6.22.3 Tensor Covariant Contravariant Split

Tensor Covariant Contravariant Split explains how tensors split into covariant and contravariant parts, key for transformation behavior in coordinates.

Tensor Covariant Contravariant Split is the division of a tensor's index slots into two distinguishable kinds — contravariant slots, which transform using the change-of-basis matrix directly, and covariant slots, which transform using its inverse — and the requirement that every index a tensor carries be assigned to exactly one of these two kinds. This split is the mechanism that turns a bare count of indices (the order) into a meaningful type (p, q), and it is the reason a tensor's behavior under a change of basis can be predicted from its notation alone.

The Two Kinds of Transformation Behavior

Contravariant Slots Transform With the Basis Change Matrix

A contravariant slot holds a component that changes oppositely to the basis vectors themselves, in the sense that if the basis vectors are stretched by a factor, the contravariant components shrink by the same factor to keep the represented vector fixed. Concretely, if e′ᵢ = Σₖ Aₖᵢeₖ, a vector's contravariant components transform as

v^{i}' = \sum_{k} C_{k}^{i} v^{k}

where C denotes the inverse of the transition matrix A, which is why ordinary vector components are called contravariant: they vary "contrary to" the transformation applied to the basis.

Covariant Slots Transform With the Basis Vectors Directly

A covariant slot holds a component that changes in the same way as the basis vectors, using the transition matrix A itself rather than its inverse:

ω_{i}' = \sum_{k} A_{i}^{k} ω_{k}

Covectors (elements of the dual space V*) and the covariant slots of general tensors follow this rule, which is why they are called covariant: they vary "along with" the basis transformation.

Assigning the Split Across All of a Tensor's Indices

Each Index Gets Exactly One Assignment

For a tensor of order n, the covariant contravariant split assigns each of the n index slots independently to either the contravariant category or the covariant category, with no slot left unassigned and no slot assigned to both. The total counts of each category, p contravariant and q covariant, together constitute the tensor's type, and the split is what makes this pair of counts meaningful rather than arbitrary.

The Split Is Fixed by the Tensor's Definition, Not by Convenience

The split is not a matter of notational choice made after the fact; it is determined by which vector space each tensor-product factor belongs to. A factor of V in V ⊗ V* ⊗ V contributes a contravariant slot, and a factor of V* contributes a covariant slot, so the split is read directly off the definition of the tensor product space the tensor lives in.

Diagram of the Split

Why the Split Is Preserved Under Tensor Operations

Tensor Product Concatenates Splits

Forming the tensor product of two tensors concatenates their index lists, and correspondingly concatenates their covariant-contravariant splits: a (p₁, q₁) tensor combined with a (p₂, q₂) tensor via tensor product yields a (p₁ + p₂, q₁ + q₂) tensor, since none of the original slots change category during this operation.

Contraction Removes a Matched Pair

A valid contraction removes exactly one contravariant slot and one covariant slot together, which is precisely why contraction requires pairing an upper index with a lower index: only such a pair transforms in mutually cancelling ways (A and A⁻¹ multiplying together to give the identity), producing a result that no longer depends on the choice of basis. Attempting to contract two slots of the same category — two contravariant or two covariant indices — does not produce this cancellation and does not yield a basis-independent result, which is why the covariant contravariant split is the rule that governs which contractions are legitimate.