# Contorsion tensor

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{{Short description|Object in differential geometry}}
The '''contorsion tensor''' (or '''contortion tensor''') in [differential geometry](/source/differential_geometry) is the difference between a [connection](/source/connection_(vector_bundle)) with and without [torsion](/source/Torsion_tensor) in it.  It commonly appears in the study of [spin connection](/source/spin_connection)s. Thus, for example, a [vielbein](/source/vielbein) together with a spin connection, when subject to the condition of vanishing torsion, gives a description of Einstein gravity. For [supersymmetry](/source/supersymmetry), the same constraint, of vanishing torsion, gives (the field equations of) [eleven-dimensional supergravity](/source/eleven-dimensional_supergravity).<ref>Urs Schreiber, "[https://www.physicsforums.com/insights/11d-gravity-just-torsion-constraint/ 11d Gravity From Just the Torsion Constraint]" (2016)</ref> 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](/source/Cartan's_equivalence_method) for establishing the equivalence of geometric structures.

==Definition in metric geometry==
In [metric geometry](/source/metric_geometry), the contorsion tensor expresses the difference between a [metric-compatible](/source/Metric_connection)  [affine connection](/source/affine_connection) with [Christoffel symbol](/source/Christoffel_symbol) <math> {\Gamma^{k}}_{ij} </math>  and the unique torsion-free [Levi-Civita connection](/source/Levi-Civita_connection) for the same metric.

The contorsion tensor <math>K_{kji}</math> is defined in terms of the [torsion tensor](/source/torsion_tensor), so the latter tensor's definition is important:

:<math>{T^{l}}_{ij}= {\Gamma^{l}}_{ij} -{\Gamma^{l}}_{ji} </math>

Notice that the torsion tensor is antisymmetric in its ''last'' two indices.  Now the contorsion tensor can be defined, and it is obviously antisymmetric in its ''first'' two indices:

:<math> K_{ijk} =  \tfrac{1}{2} (T_{ijk} - T_{jik} - T_{kij}) </math>

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

:<math>T_{ijk} \equiv  g_{il} {T^{l}}_{jk}</math>.

The reason for the non-obvious sum in the definition of the contorsion tensor is due to the sum-sum difference that enforces metric compatibility.  The full metric compatible affine connection can be written as: 

:<math> {\Gamma^{l}}_{ij} =\bar\Gamma^{l}{}{}_{ij} + {K^{l}}_{ij},</math>

where <math> \bar\Gamma^{l}{}{}_{ji} </math> is the torsion-free Levi-Civita connection:

:<math> \bar\Gamma^{l}{}{}_{ji} = \tfrac{1}{2} g^{lk} (\partial_{i}g_{jk} + \partial_{j}g_{ki} - \partial_{k}g_{ij}) </math>

Because the contorsion tensor is antisymmetric in its first two indices (not necessarily its last two indices), the contorsion tensor will typically add (to the Levi-Civita connection) not just a term that is antisymmetric in "i" and "j", but will also add a term that is symmetric in "i" and "j".

==Definition in affine geometry==
In [affine geometry](/source/affine_geometry), one does not have a metric nor a metric connection, and so one is not free to raise and lower indices on demand. One can still achieve a similar effect by making use of the [solder form](/source/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](/source/vertical_and_horizontal_bundles) of a [fiber bundle](/source/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](/source/one-form) on the [tangent bundle](/source/tangent_bundle). 

Recall that the [torsion](/source/torsion_form) of a connection <math>\omega</math> can be expressed as 
:<math>\Theta_\omega = D\theta = d\theta + \omega \wedge \theta</math>

where <math>\theta</math> is the [solder form](/source/solder_form) ([tautological one-form](/source/tautological_one-form)). The subscript <math>\omega</math> serves only as a reminder that this torsion tensor was obtained from the connection.

By analogy to the lowering of the index on torsion tensor on the section above, one can perform a similar operation with the solder form, and construct a tensor 
:<math>\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
</math>

Here <math>\langle,\rangle</math> is the scalar product. This tensor can be expressed as<ref>David Bleecker, "[https://zulfahmed.files.wordpress.com/2014/05/88623149-bleecker-d-gauge-theory-and-variational-principles-aw-1981-ka-t-201s-pqgf.pdf Gauge Theory and Variational Principles] {{Webarchive|url=https://web.archive.org/web/20210709185749/https://zulfahmed.files.wordpress.com/2014/05/88623149-bleecker-d-gauge-theory-and-variational-principles-aw-1981-ka-t-201s-pqgf.pdf |date=2021-07-09 }}" (1982) D. Reidel Publishing ''(See theorem 6.2.5)''</ref>

:<math>\Sigma_\omega(X,Y,Z)=2\langle\theta(Z), \sigma_\omega(X)\theta(Y)\rangle</math>

The quantity <math>\sigma_\omega</math> 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](/source/Ehresmann_connection) <math>\omega</math>, there is another connection <math>\omega+\sigma_\omega</math> that is torsion-free.

The vanishing of the torsion is then equivalent to having 

:<math>\Theta_{\omega+\sigma_\omega} = 0</math>
or 
:<math>d\theta = - (\omega +\sigma_\omega) \wedge \theta</math>

This can be viewed as a [field equation](/source/field_equation) relating the dynamics of the connection to that of the contorsion tensor.

==Derivation==
One way to quickly derive a metric compatible affine connection is to repeat the sum-sum difference idea used in the derivation of the Levi–Civita connection but not take torsion to be zero. Below is a derivation.

