11.19.1 Tensor Variance Convention Index Placement

Tensor Variance Convention Index Placement defines how indices are placed to indicate covariant and contravariant components in tensor notation.

Tensor Variance Convention Index Placement is the specific rule within the broader variance convention that assigns each tensor index to either the superscript position or the subscript position based strictly on its transformation behavior, upper for contravariant indices and lower for covariant indices, and that further governs how these positions must be arranged and read consistently across an entire expression.

Foundational Setting

Placement as a Behavioral Declaration

Index placement is the single notational choice that carries the entire burden of specifying how a tensor component transforms. Writing an index as a superscript, as in $v^{i}$ , versus as a subscript, as in $ω_{i}$ , is the entirety of what distinguishes a contravariant component from a covariant one at the level of notation.

Placement Determines the Transformation Factor

Once an index's placement is fixed, the matrix factor it receives under a change of basis is fixed as well: an upper index receives a factor of the inverse basis-change matrix, while a lower index receives a factor of the direct basis-change matrix, with no ambiguity remaining once the placement is read.

Rules Governing Consistent Placement

Free Indices Must Match in Position

Any index that is not summed, a free index, must appear in the same vertical position, upper or lower, on every additive term of a well-formed tensor equation. An expression combining a term with a free upper index $i$ and another term with the same letter as a free lower index would violate this placement rule and is not a valid tensor equation.

Repeated Indices Must Appear in Opposite Positions

When the same index letter appears twice within a single term, the placement convention requires the two occurrences to be in opposite positions, one upper and one lower, in order to trigger the implicit summation and guarantee an invariant result:

ω_{i} v^{i} = \sum_{i} ω_{i} v^{i}

A repeated letter appearing twice in the same position, both upper or both lower, is not sanctioned by the placement convention and typically signals a notational error.

Placement in Mixed Tensors

Independent Slots for Each Index

A mixed tensor such as $T_{j}^{i}$ places its upper index $i$ and its lower index $j$ in fixed, independent slots, each following its own placement rule regardless of the other, so that a single tensor symbol may carry any number of upper and lower indices simultaneously without conflict.

Order Among Same-Type Indices

When a tensor carries multiple indices of the same type, their left-to-right or otherwise fixed relative order is itself part of the placement convention adopted for that tensor, and must be tracked consistently, since swapping the order of two indices of the same type can correspond to a genuinely different tensor unless the tensor is known to be symmetric in that pair.

Visual Overview

Diagram of Placement Rules

Practical Consequences of the Placement Rule

Reading Validity at a Glance

Because placement is fixed and rule-governed, scanning a tensor equation for index positions is often sufficient to check its validity: every free index must line up in position across all terms, and every summed index must appear once upper and once lower, without needing to inspect the numerical content of the tensors at all.

Guiding Correct Manipulation

When raising or lowering an index using a metric tensor, the placement convention dictates that the index physically moves from subscript to superscript, or vice versa, on the symbol, reflecting the change in transformation behavior that the operation produces.

v^{i} = \sum_{j} g^{i j} v_{j}

Summary of Key Traits

Defining Characteristics

Index placement, upper versus lower, is the sole notational signal of a component's transformation behavior.
Free indices must retain the same position across every term of a valid equation.
Repeated indices must occupy opposite positions to trigger the implicit summation rule correctly.
The relative order of same-type indices on a mixed tensor is part of its fixed placement convention and generally cannot be swapped without changing the tensor, absent symmetry.