7.22.3 Tensor Component Subscript Index

The Tensor Component Subscript Index denotes the position of a tensor's component in a multidimensional space, using subscripts to identify its coordinate placement.

Tensor Component Subscript Index is the lower index attached to a tensor component symbol, such as the i in Tᵢ, that identifies the component as transforming covariantly under a change of basis or coordinates. A subscript index marks a slot of the tensor that is dual to a vector slot: it is the index carried by the tensor's covector-like (covariant) arguments, and it is precisely the index type that a superscript index is built to pair with and cancel in the summation convention.

Role in Tensor Component Notation

Distinguishing Subscript from Superscript

A general tensor of type (p, q) is written with p superscript indices and q subscript indices, T^{i₁...iₚ}_{j₁...j_q}. The subscript indices label the covariant slots — those that accept ordinary vectors as arguments — while the superscript indices label the contravariant slots, which accept covectors (dual vectors). The subscript index is not merely a list label or array offset; its vertical position is meaningful notation that encodes a specific transformation behavior.

The Covariant Transformation Law

Under a change of basis described by a Jacobian matrix, a tensor component carrying a subscript index transforms with the Jacobian itself, in contrast to a superscript index, which transforms with the inverse Jacobian. For a covector component wᵢ, the law reads

w_{i'} = \sum_{i} \frac{\partial x^{i}}{\partial x^{i'}} w_{i}

where x^{i} and x^{i′} denote the old and new coordinate functions. The presence of the partial derivative of the old coordinate with respect to the new — the opposite ratio from the contravariant law — is the defining feature of covariant, subscript-indexed transformation.

Subscript Indices and the Summation Convention

Pairing with a Superscript for Contraction

The Einstein summation convention stipulates that a subscript index repeated as a superscript index within the same term is summed over its full range, and the pair is said to be contracted. In T^{i}_{j} v^{j}, the subscript j on T pairs with the superscript j on v, producing a sum over j, while the superscript i remains free and survives into the result.

Why Only Subscript–Superscript Pairs Contract

A valid contraction always pairs one subscript with one superscript occurrence of the same letter; two subscripts sharing a letter do not form a legal implicit sum in this convention, because the covariant transformation factor attached to a subscript index is the matrix inverse of the contravariant factor attached to a superscript index. Summing over a subscript–superscript pair causes the two Jacobian factors to cancel, leaving an expression that is invariant under the change of coordinates. Summing two subscript indices together would leave an uncancelled Jacobian factor, so the result would depend on the coordinate system chosen, which disqualifies it as a tensorial operation.

\sum_{i} \frac{\partial x^{i}}{\partial x^{i'}} \frac{\partial x^{j'}}{\partial x^{i}} = δ_{i'}^{j'}

Free Versus Dummy Subscript Indices

Free Subscript Indices

A subscript index that appears exactly once in a term, with no matching superscript occurrence, is a free index: it is not summed, and it labels one component among the several components of the resulting tensor. The set of free subscript and superscript indices in an equation must match on both sides, term by term, for the equation to be a valid tensor equation.

Dummy Subscript Indices

A subscript index that is paired with a matching superscript index in the same term is a dummy (or bound) index: it is summed out and does not survive to label the result. Dummy indices may be renamed to any unused letter without changing the meaning of the expression, since only the pattern of pairing between a subscript and a superscript matters, not the particular letter chosen.

Diagram of Index Placement

Subscript Indices in Common Tensor Objects

Covector and Basis Components

A covector (one-form) ω expressed in a coordinate basis has components ωᵢ carrying a single subscript index, reflecting that a covector is the archetypal covariant object. Likewise, the coordinate basis vectors themselves eᵢ are subscript-indexed, so that a vector's contravariant components v^{i} combine with them as v = v^{i} eᵢ, a contraction between a superscript and a subscript.

The Metric Tensor

The metric tensor g_{ij} carries two subscript indices and is used to lower indices, converting a superscript (contravariant) index into a subscript (covariant) one: v_{i} = g_{ij} v^{j}. This operation is the standard mechanism by which a contravariant component is re-expressed as a covariant one relative to a chosen metric, and it is the inverse operation to raising an index with g^{ij}.

Mixed Tensors

Tensors such as the Kronecker delta δ^{i}_{j} or the Riemann curvature tensor R^{i}_{jkl} combine subscript and superscript indices in a single object, with each subscript slot transforming according to the covariant law independently of the superscript slots; the total count of subscripts fixes part of the tensor's type (p, q).

Consequences of Misplacing a Subscript Index

Loss of Tensorial Character

Writing an index as a superscript when it should be a subscript, or vice versa, changes the transformation law attached to that slot; an expression with a misplaced index generally fails to transform as a tensor under a change of coordinates, even if it happens to agree numerically with a correctly written tensorial expression in one particular coordinate system.

Detecting Errors via Balance

Because free indices must match in type and letter on both sides of a valid tensor equation, checking that every subscript on the left appears as a subscript on the right (and likewise for superscripts) is a standard and effective way to catch notational errors before any numerical computation is performed.