Raising and lowering indices

In mathematics and mathematical physics, raising and lowering indices are operations on tensors which change their type. Raising and lowering indices are a form of index manipulation in tensor expressions.

Tensor type

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 to change a type (a, b) tensor to a (a + 1, b − 1) tensor (raise index) or to a (a − 1, b + 1) tensor (lower index), where the notation (a, b) has been used to denote the tensor order a + b with a upper indices and b 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 indices (applying Einstein notation). See examples below.

Vectors (order-1 tensors)

Multiplying by the contravariant metric tensor $g^{ij}$ and contracting produces another tensor with an upper index:

$g^{{ij}}A_{j}=B^{i},$

The same base symbol is typically used to denote this new tensor, and repositioning the index is typically understood in this context to refer this new tensor, and is called raising the index, which would be written

$g^{{ij}}A_{j}=A^{i}.$

Similarly, multiplying by the covariant metric tensor and contracting lowers an index (with the same understanding about the reuse of the base symbol):

$g_{{ij}}A^{j}=A_{i}.$

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.

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:

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

where δⁱ_k 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.

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

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

Again, while a helpful guide, this is only mnemonical and not a property of tensors since the indices do not cancel like in equations, it is only a concept of the notation. The results are continued below, for higher order tensors (i.e. more indices).

When raising indices of quantities in spacetime, it helps to decompose summations into "timelike components" (where indices are zero) and "spacelike components" (where indices are 1, 2, 3, represented conventionally by Latin indices).

Example from Minkowski spacetime

The covariant 4-position is given by

${\displaystyle X_{\mu }=(-ct,x,y,z)}$

in components:

${\displaystyle X_{0}=-ct,\quad X_{j}=x_{j}}$

(where x_j are the usual Cartesian coordinates) and the Minkowski metric tensor with signature (-+++) is defined as

${\displaystyle \eta _{\mu \nu }=\eta ^{\mu \nu }={\begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}}$

in components:

${\displaystyle \eta _{00}=-1,\quad \eta _{i0}=\eta _{0i}=0,\quad \eta _{ij}=\delta _{ij}\,.}$

To raise the index, multiply by the tensor and contract:

$X^{\lambda }=\eta ^{{\lambda \mu }}X_{\mu }=\eta ^{{\lambda 0}}X_{0}+\eta ^{{\lambda i}}X_{i}$

then for λ = 0:

${\displaystyle X^{0}=\eta ^{00}X_{0}+\eta ^{0i}X_{i}=-X_{0}}$

and for λ = j = 1, 2, 3:

${\displaystyle X^{j}=\eta ^{j0}X_{0}+\eta ^{ji}X_{i}=\delta ^{ji}X_{i}=X_{j}}$

So the index-raised contravariant 4-position is:

$X^{\mu }=(ct,x,y,z)\,.$

Tensors (higher order)

Order-2

For an order-2 tensor,^[1] 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:

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

Example from classical electromagnetism and special relativity

The contravariant electromagnetic tensor in the (+−−−) signature is given by^[2]

$F^{{\alpha \beta }}={\begin{pmatrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\E_{x}/c&0&-B_{z}&B_{y}\\E_{y}/c&B_{z}&0&-B_{x}\\E_{z}/c&-B_{y}&B_{x}&0\end{pmatrix}}$

in components:

$F^{{0i}}=-F^{{i0}}=-E^{i}/c,\quad F^{{ij}}=-\varepsilon ^{{ijk}}B_{k}$

To obtain the covariant tensor F_αβ, multiply by the metric tensor and contract:

${\begin{aligned}F_{{\alpha \beta }}&=\eta _{{\alpha \gamma }}\eta _{{\beta \delta }}F^{{\gamma \delta }}\\&=\eta _{{\alpha 0}}\eta _{{\beta 0}}F^{{00}}+\eta _{{\alpha i}}\eta _{{\beta 0}}F^{{i0}}+\eta _{{\alpha 0}}\eta _{{\beta i}}F^{{0i}}+\eta _{{\alpha i}}\eta _{{\beta j}}F^{{ij}}\end{aligned}}\,$

and since F⁰⁰ = 0 and F⁰ⁱ = − Fⁱ⁰, this reduces to

$F_{{\alpha \beta }}=(\eta _{{\alpha i}}\eta _{{\beta 0}}-\eta _{{\alpha 0}}\eta _{{\beta i}})F^{{i0}}+\eta _{{\alpha i}}\eta _{{\beta j}}F^{{ij}}\,$

