8.17 Tensor Index Matching Rule

Tensor Index Matching Rule is the requirement that every free index appearing in a tensor equation must appear with the identical letter, in the identical position (upper or lower), on every single term of that equation, so that both sides — and every additive term within each side — refer to precisely the same set of retained, unsummed components. It is the master well-formedness condition governing tensor equations as wholes, distinguishing a legitimate tensor identity from a superficially similar but meaningless juxtaposition of expressions.

Statement of the Rule

Same Letters, Same Positions, Every Term

For a tensor equation to be valid, the collection of free indices — together with their positions — must be identical across every term on both sides of the equality. An equation such as

satisfies the matching rule: the free index $i$ appears in the upper position in every term ($A^{i}$, the contracted term $B^{i}{}_{j}C^{j}$, and $D^{i}$), while $j$ appears only as a dummy pair within the second term and is absent from the others, as required.

What Counts as a Violation

An equation such as $A^{i} = B_{i}$ violates the matching rule because the free index appears in the upper position on one side and the lower position on the other; despite using the same letter, the two sides carry different transformation behavior and cannot represent the same equality in every coordinate system. Likewise, $A^{i} = B^{j}$ violates the rule outright, since the two sides do not even share the same free-index letter, leaving no correspondence between the components on either side.

Why the Rule Is Necessary

Preserving the Same Transformation Law on Both Sides

Because the position of an index signals its transformation law, requiring matching positions across an equation ensures that both sides transform identically under a change of coordinates. If $A^{i} = D^{i}$ holds in one coordinate system, the matching rule guarantees that the corresponding equation $A^{i'} = D^{i'}$, obtained by applying the same Jacobian factor to both sides, holds equally in any other coordinate system — a guarantee that fails immediately if the two sides carried indices in mismatched positions, since they would then pick up different Jacobian factors under the same coordinate change.

Preserving the Number of Represented Scalar Equations

Because a free index stands for an entire family of scalar equations, one for each value in its range, the matching rule also ensures that both sides of a tensor equation represent the same number of underlying scalar equations. An equation with a mismatched free-index count between its two sides, such as one side carrying a single free index and the other carrying two, cannot correspond term by term to any consistent set of scalar equations at all.

Matching Requirements Across Multiple Terms

Consistency Within a Sum

When an equation's side consists of several terms added together, the matching rule applies to every term individually: each addend must carry exactly the same set of free indices, in the same positions, as every other addend in that sum. A sum such as $A^{i}{}{j} + B{j}{}^{i}$ fails the matching rule, since although both terms use the letters $i$ and $j$, they place them in reversed positions relative to one another, meaning the two addends cannot be added together as if they were of the same type.

Dummy Indices Are Exempt From the Rule

The matching rule applies exclusively to free indices; dummy indices, being consumed entirely within their own term through implicit summation, are not required to match in letter across different terms of an equation, and indeed are conventionally renamed to avoid collisions precisely because their specific letter carries no significance beyond their own local scope.

Using the Rule as a Diagnostic Tool

A Fast Check for Malformed Equations

Because the matching rule can be verified by inspection — comparing the free-index letters and positions present in each term without needing to know what the tensors actually represent numerically — it functions as an immediate diagnostic for catching errors in a derivation. A derivation step that introduces or drops a free index, or that flips a free index's position without justification, can be flagged as erroneous purely by checking index matching, well before any numerical or geometric argument is needed.

Interaction With Metric Operations

Because raising or lowering an index changes its position, an equation that appears to violate the matching rule may sometimes be correctable by explicitly inserting a factor of the metric tensor to raise or lower the mismatched index into agreement, rather than by treating the mismatch as an unrecoverable error; the matching rule signals exactly where such a correction is needed, even if it does not by itself supply the correction.

Role Within Tensor Index Notation

The index matching rule is the equation-level counterpart to the term-level rules governing individual repeated indices: where the repeated index limit and collision-avoidance rules govern what is permissible within a single term, the matching rule governs what is permissible across the terms of an entire equation. Together, these rules constitute the complete grammar by which a string of symbols in tensor index notation is judged to be, or not to be, a coherent and coordinate-consistent mathematical statement.