So guys I was trying to show some expressions in general relativity and I have the contraction of two tensor objects, those being:
$$\sigma_{\mu\nu} = \sigma_{\nu\mu},$$
and,
$$\omega_{\mu\nu} = -\omega_{\nu\mu}.$$
The contraction of a symmetric object with an antissymetric one is zero, like this:
$$\sigma_{\alpha\beta}\omega^{\alpha\beta}=0.$$
The question is: "Can I also say that the following contraction is zero?"
$$\sigma_{\mu\beta}\omega^{\beta}_{\nu} = ?$$