8.2.2 Tensor Free Index Area

Tensor Free Index Area explores the algebraic structure of tensors without index dependencies, focusing on abstract properties and operations in multilinear algebra.

Tensor Free Index Area is the conceptual domain of tensor index notation concerned specifically with free indices: those index letters occurring exactly once, without a matching occurrence of the opposite vertical type, in a given term of a tensor expression. It covers the rules governing how free indices determine the rank and type of the object an expression represents, how they must balance across a valid equation, and how they behave differently from dummy indices, which fall instead under summation and contraction.

Identifying a Free Index

The Single-Occurrence Rule

An index is free within a term precisely when it appears once in that term, either as a lone superscript or a lone subscript, with no matching occurrence of the opposite type available to pair with it under the summation convention. A free index is therefore never summed; it survives into the result and labels one of the result's own index slots.

T_{j}^{i} v^{j} = w^{i}

In this contraction, j is repeated once up and once down and is therefore dummy, summed away; i occurs only once, as a superscript, in both the term on the left and the result on the right, and is therefore free.

Free Indices Fix the Rank of the Result

The number of free indices remaining after all contractions in an expression have been carried out equals the rank of the resulting tensor, and the vertical position of each free index — superscript or subscript — fixes whether that particular slot of the result is contravariant or covariant, together determining the result's full type (p, q).

The Balance Requirement Across an Equation

Matching Free Indices Term by Term

A tensor equation is well-formed only if every term on both sides carries exactly the same set of free indices, matched in letter and in vertical position; a term missing a free index present elsewhere in the equation, or carrying an index in the wrong position, signals that the equation does not represent a consistent tensor identity.

R_{j}^{i} = A_{j}^{i} + B_{j k}^{i k}

Here i and j are the shared free indices across every term, while k in the last term is dummy, appearing once up and once down within that term alone, and therefore does not disturb the free-index balance required between the two sides.

Why the Requirement Exists

Because a tensor equation with r free indices stands for a whole family of n^{r} scalar equations — one for each assignment of values to the free indices — every term must supply a value under that same assignment; a term lacking one of the shared free indices would fail to specify which member of the family it belongs to, breaking the correspondence the equation is meant to express.

Diagram of the Free Index Area

Free Indices Across Different Notation Areas

In Differential Geometry and Relativity

A free index in an expression like R^{i}_{jkl} for the Riemann curvature tensor labels which component of the tensor's several free slots is being referenced; a physical statement such as the vacuum field equation R_{μν} = 0 is a compact way of asserting an entire family of equalities, one for each pair of values the free indices μ and ν may independently take.

In Continuum Mechanics

The traction vector formula tᵢ = σᵢⱼ nⱼ, relating stress to the force per unit area on a surface, has i as a free index on both sides while j is dummy; the free index area governs exactly the requirement that this same i appears matched on both sides, guaranteeing the formula gives a consistent vector regardless of which coordinate direction i is later set to.

Common Errors Within the Free Index Area

Free-Index Mismatch

Writing an expression where a free index appears in one term but not another, or appears as a superscript in one term and a subscript in another sharing the same letter, produces a formula with no consistent tensorial meaning; this is the single most common notational error identifiable purely by inspecting free indices.

Overusing a Letter Meant to Be Free

Accidentally repeating a free index's letter elsewhere in the same term in a way that creates an unintended one-up-one-down pairing converts what was meant to be a free index into a dummy one, silently altering the rank of the result; careful bookkeeping of which letters are committed to a free role, distinct from those available for dummy use, prevents this class of mistake.