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Eris
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The torsion tensor in 4 dimensions $S_{ab}^{\hphantom0\hphantom0 c}$ has 24 components and it can be split into a vector part $\hphantom0^{V}S_{ab}^{\hphantom0\hphantom0 c}=\frac{1}{3}(S_a\delta^c_b-S_b\delta^c_a)$ (4 components) and an axial vector-vector part $\hphantom0^AS_{ab}^{\hphantom0\hphantom0 c}=g^{cd}S_{[abd]}$ (4 components) and a traceless part $\hphantom0^TS_{ab}^{\hphantom0\hphantom0c}=S_{ab}^{\hphantom0\hphantom0 c}-\hphantom0^AS_{ab}^{\hphantom0\hphantom0 c}-\hphantom0^VS_{ab}^{\hphantom0\hphantom0 c}$ (16 components).

Torsion is usually interpreted physically as a twist in space because it causes a parallelogram to fail to close. How does the vector components particularly contribute to that twist geometrically? and what happen if it is a complex quantity i.e. $S_a=S_b=i\phi(t)$?

The torsion tensor $S_{ab}^{\hphantom0\hphantom0 c}$ has 24 components and it can be split into a vector part $\hphantom0^{V}S_{ab}^{\hphantom0\hphantom0 c}=\frac{1}{3}(S_a\delta^c_b-S_b\delta^c_a)$ (4 components) and an axial vector part $\hphantom0^AS_{ab}^{\hphantom0\hphantom0 c}=g^{cd}S_{[abd]}$ (4 components) and a traceless part $\hphantom0^TS_{ab}^{\hphantom0\hphantom0c}=S_{ab}^{\hphantom0\hphantom0 c}-\hphantom0^AS_{ab}^{\hphantom0\hphantom0 c}-\hphantom0^VS_{ab}^{\hphantom0\hphantom0 c}$ (16 components).

Torsion is usually interpreted physically as a twist in space because it causes a parallelogram to fail to close. How does the vector components particularly contribute to that twist geometrically? and what happen if it is a complex quantity?

The torsion tensor in 4 dimensions $S_{ab}^{\hphantom0\hphantom0 c}$ has 24 components and it can be split into a vector part $\hphantom0^{V}S_{ab}^{\hphantom0\hphantom0 c}=\frac{1}{3}(S_a\delta^c_b-S_b\delta^c_a)$ (4 components) and an axial-vector part $\hphantom0^AS_{ab}^{\hphantom0\hphantom0 c}=g^{cd}S_{[abd]}$ (4 components) and a traceless part $\hphantom0^TS_{ab}^{\hphantom0\hphantom0c}=S_{ab}^{\hphantom0\hphantom0 c}-\hphantom0^AS_{ab}^{\hphantom0\hphantom0 c}-\hphantom0^VS_{ab}^{\hphantom0\hphantom0 c}$ (16 components).

Torsion is usually interpreted physically as a twist in space because it causes a parallelogram to fail to close. How does the vector components particularly contribute to that twist geometrically? and what happen if it is a complex quantity i.e. $S_a=S_b=i\phi(t)$?

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YCor
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The torsion tensor $S_{ab}^{\hphantom0\hphantom0 c}$ has 24 components and it can be split into a Vectorvector part $\hphantom0^{V}S_{ab}^{\hphantom0\hphantom0 c}=\frac{1}{3}(S_a\delta^c_b-S_b\delta^c_a)$ (4 components) and an Axialvectoraxial vector part $\hphantom0^AS_{ab}^{\hphantom0\hphantom0 c}=g^{cd}S_{[abd]}$ (4 components) and a traceless part $\hphantom0^TS_{ab}^{\hphantom0\hphantom0c}=S_{ab}^{\hphantom0\hphantom0 c}-\hphantom0^AS_{ab}^{\hphantom0\hphantom0 c}-\hphantom0^VS_{ab}^{\hphantom0\hphantom0 c}$ (16 components).

Torsion is usually interpreted physically as a twist in space because it causes a parallelogram to fail to close. How does the vector components particularly contribute to that twist geometrically? and what happen if it is a complex quantity?

The torsion tensor $S_{ab}^{\hphantom0\hphantom0 c}$ has 24 components and it can be split into a Vector part $\hphantom0^{V}S_{ab}^{\hphantom0\hphantom0 c}=\frac{1}{3}(S_a\delta^c_b-S_b\delta^c_a)$ (4 components) and an Axialvector part $\hphantom0^AS_{ab}^{\hphantom0\hphantom0 c}=g^{cd}S_{[abd]}$ (4 components) and a traceless part $\hphantom0^TS_{ab}^{\hphantom0\hphantom0c}=S_{ab}^{\hphantom0\hphantom0 c}-\hphantom0^AS_{ab}^{\hphantom0\hphantom0 c}-\hphantom0^VS_{ab}^{\hphantom0\hphantom0 c}$ (16 components).

Torsion is usually interpreted physically as a twist in space because it causes a parallelogram to fail to close. How does the vector components particularly contribute to that twist geometrically? and what happen if it is a complex quantity?

The torsion tensor $S_{ab}^{\hphantom0\hphantom0 c}$ has 24 components and it can be split into a vector part $\hphantom0^{V}S_{ab}^{\hphantom0\hphantom0 c}=\frac{1}{3}(S_a\delta^c_b-S_b\delta^c_a)$ (4 components) and an axial vector part $\hphantom0^AS_{ab}^{\hphantom0\hphantom0 c}=g^{cd}S_{[abd]}$ (4 components) and a traceless part $\hphantom0^TS_{ab}^{\hphantom0\hphantom0c}=S_{ab}^{\hphantom0\hphantom0 c}-\hphantom0^AS_{ab}^{\hphantom0\hphantom0 c}-\hphantom0^VS_{ab}^{\hphantom0\hphantom0 c}$ (16 components).

Torsion is usually interpreted physically as a twist in space because it causes a parallelogram to fail to close. How does the vector components particularly contribute to that twist geometrically? and what happen if it is a complex quantity?

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Eris
Eris
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What is the physical meaning of torsion

The torsion tensor $S_{ab}^{\hphantom0\hphantom0 c}$ has 24 components and it can be split into a Vector part $\hphantom0^{V}S_{ab}^{\hphantom0\hphantom0 c}=\frac{1}{3}(S_a\delta^c_b-S_b\delta^c_a)$ (4 components) and an Axialvector part $\hphantom0^AS_{ab}^{\hphantom0\hphantom0 c}=g^{cd}S_{[abd]}$ (4 components) and a traceless part $\hphantom0^TS_{ab}^{\hphantom0\hphantom0c}=S_{ab}^{\hphantom0\hphantom0 c}-\hphantom0^AS_{ab}^{\hphantom0\hphantom0 c}-\hphantom0^VS_{ab}^{\hphantom0\hphantom0 c}$ (16 components).

Torsion is usually interpreted physically as a twist in space because it causes a parallelogram to fail to close. How does the vector components particularly contribute to that twist geometrically? and what happen if it is a complex quantity?