8.17.1 Tensor Equation Free Index Matching

Tensor Equation Free Index Matching is the finest-grained level of the index matching rule, requiring that the exact same letters, not merely the same count or type, be used for every free index across every term of a complete tensor equation, so that a specific free index appearing as $i$ in one term is denoted by that same letter $i$, in the same position, in every other term of the equation rather than by some other letter that merely happens to have a compatible type. Where type matching checks that the number of upper and lower free indices agrees between terms, equation-level free index matching checks the stricter condition that the individual named slots line up one-to-one across the whole equation.

The Full Requirement Stated

Letter-for-Letter Correspondence

Consider a candidate tensor equation with multiple terms on one side:

Free index matching requires that the letter $i$, occupying the upper position, be present in every term: on $D^{i}$, on $A^{i}$, and on the contracted term $B^{i}{}_{j}C^{j}$ (where $i$ survives as the term's free index after $j$ is contracted away). The letter $j$, by contrast, appears only as a dummy pair within the third term and is correctly absent, as a free index, from the other two.

A Failing Example Despite Correct Type

An expression such as $D^{i} = A^{k} + B^{i}{}_{j}C^{j}$ has every term individually of type $(1,0)$, satisfying type matching, yet fails full free index matching because the second term uses $k$ where the first and third terms use $i$; nothing in the equation establishes any relationship between $k$ and $i$, so the equation as written does not assert a coherent identity between three quantities that are supposed to share the same free slot.

Why Letter Identity, Not Just Type, Is Required

Free Indices Name a Specific Retained Slot

A free index does more than mark that a slot of a certain transformation type exists in a term; it identifies which particular slot of the resulting tensor that term's contribution belongs to. Two terms sharing the same type but using different letters for their free indices have not actually been shown to contribute to the same slot of the result, even though both terms are individually well-typed tensors of the correct kind.

Consequence for the Number of Represented Scalar Equations

Because each value of a free index corresponds to a distinct scalar equation, requiring letter-for-letter matching across a tensor equation guarantees that, once every free index is assigned a specific value, every term of the equation is evaluated at that same specific value — not at some independently and arbitrarily chosen value for each term. Free index matching is what allows the compact tensor equation to be legitimately unpacked into $n$ separate, internally consistent scalar equations, one for each value the shared free index takes.

Matching Across Multiple Free Indices

Independent Matching for Each Distinct Free Letter

When an equation carries more than one free index, each distinct free letter must independently satisfy the matching requirement across every term. An equation of type $(1,1)$ in both free indices, such as $T^{i}{}{j} = A^{i}B{j} + C^{i}{}_{j}$, requires that $i$ match across all three terms and, separately, that $j$ match across all three terms; a failure in either letter independently invalidates the equation, even if the other letter matches correctly throughout.

Order Sensitivity for Multiple Free Indices

When multiple free indices are present, their relative left-to-right ordering within the staggered notation of a mixed tensor must also agree across terms whenever that ordering is significant; $T^{i}{}{j}{}^{k}$ matching against $A^{i}{}{j}{}^{k}$ term by term is a stronger and more specific requirement than merely confirming that $i$, $j$, and $k$ each individually appear somewhere in the correct position across every term.

Detecting and Correcting Mismatches

Symbol-by-Symbol Comparison Across Terms

The practical method for checking equation-level free index matching is to list, for every term in the equation, the free indices present along with their positions, and to compare these lists across all terms; any letter present in one term's list but absent, or present under a different letter, in another term's list identifies a specific point of failure.

Correcting a Mismatch Through Careful Rederivation, Not Renaming

Unlike a collision involving dummy indices, a mismatch in free index matching across an equation cannot be fixed by an arbitrary renaming operation, since free indices are not bound variables free to be relabeled at will; a genuine mismatch indicates either an error in how one of the terms was derived or a genuine incompatibility between the terms being equated, and resolving it requires revisiting the derivation of the mismatched term rather than merely substituting a different letter.

Role Within the Index Matching Rule

Equation-level free index matching represents the complete and most exacting form of the broader index matching rule, incorporating both the coarse check of type matching and the finer check of position agreement into a single requirement applied across every term of an entire equation rather than to any one term in isolation. Satisfying this full requirement is both necessary and sufficient for a tensor equation to correspond to a coherent, coordinate-consistent set of scalar equations, making it the ultimate test of whether a proposed tensor identity is genuinely well-formed.