Convention for derivation (Choose to define connection coefficients this way. The motivation is that of connection-one forms in [gauge theory](/source/gauge_theory)):

:<math>\nabla_{i}v^{j} = \partial_{i}v^{j} + {\Gamma^{j}}_{ki} v^{k},</math>

:<math>\nabla_{i}\omega_{j} = \partial_{i}\omega_{j} - {\Gamma^{k}}_{ji} \omega_{k},</math>

We begin with the Metric Compatible condition: 

:<math>\nabla_{i}g_{jk} = \partial_{i}g_{jk} - {\Gamma^{l}}_{ji}g_{lk} - {\Gamma^{l}}_{ki}g_{jl} = 0,</math>

Now we use sum-sum difference (Cycle the indices on the condition):
:<math>\partial_{i}g_{jk} - {\Gamma^{l}}_{ji}g_{lk} - {\Gamma^{l}}_{ki}g_{jl} + \partial_{j}g_{ki} - {\Gamma^{l}}_{kj}g_{li} - {\Gamma^{l}}_{ij}g_{kl} - \partial_{k}g_{ij} + {\Gamma^{l}}_{ik}g_{lj} + {\Gamma^{l}}_{jk}g_{il} = 0</math>

:<math>\partial_{i}g_{jk} + \partial_{j}g_{ki} - \partial_{k}g_{ij} - \Gamma_{kji} - \Gamma_{jki}  - \Gamma_{ikj} - \Gamma_{kij} + \Gamma_{jik} + \Gamma_{ijk} = 0</math>
We now use the below torsion tensor definition (for a holonomic frame) to rewrite the connection:
:<math>{T^k}_{ij}= {\Gamma^k}_{ij} - {\Gamma^k}_{ji} </math>
:<math>\Gamma_{kij} = T_{kij} + \Gamma_{kji} </math>

Note that this definition of torsion has the opposite sign as the usual definition when using the above convention <math>\nabla_{i}v^{j} = \partial_{i}v^{j} + {\Gamma^{j}}_{ki} v^{k}</math> for the lower index ordering of the connection coefficients, i.e. it has the opposite sign as the [coordinate-free](/source/coordinate-free) definition <math>\Theta_\omega = D\theta</math> in the below section on geometry. Rectifying this inconsistency (which seems to be common in the literature) would result in a contorsion tensor with the opposite sign.

Substitute the torsion tensor definition into what we have:
:<math>\partial_{i}g_{jk} + \partial_{j}g_{ki} - \partial_{k}g_{ij} - (T_{kji} + \Gamma_{kij}) - \Gamma_{jki}  - (T_{ikj} + \Gamma_{ijk}) - \Gamma_{kij} + (T_{jik} + \Gamma_{jki}) + \Gamma_{ijk} = 0</math>

Clean it up and combine like terms 

:<math>2\Gamma_{kij} = \partial_{i}g_{jk} + \partial_{j}g_{ki} - \partial_{k}g_{ij} - T_{kji} - T_{ikj} + T_{jik} </math>

The torsion terms combine to make an object that transforms tensorially. Since these terms combine together in a metric compatible fashion, they are given a name, the Contorsion tensor, which determines the skew-symmetric part of a metric compatible affine connection.  

We will define it here with the motivation that it match the indices of the left hand side of the equation above.

:<math> K_{kij} =  \tfrac{1}{2} (- T_{kji} - T_{ikj} + T_{jik}) </math>

Cleaning by using the anti-symmetry of the torsion tensor yields what we will define to be the contorsion tensor:

:<math> K_{kij} =  \tfrac{1}{2} (T_{kij} + T_{ijk} - T_{jki}) </math>

Subbing this back into our expression, we have:

:<math>2\Gamma_{kij} = \partial_{i}g_{jk} + \partial_{j}g_{ki} - \partial_{k}g_{ij} + 2 K_{kij} </math>

Now isolate the connection coefficients, and group the torsion terms together:

:<math>{\Gamma^{l}}_{ij} = \tfrac{1}{2} g^{lk} (\partial_{i}g_{jk} + \partial_{j}g_{ki} - \partial_{k}g_{ij}) + \tfrac{1}{2} g^{lk} (2 K_{kij}) </math>

Recall that the first term with the partial derivatives is the Levi-Civita connection expression used often by relativists.

Following suit, define the following to be the torsion-free Levi-Civita connection:

:<math> \bar\Gamma^{l}{}{}_{ij} = \tfrac{1}{2} g^{lk} (\partial_{i}g_{jk} + \partial_{j}g_{ki} - \partial_{k}g_{ij}) </math> 

Then we have that the full metric compatible affine connection can now be written as: 

:<math> {\Gamma^{l}}_{ij} =\bar\Gamma^{l}{}{}_{ij} + {K^{l}}_{ij},</math>

==Relationship to teleparallelism==
In the theory of [teleparallelism](/source/teleparallelism), one encounters a connection, the [Weitzenböck connection](/source/Weitzenb%C3%B6ck_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 [supermanifold](/source/supermanifold)s.<ref name="dewitt">[Bryce DeWitt](/source/Bryce_DeWitt), ''Supermanifolds'', (1984) Cambridge University Press {{ISBN|0521 42377 5}} ''(See the subsection "distant parallelism" of section 2.7.)''</ref>

== See also==
* [Belinfante–Rosenfeld stress–energy tensor](/source/Belinfante%E2%80%93Rosenfeld_stress%E2%80%93energy_tensor)

==References==
{{reflist}}
Category:Tensors
Category:Riemannian geometry
Category:Connection (mathematics)

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Adapted from the Wikipedia article [Contorsion tensor](https://en.wikipedia.org/wiki/Contorsion_tensor) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Contorsion_tensor?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.