8.11.4 Tensor Explicit Contraction Expansion

Tensor Explicit Contraction Expansion is a technique in algebra that expands tensor contractions to clarify structural relationships and simplify computations.

Tensor Explicit Contraction Expansion is the process of taking a tensor contraction written compactly, whether under the Einstein convention or with an explicit sigma symbol, and writing it out fully as a finite sum of individually computed products, one for each value the contracted index takes across its range, exposing every term that the compact notation had folded together.

The Expansion Process

Substituting Each Index Value in Turn

Explicit contraction expansion proceeds by substituting, one at a time, each admissible value of the contracted index into the compact expression, producing a separate explicit term for every such value, and then joining all of these terms together with addition.

A_{i} B^{i} = A_{1} B^{1} + A_{2} B^{2} + A_{3} B^{3}

Preserving Any Surviving Free Indices

When the original contraction also carries a free index, that free index is preserved, unexpanded, throughout every term of the expansion, since it is only the contracted index that is stepped through its range and substituted with concrete values.

R_{k} = A_{1 k} B^{1} + A_{2 k} B^{2} + A_{3 k} B^{3}

Expansion of Multiple Contracted Indices

Nested Expansion

When more than one index is contracted within a single expression, expansion must be carried out for each contracted index independently, producing a number of explicit terms equal to the product of the ranges of every contracted index involved.

s = A_{i j} B^{i j}

Expanding this double contraction in a three-dimensional space produces nine individual product terms, one for each combination of values that (i) and (j) may jointly take, all added together into the single scalar (s).

Purposes of Expansion

Verifying a Compact Identity

Explicit expansion is often used as a verification technique: by writing out every term of both sides of a proposed tensor identity for a small, concrete dimension, one can confirm numerically that the compact notation on each side genuinely represents the same underlying computation.

Preparing for Direct Numerical Computation

Expansion is also the necessary final step before any numerical evaluation, since a computer or a hand calculation ultimately must carry out each individual product and addition that the compact contraction notation only implies.

Reversing the Process

Recompressing an Expansion into Compact Notation

The expansion process can, in principle, be reversed: a fully written-out sum of products sharing a consistent structural pattern across every term can often be recognized and rewritten back into a compact contraction, provided the pattern of repeated indices and free indices can be identified across all the explicit terms.

Practical Illustration

Carrying out an explicit contraction expansion, even as a mental exercise, is one of the most reliable ways to confirm that a compact tensor expression has been correctly understood, since the expansion exposes every individual term the compact notation was silently representing all along.