Index Raising and Lowering

Relevant parts to questions...

• $g_{pq}$ lowers, $g^{pq}$ raises.
• One metric factor per index you are moving.
• The metric’s dummy index pairs with the target dummy; its free index replaces it at the new level.
• In Cartesian coordinates nothing happens - $g_{pq}={\delta }_{pq}$.

The Metric Tensor performs two basic operations on a tensor’s indices:

$$\boxed{g_{pq}\text{ lowers an index},g^{pq}\text{ raises an index}.}$$

Each contraction with a metric tensor replaces a dummy index with a new free index at the opposite level (raised ↔ lowered).

The Recipe

For every index you want to raise or lower:

1. Identify the repeated (dummy) index on the right-hand side.
2. Contract with one factor of the metric: $g_{pq}$ to lower, $g^{pq}$ to raise.
3. Replace the dummy with the new free index, at the level dictated by the metric.

Rank 1

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

Rank 2 - raising one index

Given $A_{rq}$, raise the first:

$$A_{\cdot q}^p=g^{rp}{A}_{rq}$$

The dummy $r$ is replaced by the raised $p$. Similarly, $A_p^{\cdot q}=g_{rp}{A}^{rq}$ lowers the first index of $A^{rq}$.

Rank 2 - raising both indices

$$A^{pq}=g^{rp}{g}^{sq}{A}_{rs}$$

One metric per index being moved; dummies $r,s$ become $p,q$.

Higher Ranks

A single expression can move several indices at once - use one factor of $g$ per index. For example:

$$A_{\cdot \cdot n}^{qm\cdot tk}=g^{pk}{g}_{sn}{g}^{rm}{A}_{\cdot r\cdot \cdot p}^{q\cdot st}$$

Here $g^{pk}$ raises $k$, $g^{rm}$ raises $m$, and $g_{sn}$ lowers $n$. The other indices $q,s,t$ stay where they are.

Properties

• Each move is reversible: applying $g_{pq}$ and then $g^{pr}$ returns the original (since $g^{pr}{g}_{pq}={\delta }_q^r$).
• Position matters in the metric: $g_{pq}{A}^q$ gives $A_p$ (with the new index in the first slot of $g$).
• Orthogonal basis shortcut::$g_{ii}=1/g^{ii}$ (no sum), so raising or lowering reduces to division/multiplication by a single scale factor.
• Cartesian collapse::$g_{ij}={\delta }_{ij}$, so raising/lowering is the identity and all flavours coincide.

Applications

1. Producing Associated Tensors in any flavour from a starting one.
2. Making contractions well-defined: to pair indices you need one raised with one lowered (e.g., $A^i{B}_i$).
3. Converting between “physical” and “tensor” components in curvilinear systems - physical components carry scale factors $h_i$ while tensor components are pure numbers.

Lowering a mixed tensor's second index

Given $A_{\cdot \cdot s}^{pq}$, lower $q$:

$A_{\cdot rs}^p=g_{rq}{A}_{\cdot \cdot s}^{pq}$.

Dummy $q$ disappears, new free index $r$ sits in the lowered position. $\boxed{A_{\cdot rs}^p=g_{rq}{A}_{\cdot \cdot s}^{pq}}$ ✓

One metric per index you move.