<General Relativity

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

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

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 ${\displaystyle w^{\alpha }=g^{\alpha \mu }w_{\mu }}$ and ${\displaystyle w_{\alpha }=g_{\alpha \mu }w^{\mu }}$. Here are some examples:

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

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