8.20.5 Tensor Expanded Expression Equivalence

Tensor Expanded Expression Equivalence is the criterion by which two tensor expressions, possibly written with different index letters, different orderings of terms, or different intermediate groupings, are recognized as denoting exactly the same object because their fully expanded forms — every free-index equation written out and every implicit sum written out as an explicit sum of products — agree term for term after accounting for the commutativity and associativity of ordinary addition and multiplication. It is the standard of proof that expansion is used to establish: rather than manipulating compact index expressions and hoping an identity holds, equivalence is settled by expanding both sides down to plain arithmetic and checking they match exactly.

What Must Match After Expansion

Same Free Indices, Same Number of Equations

Two expressions can be equivalent only if, after expansion, they resolve into the same set of free-index equations — the same number of them, indexed the same way, one for each combination of values the shared free indices take. An expression with a different set of free indices than another simply is not a candidate for equivalence, since the two would produce results of different orders or different labelings.

Term-for-Term Agreement of the Expanded Sums

Within each corresponding free-index equation, expanded expression equivalence requires that the two explicit sums of scalar products, once both are fully written out, are equal as sums — not necessarily as literally identical sequences of terms, but as sums whose terms can be matched up (reordered, regrouped, or combined by ordinary algebra) to show the two totals coincide:

illustrating that renaming the dummy index from i to k, and reversing the order of the two factors in each product, does not change the sum, so the two written forms are expanded-expression equivalent even though the compact expressions differ superficially.

Sources of Superficial, Non-Substantive Difference

Dummy Index Renaming

Because a dummy index does not appear in the final result, replacing it consistently with any other unused letter throughout a term leaves the expanded sum completely unchanged; expanded expression equivalence therefore treats TⁱSᵢ and TʲSⱼ as denoting the identical object, since expansion produces literally the same list of terms regardless of which letter was chosen for the summed index.

Reordering of Factors and Terms

Ordinary commutativity of multiplication and addition of scalars means that reordering the factors within a single product term, or reordering the terms of an expanded sum, does not change the total; expanded expression equivalence absorbs all such reordering as immaterial, since after full expansion both sides reduce to sums of the same multiset of numerical or symbolic products.

Different Intermediate Groupings

Two expressions that reach the same fully expanded form by different intermediate routes — for instance, contracting indices in a different order, or introducing a different intermediate named tensor along the way — are equivalent in this sense even though the compact symbolic derivations look different, since equivalence is judged solely by the final expanded content, not by the path used to reach it.

Diagram of Two Routes Converging After Expansion

Distinguishing Genuine From Spurious Equivalence Claims

Equivalence Must Hold for the Stated Range and Type

An equivalence established by expansion is only as general as the index ranges and tensor types used in the check; confirming that two expressions expand identically in three dimensions for a symmetric tensor does not establish equivalence in general dimension or for an unrestricted (non-symmetric) tensor, so expanded expression equivalence checked in a specific low-dimensional or specially structured case is a necessary but not sufficient step toward a fully general proof.

Distinguishing True Identities From Coincidental Matches

Because expansion in a very small dimension, such as n = 1, can make many genuinely different expressions collapse to the same trivial numerical value by coincidence, expanded expression equivalence is most reliably established either by an expansion carried out in a generic symbolic dimension n, or by confirming the match across more than one distinct small dimension, so that an accidental agreement at a single special value of n is not mistaken for a true general identity.

Relationship to Formal Tensor Identities

Expanded expression equivalence is the concrete, computational counterpart of an abstract tensor identity stated without reference to any particular basis or dimension; a formal identity proved abstractly (for example from the definition of the metric or from a symmetry property) implies expanded expression equivalence for every choice of dimension and basis, while a verified expanded expression equivalence in one or several cases provides strong supporting evidence for, but does not by itself constitute, such a fully general abstract identity.