8.15.3 Tensor Renaming Free Index Protection

Tensor Renaming Free Index Protection is the rule that a free index appearing in a tensor expression must never be altered by a renaming operation targeting a different (dummy) index, and, conversely, that a free index itself is not eligible for the kind of arbitrary relabeling permitted for dummy indices unless it is renamed consistently across every single term of the entire equation in which it appears. Free index protection exists to preserve the identity of the retained, unsummed slot that a free index represents, ensuring that operations intended to simplify or recombine dummy indices never accidentally interfere with the structural backbone of the equation.

Why Free Indices Require Protection

Free Indices Denote Retained Output Slots

A free index labels a component of the tensor that survives into the result of an expression, rather than being summed away; in $A^{i} = B^{i}{}_{j}C^{j}$, the index $i$ is free and marks which component of the resulting rank-1 tensor each side of the equation refers to. Because this labeling must correspond exactly between the left- and right-hand sides of an equation, a free index cannot be treated as an interchangeable placeholder the way a dummy index can.

Consequences of Failing to Protect a Free Index

If a renaming operation aimed at simplifying a dummy index were to inadvertently alter a free index instead, the resulting expression would refer to a different component than the one originally intended, silently invalidating the equation. Given

renaming the free index $i$ on the left alone, without making the identical change to every occurrence of $i$ as a free index on the right, breaks the correspondence between the two sides and produces a false statement.

Distinguishing Protected Renaming From Prohibited Renaming

Dummy Index Renaming Leaves Free Indices Untouched

A correctly performed dummy index renaming operation is defined so that it acts exclusively on the repeated pair it targets, leaving every free index in the same term completely unchanged. Renaming the dummy index $j$ in

to $k$ produces

with the free index $i$ preserved in its original form on both sides, exactly as free index protection requires.

Global Relabeling of a Free Index Is a Separate, Stricter Operation

A free index can be relabeled, but only as a wholesale substitution applied identically to every occurrence of that index as a free index throughout the entire equation — left-hand side, right-hand side, and any other terms present — never as a local change confined to a single term. This is fundamentally different from dummy renaming, which is always local to a single term's scope; relabeling a free index is, in effect, a renaming of the entire equation's variable of reference, not an operation internal to any one expression.

Protection During Collision Avoidance

Free Indices Are Off-Limits as Renaming Targets

When selecting a new letter to rename a dummy index, free index protection requires that the letter chosen never coincide with a free index already present in the same scope, since doing so would conflate the dummy index with the protected free index, corrupting both. This requirement is the mirror image of ordinary collision avoidance between two dummy indices: it specifically singles out free indices as symbols that must never be treated as available for reuse during a renaming operation targeting something else.

Free Indices Are Also Protected From Each Other

Within a single term, distinct free indices must remain distinct from one another; a renaming operation must never cause two previously separate free indices to become identical, since this would eliminate a genuine degree of freedom from the equation and change how many independent scalar equations it represents. Protecting each free index's distinctness from every other free index is as essential as protecting it from dummy indices.

Verifying Free Index Protection in Practice

Index Balance as a Diagnostic

A tensor equation in which free index protection has been correctly maintained will exhibit index balance: the same set of free indices, in the same positions, appears on every term of the equation, unchanged by any renaming performed on the dummy indices within those terms. Checking index balance after a renaming operation is therefore a direct and reliable way to confirm that free index protection was respected.

Isolating Which Indices Are Free Before Renaming

Because free index protection depends entirely on correctly identifying which indices in an expression are free before any renaming is attempted, a necessary preliminary step to any dummy index renaming is to enumerate the free indices of the term in question, so that the pool of letters available for renaming a dummy index can be restricted to exclude them.

Role Within the Index Renaming Operation

Free index protection is the condition that keeps the index renaming operation confined to the parts of an expression where relabeling is actually harmless. Together with scope preservation and collision avoidance, it completes the set of safeguards that make dummy index renaming a value-preserving transformation: scope preservation bounds where a renaming acts, collision avoidance prevents it from colliding with other indices, and free index protection specifically shields the retained, meaningful slots of an expression from ever being altered by a renaming meant only for its summed-away dummy indices.