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How we can swap indices while finding contorsion tensor

\begin{equation} {K^{\rho}}_{\mu\nu} = \frac{1}{2} \left( {T^{\rho}}_{\mu\nu} - {{T_{\mu}}^{\rho}}_{\nu} - {{T_{\nu}}^{\rho}}_{\mu} \right) \end{equation}

I know the formula for torsion scalar i.e. \begin{equation} {T^{\rho}}_{\mu\nu} \equiv 2 {\bar{\Gamma}^\rho}_{[\mu\nu]} = {\bar{\Gamma}^\rho}_{\mu\nu} - {\bar{\Gamma}^\rho}_{\nu\mu} \end{equation}

But how I will find the other two components of torsion used in contorsion tensor formula.

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  • $\begingroup$ en.wikipedia.org/wiki/Contorsion_tensor $\endgroup$
    – MBN
    Commented Oct 26, 2021 at 11:25
  • $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$
    – Community Bot
    Commented Oct 26, 2021 at 11:26

1 Answer 1

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If I've understood your question, the salient point is$$T_{\mu\;\;\nu}^{\;\rho}=\delta_\mu^\alpha g^{\beta\rho}\delta_\nu^\gamma T_{\alpha\beta\gamma}$$and similarly with other terms, so$$K_{\;\;\mu\nu}^\rho=\frac12T_{\alpha\beta\gamma}(g^{\alpha\rho}\delta_\mu^\beta\delta_\nu^\gamma-\delta_\mu^\alpha g^{\beta\rho}\delta_\nu^\gamma-\delta_\nu^\alpha g^{\beta\rho}\delta_\mu^\gamma).$$

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