6.3. General Bianchi identities

Let us now check the Bianchi identities for general covariant derivatives with torsion. Note that these identities are not directly encoded in xTensor`, but can be easily computed.

Define a covariant derivative with torsion. Note the special symmetry properties of the defined tensors:

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This is the general form of the first Bianchi identity:

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The computation can be performed by transforming the derivative CD and its associated tensors into PD and Christoffel tensors:

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This is the general form of the second Bianchi identity:

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The computation proceeds along the same lines:

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Clean up:

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Created by Mathematica (May 16, 2008) |