Definitions

Raising and lowering indices

In mathematics and mathematical physics, given a tensor on a manifold M, in the presence of a nonsingular form on M (such as a Riemannian metric or Minkowski metric), one can raise or lower indices: change a (k, l) tensor to a (k + 1, l − 1) tensor (raise index) or to a (k − 1, l + 1) tensor (lower index).

One does this by multiplying by the covariant or contravariant metric tensor and then contracting.

Multiplying by the contravariant metric tensor (and contracting) raises indices:

$g^\left\{ij\right\}A_j=A^i,$

and multiplying by the covariant metric tensor (and contracting) lowers indices:

$g_\left\{ij\right\}A^j=A_i,$

Raising and then lowering the same index (or conversely) are inverse, which is reflected in the covariant and contravariant metric tensors being inverse:

$g^\left\{ij\right\}g_\left\{ji\right\}=g_\left\{ij\right\}g^\left\{ji\right\}=g_\left\{i\right\}^\left\{i\right\}=Tr g = N.$

where N is the dimension of the manifold. Note that you don't need the form to be nonsingular to lower an index, but to get the inverse (and thus raise indices) you need nonsingular.