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I'm working out a problem where at some point get the following product of metric tensors and momenta: $$g^{\mu\beta}g^\nu_{\space\space\alpha}(2k+\frac{q}{2})^\alpha(\frac{q}{2}-k)_\beta$$ How can I work out the metric tensor's indexes. What is the difference between a metric tensor with both indexes either up or down $g^{\alpha\beta}$ and another with one index up and the other down $g^\alpha_{\space\space\beta}$?

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  1. The components $g^{\alpha\beta}:=(g^{-1})^{\alpha\beta}$ are the components of the inverse metric tensor, which is a symmetric (2,0) contravariant tensor.

  2. Since one lowers (raises) indices with the metric (inverse metric), respectively, the mixed tensor components $g^{\alpha}{}_{\beta}=\delta^{\alpha}_{\beta}$ are nothing but the Kronecker delta.

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