13.21 Tensor Contraction Notation

Tensor Contraction Notation is the collection of symbolic conventions used to write down a tensor contraction in indexed form, specifying how upper and lower indices are placed, how repeated indices signal an implied summation, and how the resulting free indices identify the type and components of the contracted tensor, without requiring an explicit summation symbol to be written.

Definition

A tensor contraction is written by placing the indices of a tensor as superscripts for contravariant components and subscripts for covariant components, and by repeating a single index symbol once in each position to indicate that it is summed over:

denotes the contraction of the first upper index of $$T$$ against its lower index, leaving $$b$$ as the sole free index of the result.

Core Conventions

Einstein Summation Convention

Under the standard convention adopted throughout tensor contraction notation, any index symbol repeated exactly once as an upper index and once as a lower index within a single term is automatically summed over its full range, without an explicit summation sign:

Free Versus Dummy Index Notation

Indices appearing exactly once in an expression are free indices, retained in the notation for the resulting tensor; indices appearing exactly twice, once up and once down, are dummy indices, consumed by the implied summation and absent from the final expression's index list.

Variance Placement

Contravariant indices are consistently written as superscripts and covariant indices as subscripts, so that the placement itself, rather than an additional label, conveys the variance of each index throughout the notation.

Notational Variants

Explicit Summation Form

In contexts where the Einstein convention is not assumed, the same contraction may be written with an explicit summation sign over the repeated index, making the implied sum visually apparent at the cost of additional symbols:

Diagrammatic Form

The same contraction can be expressed using contraction diagram representation, in which the repeated index corresponds to an internal edge and the free index corresponds to an open leg, offering a visual alternative to the indexed algebraic form.

Notation Diagram

Purpose and Scope

Tensor contraction notation provides the written foundation upon which the rest of contraction theory operates: the type pair update rule, the simplification procedure, the verification procedure, and the diagrammatic representation are all defined in reference to how indices are placed and repeated according to these conventions, making consistent adherence to the notation a prerequisite for correctly applying every other concept in the study of tensor contractions.