<General Relativity
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:
1.
Finally, here is a useful trick: thinking of the components of the metric as a matrix, it is true that
since
.