13.6.3 Tensor Full Contraction Index Exhaustion

Tensor Full Contraction Index Exhaustion is the structural condition under which a full contraction of a tensor consumes every one of its indices in matched covariant–contravariant pairs, leaving no free index behind and producing a scalar as the final result. It describes the exhaustive pairing process that must occur across all upper and lower indices of a tensor, or across the combined index sets of several tensors joined by contraction, so that the operation terminates in a rank-zero object rather than a tensor of reduced but nonzero rank.

Conceptual Basis

Contraction as Index Pairing

A tensor contraction pairs one contravariant (upper) index with one covariant (lower) index and applies the Einstein summation convention over that shared index, reducing the total rank of the tensor by two for each pairing performed.

Exhaustion as a Terminal State

Index exhaustion occurs specifically when the number of contraction operations performed equals half the total number of indices present, so that every index has been assigned a summation partner and none remain to characterize components along any remaining direction.

Distinction from Partial Contraction

A partial contraction pairs only some of the available indices, leaving a tensor of lower rank that still transforms under coordinate changes. Full contraction index exhaustion is reached only when this reduction is carried through completely, yielding an invariant scalar with no transformation dependence on basis choice.

Formal Description

Rank and Index Counting

For a tensor of type $$(p,q)$$ with $$p$$ contravariant indices and $$q$$ covariant indices, full contraction index exhaustion is only possible when $$p=q$$, since each contraction step removes exactly one upper and one lower index simultaneously.

The Exhaustion Condition

The condition is satisfied when the number of performed contractions $$k$$ equals:

so that the resulting rank is:

Example: Exhaustion in a Mixed Tensor

For a tensor $$T_j^i$$, full contraction index exhaustion is expressed as:

where the repeated index $$i$$ is summed over its full range, consuming both the upper and lower index and leaving the trace, a scalar quantity.

Conditions for Exhaustion

Index Availability

Every index scheduled for contraction must have a valid partner of opposite variance available within the same tensor or across the tensors being combined. If an index lacks a suitable partner, exhaustion cannot occur and the operation instead produces a tensor with residual free indices.

Dimensional Compatibility

The vector spaces associated with each paired index must share the same dimension, since summation over an index requires that both the upper and lower slots range over an identical index set.

Sequential Independence

Because contraction is commutative and associative with respect to the order in which index pairs are summed, exhaustion depends only on the total count of matched pairs, not on the sequence in which they are contracted.

Consequences of Exhaustion

Scalar Invariance

Once index exhaustion is reached, the resulting scalar is invariant under changes of basis, since no free index remains to carry transformation coefficients from a change-of-coordinates matrix.

Loss of Directional Information

All directional or component-wise information encoded by the original tensor is collapsed into a single numerical value, meaning the exhausted result cannot be inverted to recover the original tensor without additional data.

Role in Physical and Geometric Quantities

Fully exhausted contractions correspond to familiar invariant quantities such as the trace of a matrix, the squared norm of a vector obtained by contracting a vector with itself through a metric tensor, and scalar curvature obtained by successive contractions of the Riemann curvature tensor.

Non-Exhaustive Cases

Odd Total Index Count

A tensor whose total index count is odd can never reach full exhaustion, since contraction removes indices in pairs and an unpaired index must always remain.

Unequal Upper and Lower Counts

When $$p\ne q$$, exhaustion is bounded by the smaller of the two counts, and the surplus indices of the larger variance type remain free regardless of how many contractions are performed.

Mismatched Index Ranges

If paired indices range over spaces of different dimension, the summation convention cannot be applied consistently, and the contraction is undefined rather than merely incomplete.