General Relativity/Raising and Lowering Indices

$$, we get components $$g_{\mu \nu}=<\mathbf{e}_\mu \ |\ \mathbf{e}_\nu>$$ and $$g^{\mu \nu}=<\mathbf{d}x^\mu \ |\ \mathbf{d}x^\nu>$$. Note that $$g^\mu_{\ \nu}=g_\mu^{\ \nu}=\delta^\mu_\nu$$ since $$<\mathbf{e}_\mu \ |\ \mathbf{d}x^\nu>=<\mathbf{d}x^\mu \ |\ \mathbf{e}_\nu>=\delta^\mu_\nu$$.

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

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 $$g^{\mu \sigma}g_{\sigma \nu}=g^\mu_{\ \nu}=\delta^\mu_\nu$$.