Associated Tensors

Relevant parts to questions...

• Identify the dummy index on the RHS - that’s what gets replaced.
• Use $g_{pq}$ to lower, $g^{pq}$ to raise.
• For each index you’re moving, contract with one factor of the Metric Tensor.

Two tensors are Associated Tensors if one is obtained from the other by raising or lowering its indices with the Metric Tensor. Every tensor in a generalised coordinate system has a whole family of associated tensors - covariant, contravariant, and mixed - and they are all considered the same tensor expressed in different components.

The Raising/Lowering Recipe

To raise or lower an index:

1. Identify the repeated (dummy) index on the right-hand side.
2. Contract with $g_{pq}$ (to lower) or $g^{pq}$ (to raise): the metric replaces that dummy index with the new free index.
3. The position of the indices in the metric tells you whether the new index ends up raised or lowered.

First-Rank

$$\boxed{A_p=g_{pq}{A}^q,A^p=g^{pq}{A}_q}$$

The dummy $q$ is absorbed by the metric and replaced by the free index $p$, either raised or lowered according to the position of the $p$ in the metric.

Rank 2

All four flavours of components of a second-rank tensor $A_{ik},A^{ik},A_i^k,A_k^i$ are associated:

$$\begin{array}{rl}{A}_{ik}& =g_{i\ell }{g}_{km}{A}^{\ell m}=g_{k\ell }{A}_i^{\ell }=g_{i\ell }{A}_k^{\ell }\\ A^{ik}& =g^{i\ell }{g}^{km}{A}_{\ell m}=g^{i\ell }{A}_{\ell }^k=g^{k\ell }{A}_{\ell }^i\\ A_i^k& =g^{k\ell }{A}_{i\ell }=g_{i\ell }{A}^{\ell k}\\ A_k^i& =g^{i\ell }{A}_{\ell k}=g_{k\ell }{A}^{i\ell }\end{array}$$

Once you know any one flavour and the metric, you know all four.

Higher Ranks

The pattern is entirely mechanical: one factor of $g$ per index being raised/lowered. For example, raising both indices of $A_{rs}$ gives $A^{pq}=g^{rp}{g}^{sq}{A}_{rs}$, while lowering the second contravariant index of $A_s^{pq}$ gives $A_{rs}^p=g_{rq}{A}_s^{pq}$.

Properties

• Same object, different components::moving an index does not change the underlying tensor, just the basis it is expressed against.
• Cartesian collapse::since $g_{ij}={\delta }_{ij}$ in Cartesian coordinates, all flavours coincide: $A_{ij}=A^{ij}=A_i^j=A_j^i$.
• Symmetry can break under moves::see Symmetry and Antisymmetry - raising an index of a symmetric covariant tensor does not necessarily produce a symmetric contravariant tensor.

Applications

1. Building contravariant components from covariant ones (and vice versa), without recomputing from scratch - just apply the metric.
2. Simplifying contractions::pair a raised index with a lowered one so you can contract cleanly (e.g., $A_i{B}^i$).
3. Tensor-algebra shortcuts::many identities are easier to prove in one flavour and transported to the others by raising/lowering.

Raising the first index of $A_{rq}$

$A_{rq}$ has two covariant indices. To raise the first, multiply by $g^{rp}$ (dummy index $r$, new free index $p$, raised):

$A_q^p=g^{rp}{A}_{rq}$. ✓ The dummy $r$ is gone; $p$ is now the first index and is contravariant.

Pick the dummy; metric raises or lowers it.