General Relativity/Raising and Lowering Indices

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Given a tensor $𝐓$, the components $T_{\beta }^{\alpha \mu \nu }$ are given by $T_{\beta }^{\alpha \mu \nu }=𝐓(𝐝x^{\alpha },{𝐞}_{\beta },𝐝x^{\mu },𝐝x^{\nu })$ (just insert appropriate basis vectors and basis one-forms into the slots to get the components).

So, given a metric tensor $𝐠(𝐮,𝐯)=<𝐮|𝐯>$, we get components $g_{\mu \nu }=<{𝐞}_{\mu }|{𝐞}_{\nu }>$ and $g^{\mu \nu }=<𝐝x^{\mu }|𝐝x^{\nu }>$. Note that $g_{\nu }^{\mu }=g_{\mu }^{\nu }={\delta }_{\nu }^{\mu }$ since $<{𝐞}_{\mu }|𝐝x^{\nu }>=<𝐝x^{\mu }|{𝐞}_{\nu }>={\delta }_{\nu }^{\mu }$.

Now, given a metric, we can convert from contravariant indices to covariant indices. The components of the metric tensor act as "raising and lowering operators" according to the rules $w^{\alpha }=g^{\alpha \mu }{w}_{\mu }$ and $w_{\alpha }=g_{\alpha \mu }{w}^{\mu }$. Here are some examples:

1. $T_{\beta }^{\alpha \gamma }=g_{\beta \mu }{T}^{\alpha \mu \gamma }$

Finally, here is a useful trick: thinking of the components of the metric as a matrix, it is true that $\left(g^{\mu \nu }\right)={\left(g_{\mu \nu }\right)}^{-1}$ since $g^{\mu \sigma }{g}_{\sigma \nu }=g_{\nu }^{\mu }={\delta }_{\nu }^{\mu }$.