8.6 Tensor Free Index Notation Role

Tensor Free Index Notation Role is the function performed by an index that appears exactly once in a tensor expression, is not repeated within that expression, and therefore is not summed over under the Einstein summation convention. A free index remains present on both sides of an equation involving tensors, labels an unspecified but fixed slot of the resulting tensor, and determines the rank and transformation behavior of the expression as a whole.

Defining Property of a Free Index

Single Occurrence

A free index is characterized by appearing only once within a given term of an expression, in contrast to a dummy or repeated index, which appears exactly twice, once as an upper index and once as a lower index, and is summed over. In the expression below, (i) is free while (j) is a dummy index subject to summation.

Persistence Across an Equation

Because a free index is not summed away, it must appear, with the same name, the same variance (upper or lower), and the same position, on every additive term of a valid tensor equation. This persistence is what allows the free index to identify which component of the resulting tensor a given equation describes.

Role in Determining Expression Rank

Counting Free Indices

The number of distinct free indices in a tensor expression equals the rank of the object the expression produces. An expression with one free index produces a vector-like object with one component per value of that index; an expression with two free indices produces a matrix-like object.

Here (j) is a dummy index that disappears upon summation, while (i) and (k) are free indices, so the resulting object (W) carries exactly two free slots and is understood as a rank-two tensor.

Free Indices as Placeholders

A free index acts as a placeholder ranging implicitly over all admissible values in the working dimension, so a single symbolic equation with a free index compactly represents an entire family of component equations, one for each value the free index may take.

Consistency Requirements Imposed by Free Indices

Matching Across Terms

A tensor equation is only well formed if every term carries the identical set of free indices, in the same variance. An expression that sums a term with a free lower index against a term with a free upper index of the same name is not a legitimate tensor equation, since the two terms would not transform identically under a change of basis.

Renaming Freedom

A free index may be renamed consistently throughout an entire expression without altering its meaning, provided the new name does not collide with any other index already present in that expression. This freedom distinguishes a free index from a dummy index only in that a free index's name must remain synchronized across every term, whereas a dummy index's name is local to the single term in which it is summed.

Distinguishing Free Indices from Dummy Indices

The clearest operational test for the role of an index is a direct occurrence count within a single term: an index occurring once is free and survives into the result, while an index occurring twice, split between an upper and a lower position, is a dummy index and vanishes through summation. Correct identification of which indices are free is a prerequisite for correctly interpreting the rank, symmetry, and transformation law of any tensor expression.