9.13.2 Tensor Component Expansion Index Pattern

The Tensor Component Expansion Index Pattern explains how tensor components are indexed and expanded in algebraic notation.

Tensor Component Expansion Index Pattern is the fixed arrangement of upper and lower indices that characterizes how a tensor's components are labeled once the tensor is expanded relative to a basis. It describes the shape of the index signature itself, distinct from the particular values the components take, and it dictates how the indices must appear in every valid expression involving the tensor.

What the Pattern Describes

Number and Position of Indices

The index pattern of a tensor of type (p, q) consists of exactly p upper indices and q lower indices, arranged in a definite order. This pattern is fixed by the type of the tensor and does not change regardless of which basis is used to compute the numerical values of the components.

T_{j_{1} \dots j_{q}}^{i_{1} \dots i_{p}}

Distinct from Numerical Value

Two tensors of the same type share the same index pattern even though their actual component values differ entirely. The pattern is a statement about structure and index bookkeeping, while the values are a separate matter of coefficient assignment.

Consistency Requirements

Matching Patterns in Addition

Tensors can only be added together component by component when their index patterns match exactly in number and placement of upper and lower indices. Adding components with mismatched patterns produces an expression with no consistent tensorial meaning.

Patterns Under Tensor Product

Forming the tensor product of two tensors concatenates their index patterns, placing the upper indices of both factors together and the lower indices of both factors together, so that the pattern of the product tensor can be read off directly from the patterns of its factors.

A^{i} B_{j} = C_{j}^{i}

Patterns Under Contraction

Contracting a tensor removes one upper index and one lower index from the pattern, reducing the total count of free indices by two while leaving the relative order of the remaining indices unchanged.

Reading the Pattern

Determining Tensor Type from the Pattern

The type of a tensor can be read directly from its index pattern by counting the upper indices to obtain the contravariant order and counting the lower indices to obtain the covariant order. This makes the pattern a compact summary of the tensor's algebraic classification.

Recognizing Valid Operations

Because every legitimate tensor operation acts on index patterns in a predictable way, inspecting the pattern of an expression is often sufficient to determine, without computing any actual values, whether an operation such as addition, contraction, or multiplication is being applied validly.

Pattern Variants

Purely Contravariant and Purely Covariant Patterns

A pattern with only upper indices and no lower indices describes a purely contravariant tensor, while a pattern with only lower indices and no upper indices describes a purely covariant tensor. These are the two extreme cases of the general mixed pattern.

Symmetric and Antisymmetric Patterns

Some index patterns carry an additional constraint beyond mere position, namely that the components must remain unchanged, or must change sign, under a permutation of indices within the same raised or lowered group. Recognizing such a constraint as part of the pattern is essential for correctly interpreting tensors with special symmetry properties.