1.11.2 Index Placement Convention

Index Placement Convention is the specific set of rules governing where, relative to a tensor's base symbol, each index is written, whether as a superscript or subscript, and in what left-to-right order, so that the placement of an index alone communicates its variance, contravariant or covariant, and its position within the tensor's argument structure. Because a tensor's type and behavior under contraction depend critically on which indices are upper and which are lower, and because multi-index tensors depend on the order in which those indices are listed, index placement is not a stylistic detail but a load-bearing part of tensor notation that must be fixed by explicit convention.

Index placement convention governs two largely independent questions: vertical placement, whether a given index sits above or below the baseline, which encodes variance, and horizontal placement, the left-to-right sequence in which multiple indices of the same variance are listed, which encodes argument order. Both questions must be settled consistently for a tensor expression to be read without ambiguity.

Vertical Placement: Upper Versus Lower

Superscripts for Contravariant Indices

An index written as a superscript, such as the i in v^i, denotes a contravariant component, one that transforms with the Jacobian of a coordinate change. This placement is used for vector components, and more generally for any index corresponding to an argument slot that accepts a covector.

Subscripts for Covariant Indices

An index written as a subscript, such as the i in ω_i, denotes a covariant component, one that transforms with the inverse Jacobian. This placement is used for covector components, gradients, and any index corresponding to an argument slot that accepts a vector.

Why the Distinction Is Preserved Notationally

Because contravariant and covariant components transform in mutually inverse ways, keeping their notational placement visually distinct allows the Einstein summation convention to be applied correctly: a valid contraction always pairs one upper index with one lower index, and this pairing is verified instantly by inspecting index placement, without needing to separately recall the transformation behavior of each individual quantity involved.

Horizontal Placement: Ordering Among Same-Variance Indices

Order Reflects Argument Position

When a tensor carries more than one index of the same variance, such as T_ijk, the left-to-right order in which those indices are written corresponds to the order of the arguments in the tensor's associated multilinear map. T_ijk and T_jik are, in general, different quantities, since swapping the written order corresponds to swapping which argument is matched to which slot, and only tensors that happen to be symmetric in those indices give the same value under such a swap.

Interleaving Upper and Lower Indices

For tensors carrying both upper and lower indices, a further placement question arises: whether to group all upper indices together followed by all lower indices, as in T^{ij}_{kl}, or interleave them to preserve a specific correspondence between argument order and index position, as in T^i_k{}^j_l. Interleaved placement is adopted specifically when the relative position between an upper and a lower index carries independent meaning, such as tracking which pair of arguments belongs to which sub-block of the tensor.

Placement Conventions for Specific Objects

Matrices as Row-Column Placement

For a matrix regarded as a (1, 1) tensor, the standard placement convention assigns the upper index to the row position and the lower index to the column position, M^i_j, matching the row-then-column reading order used throughout ordinary matrix notation, so that formulas for matrix multiplication and matrix-vector multiplication, written using the summation convention, reproduce the standard results exactly.

The Metric's Symmetric Placement

The metric tensor is conventionally written with both indices in the same vertical position, either both lower, g_ij, for the metric itself, or both upper, g^ij, for its inverse, reflecting that the metric is a symmetric bilinear form with two arguments of the same type, rather than a mixed object like a linear map.

Consequences of Inconsistent Placement

Broken Summation Convention

If an index intended to be summed is inadvertently placed twice in the same vertical position, both upper or both lower, the Einstein summation convention no longer applies automatically, since the convention is defined specifically for a matched upper-lower pair; such an inconsistency typically indicates a transcription error or a conflation of two indices that should have carried distinct labels.

Misidentified Tensor Type

Because a tensor's type, (p, q), is read directly from the count and placement of its upper and lower indices, an error in vertical placement silently changes the apparent type of the tensor being described, which can propagate into incorrect conclusions about how the tensor transforms or which operations may legitimately be applied to it.

Placement as the First Check on a Tensor Expression

A Fast Diagnostic

Verifying that index placement is used consistently, upper indices consistently marking contravariant slots, lower indices consistently marking covariant slots, and that the order of same-variance indices is preserved throughout a derivation, is typically the fastest available check for catching structural errors in a tensor computation, since it requires only inspecting the visual arrangement of the expression rather than evaluating it numerically.

A Prerequisite for the Summation Convention

Index placement convention underlies the Einstein summation convention itself: the entire rule that a repeated index triggers summation depends on being able to recognize, purely from placement, that one occurrence is upper and the other is lower. Without a firmly fixed placement convention, the summation convention has no unambiguous trigger condition to detect, which is why placement is treated as logically prior to summation within the broader system of tensor algebra conventions.