8.13 Tensor Index Position Notation

Tensor Index Position Notation is the convention within tensor index notation that assigns meaning to whether an index is written as a superscript or a subscript on a tensor symbol, using vertical placement to encode how the corresponding component transforms under a change of basis or coordinates. Rather than being a typographic detail, the position of an index — above the baseline or below it — is one of the two coordinates, alongside the index's letter, that fully specify what a given symbol in a tensor equation represents.

The Two Positions and Their Meanings

Upper Position and Contravariance

An index written in the upper position, as in $A^{i}$, denotes a contravariant component. Contravariant components transform with the inverse of the matrix relating old and new basis vectors, so that under a coordinate change,

This is the notation associated with ordinary vectors and coordinate differentials.

Lower Position and Covariance

An index written in the lower position, as in $A_{i}$, denotes a covariant component. Covariant components transform directly with the same matrix used to relate the bases, without inversion:

This is the notation associated with covectors, gradients, and one-forms.

Encoding Tensor Type

The Pair (p, q)

The total pattern of upper and lower indices on a tensor symbol determines its type, conventionally written $(p,q)$, where $p$ is the number of upper (contravariant) indices and $q$ is the number of lower (covariant) indices. A symbol such as $T^{ij}{}_{k}$ has type $(2,1)$: two contravariant slots and one covariant slot. This pairing is a complete summary of how the tensor transforms, independent of any specific coordinate system.

Rank as the Sum of Positions

The total rank of a tensor is the sum $p + q$ of its upper and lower index counts. A scalar has rank $0$ (no indices of either kind), a vector or covector has rank $1$, and a tensor such as the Riemann curvature tensor, with four indices, has rank $4$ regardless of how those four indices are distributed between upper and lower positions.

Mixed Tensors

Coexistence of Both Positions

A tensor is called mixed when it carries both upper and lower indices simultaneously, such as $T^{i}{}{j}$. Each index retains the transformation law dictated by its own position, entirely independent of the other indices present on the same symbol; the upper index in $T^{i}{}{j}$ transforms contravariantly while the lower index transforms covariantly, and neither influences the other's transformation rule.

Index Ordering Among Mixed Positions

When a tensor has multiple indices of mixed position, the relative left-to-right ordering of upper and lower slots is sometimes significant, particularly for tensors that are not symmetric between an upper and a lower slot. Staggered notation, such as $T^{i}{}{j}{}^{k}$, is used precisely to preserve this ordering when it matters, distinguishing it from $T^{ik}{}{j}$, which groups the two upper indices together.

Converting Between Positions

Raising and Lowering

Given a metric tensor, an index can be moved from one position to the other: the covariant metric $g_{ij}$ lowers an upper index, and the contravariant metric $g^{ij}$, its matrix inverse, raises a lower index:

Without a metric, no such conversion exists, and upper and lower positions denote permanently distinct types of object.

Why the Distinction Persists Even With a Metric

Even in spaces equipped with a metric, where every covariant component has a corresponding contravariant counterpart, the position notation is retained because different tensor operations — contraction in particular — require indices in specific, opposite positions to be paired. The position notation therefore continues to carry operational information about which contractions are valid, independent of whether raising or lowering is available.

Position Notation and Contraction

Valid Contraction Requires Opposite Positions

A repeated index is only summed implicitly when it appears once in the upper position and once in the lower position within the same term. This requirement is a direct consequence of index position notation: contraction is defined as the pairing of a covariant slot with a contravariant slot, and the position of each index is precisely what tells an author or reader whether a given repeated letter constitutes a legitimate contraction.

Role Within Tensor Index Notation

Index position notation, together with the rules for free and dummy indices and the implicit summation convention, forms the complete grammar of tensor index notation. Where the letter of an index identifies which "slot" it occupies among several possible indices, the position of that letter — upper or lower — identifies the transformation character of that slot. Every well-formed tensor expression relies on this positional information to determine legality of contraction, type of the result, and correctness under a change of coordinates.