How does the spin-connection $\omega_{\mu}^{~~ab}$ transform under general coordinate transformation? Is it a tensor with respect to the $\mu$ index?
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1$\begingroup$ Yes, it is……….. $\endgroup$– PraharCommented Jun 29, 2021 at 9:58
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$\begingroup$ @PraharMitra Any reference in this regard will be most helpful. $\endgroup$– Dr. user44690Commented Jun 29, 2021 at 10:05
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1 Answer
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Yes, it transforms as a one-form under spacetime GCT's (but it isn't a tensor-valued one form, as you probably know it transforms inhomogeneously for its Lorentz indices).
A good way to check this for yourself is to calculate something like $\nabla_{\mu} T^{a}$ then apply the transformation rules to deduce the the transformation laws for $\omega_{\mu}{}^{a}{}_{b}$.
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$\begingroup$ Can you provide reference if possible? It will be very helpful. $\endgroup$ Commented Jun 29, 2021 at 10:03
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1$\begingroup$ @IIRC this sort've calculation is covered in Carrol's GR book arxiv.org/abs/gr-qc/9712019 (have a look through the parts on the spin connection) $\endgroup$– EletieCommented Jun 29, 2021 at 10:22
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