8.22.2 Tensor Indexed Equation Dummy Index Set

Tensor Indexed Equation Dummy Index Set is the collection of index letters within a given term or equation that occur exactly twice, once as an upper index and once as a lower index, and are therefore summed away by the Einstein convention rather than surviving as labels of the equation's result — the complement, within the full set of index letters used, of the equation's free index set. It is the bookkeeping device that tracks which letters have been "used up" internally by contraction and are consequently unavailable, and irrelevant, to how the equation's overall result is labeled or compared across an equals sign.

Defining the Dummy Index Set

The Repetition-and-Opposite-Variance Criterion

Within a single term, an index letter belongs to the dummy index set of that term precisely when it appears exactly twice, once upper and once lower. In the term g_{ij}A^i_{\ k}B^{jk}, the dummy index set is {i, j, k}, since each of these three letters appears exactly twice with opposite variance; no letter in this term is left over as a free index, so the free index set of the term is empty while its dummy index set accounts for every letter present.

Dummy Index Sets Are Local to Each Term

Because the summation convention applies term by term, the dummy index set is computed separately for each additive term in a larger sum rather than for the equation as a whole; one term's dummy index set may use entirely different letters, or a different number of contracted pairs, from another term added alongside it, so long as each term's own free index set matches the others'.

Why Dummy Index Set Membership Does Not Persist

Dummy Indices Vanish From the Result

A letter belonging to the dummy index set of a term contributes to that term's numerical value through the summation it performs, but the letter itself does not appear anywhere in the resulting quantity once the term is evaluated; unlike a free index, which labels a specific component of the output, a dummy index is entirely internal to the computation of a single number (or, together with any surviving free indices, a single component).

Renaming Members of the Dummy Index Set Is Always Permitted

Since a dummy index's specific letter carries no information beyond marking which occurrences are paired for summation, any member of a term's dummy index set may be renamed to any other unused letter throughout that term without altering the term's value; this is precisely what licenses the standard technique of relabeling a dummy index to avoid a letter collision when combining two separately derived expressions into one.

Disjointness From the Free Index Set

No Letter Belongs to Both Sets Within a Term

Within any single, correctly formed term, the dummy index set and the free index set are disjoint: a letter occurring exactly once is free, a letter occurring exactly twice with opposite variance is dummy, and no letter can satisfy both conditions simultaneously in a well-formed term. A letter occurring three or more times, or twice with the same variance, belongs to neither set as conventionally defined and signals a notational error rather than a valid dummy or free index.

Full Accounting of a Term's Indices

Every index letter appearing in a well-formed term belongs to exactly one of the two sets — dummy or free — so the two sets together account completely for all index activity within the term, and computing the term's free index set (needed to check the equation's overall balance) is most directly done by first identifying the dummy index set and removing those letters from the full list of letters used.

Diagram of Dummy Versus Free Membership Within a Term

Practical Consequences for Working With Equations

Avoiding Accidental Collisions

When combining two expressions, each with its own dummy index set, into a single larger term or equation, care must be taken that no letter belonging to one expression's dummy index set coincides with a letter belonging to the other's dummy index set (or with any free index of either), since an accidental shared letter would create an unintended additional contraction; renaming members of the dummy index set as needed before combination is the standard remedy.

Counting Contractions From the Dummy Index Set

The number of letters in a term's dummy index set equals exactly the number of independent contracted index pairs within that term, which in turn determines both the order reduction the term has undergone relative to its uncontracted factors and the number of nested summations required to expand the term fully into explicit numerical products, tying the dummy index set directly to both the term's tensorial order and its computational cost.