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Tensor calculus (tensor analysis) is a systematic extension of vector calculus to multivector and tensor fields in a form that is independent of the choice of coordinates on the relevant manifold, but which accounts for respective sub-spaces, their symmetries, and their connections.
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Minimal spin connection, and the standard content of GR
In Sec. 12.1 of 'Superstring Theory', by Green, Schwarz, and Witten, the minimal spin connection is motivated as follows:
If we are to avoid modifying the standard content of general
relativity, …
4
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answer
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Decomposition of the torsion tensor
See Update below.
Consider the torsion tensor $T_{\mu\nu\rho} = -T_{\mu\rho\nu}$. In a local Lorentz frame, as determined by a vierbein $e^{a}{}_{\mu}$, it may equivalently be given as $T_{abc} = e^{ …