Now for α = 0, β = k = 1, 2, 3:

${\begin{aligned}F_{{0k}}&=(\eta _{{0i}}\eta _{{k0}}-\eta _{{00}}\eta _{{ki}})F^{{i0}}+\eta _{{0i}}\eta _{{kj}}F^{{ij}}\\&=(0-(-\delta _{{ki}}))F^{{i0}}+0\\&=F^{{k0}}=-F^{{0k}}\\\end{aligned}}$

and by antisymmetry, for α = k = 1, 2, 3, β = 0:

$F_{{0k}}=-F_{{k0}}$

then finally for α = k = 1, 2, 3, β = ℓ = 1, 2, 3;

${\begin{aligned}F_{{k\ell }}&=(\eta _{{ki}}\eta _{{\ell 0}}-\eta _{{k0}}\eta _{{\ell i}})F^{{i0}}+\eta _{{ki}}\eta _{{\ell j}}F^{{ij}}\\&=0+\delta _{{ki}}\delta _{{\ell j}}F^{{ij}}\\&=F^{{k\ell }}\\\end{aligned}}$

The (covariant) lower indexed tensor is then:

$F_{{\alpha \beta }}={\begin{pmatrix}0&E_{x}/c&E_{y}/c&E_{z}/c\\-E_{x}/c&0&-B_{z}&B_{y}\\-E_{y}/c&B_{z}&0&-B_{x}\\-E_{z}/c&-B_{y}&B_{x}&0\end{pmatrix}}$

Order-n

When a vector space is equipped with an inner product (or metric as it is often called in this context), there exist operations that convert a contravariant (upper) index into a covariant (lower) index and vice versa. A metric itself is a (symmetric) (0,2)-tensor, it is thus possible to contract an upper index of a tensor with one of lower indices of the metric. This produces a new tensor with the same index structure as the previous, but with lower index in the position of the contracted upper index. This operation is quite graphically known as lowering an index. Conversely, a metric has an inverse which is a (2,0)-tensor. This inverse metric can be contracted with a lower index to produce an upper index. This operation is called raising an index.

For a tensor of order-n, indices are raised by:^[1]

$g^{{i_{1}j_{1}}}g^{{i_{2}j_{2}}}\cdots g^{{i_{n}j_{n}}}A_{{i_{1}i_{2}\cdots i_{n}}}=A^{{j_{1}j_{2}\cdots j_{n}}}$

and lowered by:

$g_{{i_{1}j_{1}}}g_{{i_{2}j_{2}}}\cdots g_{{i_{n}j_{n}}}A^{{i_{1}i_{2}\cdots i_{n}}}=A_{{j_{1}j_{2}\cdots j_{n}}}$

and for a mixed tensor:

$g_{{i_{1}p_{1}}}g_{{i_{2}p_{2}}}\cdots g_{{i_{n}p_{n}}}g^{{j_{1}q_{1}}}g^{{j_{2}q_{2}}}\cdots g^{{j_{m}q_{m}}}A^{{i_{1}i_{2}\cdots i_{n}}}{}_{{j_{1}j_{2}\cdots j_{m}}}=A_{{p_{1}p_{2}\cdots p_{n}}}{}^{{q_{1}q_{2}\cdots q_{m}}}$

See also

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

References

1. Kay, D. C. (1988). Tensor Calculus. Schaum’s Outlines. New York: McGraw Hill. ISBN 0-07-033484-6.
2. NB: Some texts, such as: Griffiths, David J. (1987). Introduction to Elementary Particles. Wiley, John & Sons, Inc. ISBN 0-471-60386-4., 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), there are no factors of c since they are using Gaussian units. Here SI units are used.