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.

  • $\begingroup$ 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


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$$K_{\;\;\mu\nu}^\rho=\frac12T_{\alpha\beta\gamma}(g^{\alpha\rho}\delta_\mu^\beta\delta_\nu^\gamma-\delta_\mu^\alpha g^{\beta\rho}\delta_\nu^\gamma-\delta_\nu^\alpha g^{\beta\rho}\delta_\mu^\gamma).$$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.