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 type (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 an index:

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

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

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

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

$g^{ij}g_{jk}=g_{kj}g^{ji}=\delta^{i}_{k}$

where $\delta^{i}_{k}$ is the Kronecker delta or identity matrix.

Note that you don't need the form $g_{ij}$ to be nonsingular to lower an index, but to get the inverse (and thus raise an index) it must be 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, 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

$F_{\mu\nu} = g_{\mu\kappa} g_{\nu\lambda} F^{\kappa\lambda}\,$

In the following, summation is suppressed.

Note that since $g_{\mu\nu}\,$ is diagonal, many of the terms in the formula above will vanish:

$F_{\mu\nu} = g_{\mu\mu} g_{\nu\nu} F^{\mu\nu}\,$

Using the convention of Latin letters for indices 1,2 and 3:

$F_{ij} = g_{ii} g_{jj} F^{ij}=F^{ij}\,$

since both factors from the metric tensor are −1.

$F_{ii} = (g_{ii})^2 F^{ii}=F^{ii}\,$

$F_{0i} = g_{00} g_{ii} F^{0i}=-F^{0i}\,$

and similarly

$F_{i0}=-F^{i0}\,$

Putting it all together we have:

$F_{\mu\nu} = \begin{bmatrix} 0 & E_x/c & E_y/c & E_z/c \\ -E_x/c & 0 & -B_z & B_y \\ -E_y/c & B_z & 0 & -B_x \\ -E_z/c & -B_y & B_x & 0 \end{bmatrix}$

See also

• Einstein notation
• Metric tensor
• Musical isomorphism
• Bilinear products and dual spaces