8.18.2 Tensor Free Index Balance

Tensor Free Index Balance is the state in which the collection of free indices carried by a tensor expression — considered as the unbound, retained variables of that expression — is exactly conserved wherever the expression is manipulated, extended, or set equal to another, with dummy indices explicitly excluded from this accounting since they are summed away and never contribute to what is retained. It names the specific portion of the broader index balance rule that concerns free indices in particular, isolating them from the separate question of whether dummy indices are properly paired within a term.

Free Indices as the Object Being Balanced

Why Only Free Indices Are Counted

A free index marks a slot in a tensor expression that survives into the final result, functioning much like a free variable in an algebraic formula that has not been assigned a specific value or summed over any range. Dummy indices, by contrast, are consumed entirely by implicit summation and never appear in the final result at all; free index balance concerns itself exclusively with the former, since only the free indices determine what kind of object — of what type $(p,q)$ — an expression ultimately represents.

Free Index Balance as a Precondition for Meaning

Because a free index stands for an entire family of components, one for each value in its range, an expression's free-index inventory is what allows it to be interpreted as denoting a specific tensor at all; an expression whose free indices are not balanced against the other expressions it is combined with cannot be assigned a coherent value, since there is no well-defined correspondence between the components each side is supposed to be comparing.

Verifying Free Index Balance

Isolating the Free Indices of an Expression

The first step in verifying free index balance is to determine, for any given tensor expression, precisely which of its index occurrences are free and which are dummy — a distinction made by counting occurrences of each letter, with a single occurrence marking a free index and a matched upper–lower pair marking a dummy index consumed by summation. Only after this separation is made can the free-index inventory relevant to balance be correctly identified.

Comparing Inventories Across an Operation

Once isolated, an expression's free-index inventory — its set of letters together with their positions — is compared against the corresponding inventory of whatever it is being added to, equated with, or substituted into. A worked example illustrates the comparison directly: given

the free-index inventory on the left is ${i \text{ (lower)}, j \text{ (lower)}, k \text{ (upper)}}$, and both terms on the right carry that identical inventory as well, confirming free index balance across the whole equation.

Free Index Balance Distinguished From Dummy Index Concerns

Dummy Indices Follow Different Rules Entirely

Because dummy indices are governed by scope, collision avoidance, and renaming rather than by balance, an expression can have a perfectly conserved free-index inventory while still containing an internal collision among its dummy indices, or vice versa; the two concerns are checked independently, using entirely different criteria, even though both ultimately serve the goal of a well-formed tensor expression.

An Illustration of the Independence of the Two Concerns

Consider $A^{i}B_{i}C^{j} = A^{i}B_{i}D^{j}$: the free-index inventory here is simply ${j \text{ (upper)}}$ on both sides, since $i$ is dummy and contracted away identically in both terms, and this equation is free-index balanced regardless of what specific letter had been used for the internal contraction on either side, since that letter's identity is irrelevant to the free-index inventory being tracked.

Consequences of a Free Index Imbalance

A Missing or Extra Free Index Changes the Represented Type

If one side of a purported equation retains a free index that the other side has, through an intervening contraction, consumed, the two sides no longer share the same type $(p,q)$, and the imbalance manifests directly as a difference in how many independent scalar equations each side represents — a discrepancy that cannot be resolved by any relabeling, since it reflects an actual difference in the tensorial rank of the two sides.

A Mismatched Position Among Otherwise Balanced Free Indices

Even when the same free letters appear in the same count on both sides, a difference in the position (upper versus lower) assigned to a shared letter still constitutes a free index imbalance in the fuller sense, since it indicates the two sides transform by different laws despite superficially referencing what appears to be the same slot.

Role Within the Index Balance Rule

Free index balance names the aspect of the general index balance rule that applies specifically to an expression's retained, unbound indices, deliberately setting aside the separate machinery governing dummy indices so that each class of index can be verified according to the rules appropriate to it. Together with the corresponding treatment of dummy indices elsewhere in tensor index notation, free index balance completes the picture of what it means for a tensor expression, or an entire tensor equation, to be considered fully well-formed.