8 Tensor Index Notation

Tensor Index Notation is the system of writing tensors and tensor operations using labeled indices attached to a base symbol, with the vertical placement of each index — as a superscript or a subscript — and the pattern of repeated indices together encoding the tensor's type, its transformation behavior, and any contractions or sums implicit in an expression. It is the primary working notation of tensor algebra, used to write down components, perform algebraic manipulations, and state tensor identities compactly.

The Building Blocks of the Notation

Base Symbol and Index Slots

A tensor of type (p, q) — carrying p contravariant and q covariant slots — is written with a base letter followed by p superscript index slots and q subscript index slots, as in T^{i₁...iₚ}_{j₁...j_q}. Each slot is filled by an index letter drawn from a fixed range determined by the dimension of the space, and the total arrangement of slots encodes the tensor's full type.

Upper Indices for Contravariant Slots

A superscript index designates a contravariant slot: one that transforms with the Jacobian of a coordinate change applied in its inverse form, and that is naturally associated with vector-like components, such as the components v^{i} of a vector expanded in a basis.

Lower Indices for Covariant Slots

A subscript index designates a covariant slot: one that transforms with the Jacobian applied directly, and that is naturally associated with covector-like components, such as the components ωᵢ of a one-form, or the basis vectors eᵢ themselves.

The Summation Convention

Implicit Sums Over Repeated Indices

Central to the notation is the Einstein summation convention: whenever an index letter appears exactly twice in a single term, once as a superscript and once as a subscript, a sum over that index's full range is implied without an explicit summation symbol.

Free and Dummy Indices

An index appearing once in a term is free and labels a component of the result; the same free indices, matched in letter and vertical position, must appear on every term of a valid tensor equation. An index appearing twice, once up and once down, is dummy: it is summed away, does not appear in the result, and may be renamed to any unused letter without altering the expression's meaning.

Index Operations

Contraction

Setting one superscript index equal to one subscript index and summing — which the notation performs automatically whenever the same letter is written in both positions — is called contraction. Contracting a tensor of type (p, q) over one such pair produces a tensor of type (p − 1, q − 1).

Raising and Lowering

A metric tensor g_{ij} and its inverse g^{ij} provide the mechanism, within the notation, for converting a subscript index into a superscript one and vice versa:

This operation, called raising or lowering an index, is expressed purely as a contraction with the metric or its inverse, using the same repeated-index rule as any other contraction.

The Kronecker Delta

The Kronecker delta δ^{i}_{j}, equal to 1 when i = j and 0 otherwise, functions in the notation as an index-substitution tool: contracting it with another tensor's index simply replaces that index with the delta's other index, as in δ^{i}_{j} v^{j} = v^{i}.

Diagram of an Index-Notation Expression

Symmetry Notation

Symmetrization and Antisymmetrization

Parentheses and square brackets enclosing a set of indices denote, respectively, symmetrization and antisymmetrization over those indices: T_{(ij)} denotes the symmetric part of T_{ij} and T_{[ij]} denotes its antisymmetric part, each formed by averaging over the relevant permutations of the enclosed indices with appropriate signs in the antisymmetric case.

Compactly Stating Symmetry Properties

Symmetry or antisymmetry of a tensor in a pair of indices is stated directly in the notation by writing T_{ij} = T_{ji} (symmetric) or T_{ij} = −T_{ji} (antisymmetric), a statement that, because indices are free, holds identically for every value the indices may take.

Notation Variants

Abstract Index Notation

A related but distinct variant, abstract index notation, uses index letters purely as labels marking which slot of a basis-free tensor is being referred to, without implying that any particular basis or numerical components are involved; unlike component notation, the indices here do not range over numerical values, and expressions remain valid statements about the tensor as an object.

Component Notation in Practice

When indices do range over 1 through n and refer to specific numerical values, the notation is being used in its ordinary component sense, tied to a chosen basis; this is the sense used in explicit calculations and numerical work, as distinct from the basis-free abstract index variant.

Why the Notation Is Effective

Encodes Transformation Behavior Directly

Because the vertical position of every index tracks precisely how that slot transforms under a change of basis, correctly written tensor index notation carries its own built-in consistency check: matching free indices on both sides of an equation, in position and letter, is a necessary condition for that equation to be tensorially valid.

Compactness for High-Rank Objects

Tensors of high rank, carrying many indices, would require unwieldy explicit notation without the summation convention and the index-slot system; tensor index notation keeps such expressions no more cumbersome than lower-rank ones, which is a large part of its practical value throughout the subject.