schouten, J.A. (1954), Ricci Calculus, Springer-Verlag Schr dinger, E. (1950),Space-Time Structure, Cambridge University Press Sciama, D.W. (1964), "The physical structure of general relativity" (http:/ / rmp. aps. org/ abstract/ RMP/ v36/ i1/

p463_1), Rev. Mod. Phys.36 , 463. Spivak, M. (1999), A comprehensive introduction to differential geometry, Volume II , Houston, Texas: Publish or

Perish, ISBN 0-914098-71-3
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