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). Where the notation (k, l) has been used to denote a rank k + l with k upper indices and l lower indices.

One does this by multiplying by the covariant or contravariant metric tensor and then contracting (simply summing over the repeated index j in the example below).

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

g^{ij}A_{j}=A^{i},

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

g_{ij}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^{ij}g_{ji}=g_{ij}g^{ji}=g_{i}^{i}=Trg=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.

Example from Special Relativity

In a Minkowski space with the metric tensor

g_{\mu \nu }={\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}}

the contravariant electromagnetic tensor is given by

F^{\mu \nu }={\begin{bmatrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\E_{x}/c&0&-B_{z}&B_{y}\\E_{y}/c&B_{z}&0&-B_{x}\\E_{z}/c&-B_{y}&B_{x}&0\end{bmatrix}}

Note: some texts, such as Griffiths^[1], will show this tensor with an overall factor of -1. This is because they used the negative of the metric tensor used here, see metric signature. Older texts such as Jackson 2ed are missing the factors of c; they are using Gaussian units whereas here we are using SI units.

To get the covariant tensor $F_{\mu \nu }\,$ , we use