Raising and lowering indices

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In mathematics and mathematical physics, given a tensor field 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 or "perform index gymnastics"^[1]: 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 indices, meaning two indices are set equal and then summing over the repeated index j. See examples below.

Contents

1 Raising and lowering indices
- 1.1 Vectors (rank-1 tensors)
- 1.2 Tensors (higher rank)
  - 1.2.1 Rank-2
  - 1.2.2 Rank-n
2 Example from Special relativity
3 References
4 See also

Raising and lowering indices

Vectors (rank-1 tensors)

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:

$g^{ij}g_{jk}=g_{kj}g^{ji}=\delta_k^i$

where $\delta_k^i$ is the Kronecker delta or identity matrix. Since there are different choices of metric with different metric signatures (signs along the diagonal elements, i.e. tensor components with equal indices), the name and signature is usually indicated to prevent confusion. Different authors use different metrics and signatures for different reasons.

NB: The form $g_{ij}$ need not be nonsingular to lower an index, but to get the inverse (and thus raise an index) it must be nonsingular.

Mnemonically (though incorrectly), one could think of indices "cancelling" between a metric and another tensor, and the metric stepping up or down the index. In the above examples, such "cancellations" and "steps" are like

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

The results are continued below, for tensors higher ranks (i.e. more indices).

Tensors (higher rank)

Rank-2

For a rank-2 tensor^[2], twice multiplying by the contravariant metric tensor and contracting in different indices raises each index:

$A^{\alpha\beta}=g^{\alpha\gamma}g^{\beta\delta}A_{\gamma \delta}$

and twice multiplying by the covariant metric tensor and contracting in different indices lowers each index:

$A_{\alpha\beta}=g_{\alpha\gamma}g_{\beta\delta}A^{\gamma \delta}$

Rank-n

For a rank-n tensor, indices are rasied by:^[3]

$g^{i_1j_1}g^{i_2j_2}\cdots g^{i_nj_n}A_{i_1i_2\cdots i_n} = A^{j_1j_2\cdots j_n}$

and lowered by:

$g_{i_1j_1}g_{i_2j_2}\cdots g_{i_nj_n}A^{i_1i_2\cdots i_n} = A_{j_1j_2\cdots j_n}$

and for a mixed tensor:

$g_{i_1p_1}g_{i_2p_2}\cdots g_{i_np_n}g^{j_1q_1}g^{j_2q_2}\cdots g^{j_nq_m}A^{i_1i_2\cdots i_n}{}_{j_1j_2\cdots j_m} = A_{p_1p_2\cdots p_n}{}^{q_1q_2\cdots q_m}$

Example from Special relativity

In a Minkowski space, the metric tensor with signature (+−−−) is defined as

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

The contravariant electromagnetic tensor is given by

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

NB: Some texts, such as Griffiths^[4], 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. In older texts such as Jackson (2nd edition), they leave out the factors of c since they are using Gaussian units. Here SI units are used.

To obtain the covariant tensor $F_{\mu\nu}\,$ , multiply by the metric tensor according to

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

In the following, summation is suppressed (i.e. no summation is done, but the indices are kept equal).

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 (not including the time component, 0),

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

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 gives the lower indexed tensor (covariant):

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

References

^ http://mathworld.wolfram.com/IndexGymnastics.html
^ Tensor Calculus, D.C. Kay, Schaum’s Outlines, McGraw Hill (USA), 1988, ISBN 0-07-033484-6
^ Tensor Calculus, D.C. Kay, Schaum’s Outlines, McGraw Hill (USA), 1988, ISBN 0-07-033484-6
^ Griffiths, David J. (1987). Introduction to Elementary Particles. Wiley, John & Sons, Inc. ISBN 0-471-60386-4.

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