General Relativity/Raising and Lowering Indices

<General Relativity

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

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

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 $w^{\alpha }=g^{\alpha \mu }w_{\mu }$ 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 $g^{\mu \sigma }g_{\sigma \nu }=g_{\ \nu }^{\mu }=\delta _{\nu }^{\mu }$.