General Relativity/Raising and Lowering Indices
Given a tensor , the components are given by (just insert appropriate basis vectors and basis one-forms into the slots to get the components).
So, given a metric tensor , we get components and . Note that since .
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 and . Here are some examples:
Finally, here is a useful trick: thinking of the components of the metric as a matrix, it is true that since .