7.17.3 Tensor Component Repeated Index Vanishing

Tensor Component Repeated Index Vanishing is the fact that any component of a tensor exhibiting the Tensor Component Antisymmetry Pattern in a designated pair of indices must equal zero whenever the two indices of that pair take the same value, since the Tensor Component Sign Change Rule forces such a component to equal its own negative.

Derivation of the Vanishing

Setting the Two Indices Equal

Beginning from the Tensor Component Sign Change Rule for a tensor antisymmetric in the pair formed by i and j:

setting j equal to i produces:

Concluding the Value Must Be Zero

Adding the component T subscript i i to both sides of this equation gives:

so that dividing by two leaves T subscript i i equal to zero. This holds for every value of i individually, meaning that the vanishing applies at each repeated-index position separately rather than only on average across all of them.

Illustration

Every shaded position along the diagonal, where the row index and column index coincide, is forced to zero. Only positions strictly off the diagonal remain free to carry a nonzero value.

Scope of the Vanishing

Applies Only to the Antisymmetric Pair

The vanishing described here is restricted to the specific pair of indices identified as following the Tensor Component Antisymmetry Pattern. In a tensor of rank higher than two with several indices, setting two indices from outside the antisymmetric pair equal to one another produces no such forced vanishing, since no sign-change relationship links those indices.

Applies Regardless of the Values Taken by Other Indices

For a tensor of rank higher than two, the vanishing holds for every fixed choice of the remaining indices. If the antisymmetric pair is formed by the first two indices of a rank-three tensor, then T subscript i i k is zero for every value of i and for every value of k independently.

Consequences of the Vanishing

Reduction to Off-Diagonal Positions

The Tensor Component Repeated Index Vanishing is the specific mechanism responsible for excluding the diagonal from the count of independent components used in the Tensor Component Antisymmetric Reduction. Without this forced vanishing, the diagonal positions would need to be counted as additional independent values.

A Necessary Condition for Recognizing Antisymmetry

Because the vanishing follows directly and unavoidably from antisymmetric behavior, observing a nonzero component at a repeated-index position in a candidate pair is sufficient to conclude that the pair does not follow the Tensor Component Antisymmetry Pattern. The vanishing therefore also functions as a diagnostic test for antisymmetry.

Persistence Under Coordinate Change

Since the Tensor Component Repeated Index Vanishing follows purely from the Tensor Component Sign Change Rule, and that rule is preserved under every admissible coordinate transformation, the vanishing of repeated-index components likewise holds in every coordinate system, not only the one in which it was first derived. A component that vanishes at a repeated-index position in one frame vanishes at the corresponding position in every other frame as well.

Relationship to Other Tensor Concepts

Tensor Component Repeated Index Vanishing is a direct consequence of the Tensor Component Sign Change Rule and forms part of the broader Tensor Component Antisymmetry Pattern. It plays a central role in establishing the reduced count of independent components described by the Tensor Component Antisymmetric Reduction, distinguishing this reduction from the corresponding reduction under symmetric behavior, where no such forced vanishing occurs.