Contorsion tensor

The contorsion tensor in differential geometry is the difference between a connection with and without torsion in it. It commonly appears in the study of spin connections. Thus, for example, a vielbein together with a spin connection, when subject to the condition of vanishing torsion, gives a description of Einstein gravity. For supersymmetry, the same constraint, of vanishing torsion, gives (the field equations of) 11-dimensional supergravity.^[1] That is, the contorsion tensor, along with the connection, becomes one of the dynamical objects of the theory, demoting the metric to a secondary, derived role.

The elimination of torsion in a connection is referred to as the absorption of torsion, and is one of the steps of Cartan's equivalence method for establishing the equivalence of geometric structures.

Metric geometry

In metric geometry, the contorsion tensor expresses the difference between a metric-compatible affine connection with Christoffel symbol $\Gamma _{ij}{}^{k}$ and the unique torsion-free Levi-Civita connection for the same metric.

The contorsion tensor ${K_{ab}}^{c}$ is defined in terms of the torsion tensor ${T_{ij}}^{k}={\Gamma _{ij}}^{k}-{\Gamma _{ji}}^{k}$ as

K_{ijk}={\frac {1}{2}}(T_{ijk}-T_{jki}+T_{kij}),

where the indices are being raised and lowered with respect to the metric:

T_{ijk}\equiv g_{kl}{T_{ij}}^{l}

The reason for the non-obvious sum in the definition is that the contorsion tensor, being the difference between two metric-compatible Christoffel symbols, must be antisymmetric in the last two indices, whilst the torsion tensor itself is antisymmetric in its first two indices.

The connection can now be written as

{\Gamma _{kj}}^{i}={\bar {\Gamma }}_{kj}{}^{i}+{K_{kj}}^{i},

where ${\bar {\Gamma }}_{kj}{}^{i}$ is the torsion-free Levi-Civita connection.

It can be shown that the geodesics for both connections are the same: that is, the any geodesic of ${\bar {\Gamma }}_{kj}{}^{i}$ is also a geodesic of $\Gamma _{kj}{}^{i}$ and vice-versa.

Affine geometry

In affine geometry, one does not have a metric nor a metric connection, and so one is not free raise and lower indecies on demand. One can still achieve a similar effect by making use of the solder form, allowing the bundle to be related to what is happening on its base space. This is an explicitly geometric viewpoint, with tensors now being geometric objects in the vertical and horizontal bundles of a fiber bundle, instead of being indexed algebraic objects defined only on the base space. In this case, one may construct a contorsion tensor, living as a one-form on the tangent bundle.

Recall that the torsion of a connection $\omega$ can be expressed as

\Theta _{\omega }=D\theta =d\theta +\omega \wedge \theta

where $\theta$ is the solder form (tautological one-form). The subscript $\omega$ serves only as a reminder that this torsion tensor was obtained from the connection.

That is, one may construct a tensor

\Sigma _{\omega }(X,Y,Z)=\langle \theta (Z),\Theta _{\omega }(X,Y)\rangle +\langle \theta (Y),\Theta _{\omega }(Z,X)\rangle -\langle \theta (X),\Theta _{\omega }(Y,Z)\rangle

Here $\langle ,\rangle$ is the scalar product. This tensor can be expressed as^[2]

\Sigma _{\omega }(X,Y,Z)=2\langle \theta (Z),\sigma _{\omega }(X)\theta (Y)\rangle

The quantity $\sigma _{\omega }$ is the contorsion form and is exactly what is needed to add to an arbitrary connection to get the torsion-free Levi-Civita connection. That is, given an Ehresmann connection $\omega$ , there is another connection $\omega +\sigma _{\omega }$ that is torsion-free.

The vanishing of the torsion is then equivalent to having

\Theta _{\omega +\sigma _{\omega }}=0

d\theta =-(\omega +\sigma _{\omega })\wedge \theta

This can be viewed as a field equation relating the dynamics of the connection to that of the contorsion tensor.

Relationship to teleparallelism

In the theory of teleparallelism, one encounters a connection, the Weitzenböck connection, which is flat (vanishing Riemann curvature) but has a non-vanishing torsion. The flatness is exactly what allows parallel frame fields to be constructed. These notions can be extended to supermanifolds.^[3]

References

↑ Urs Schreiber, "11d Gravity From Just the Torsion Constraint" (2016)
↑ David Bleecker, "Gauge Theory and Variational Principles" (1982) D. Reidel Publishing (See theorem 6.2.5)
↑ Bryce DeWitt, Supermanifolds, (1984) Cambridge University Press ISBN 0521 42377 5 (See the subsection "distant parallelism" of section 2.7.)

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