6.16.1 Tensor Zero Two Covariant Slot Pair

The Tensor Zero Two Covariant Slot Pair defines how zero tensors act in covariant slots of multi-linear mappings.

Tensor Zero Two Covariant Slot Pair is the pair of index positions occupied by the two lower indices of a type (0,2) tensor, understood as two independent slots each capable of accepting a vector from the space V and each transforming under a change of basis according to the same covariant rule that governs ordinary dual-basis components. This slot pair is the structural core of the type (0,2) classification, and its behavior, individually and jointly, is what makes type (0,2) tensors the natural vehicle for bilinear forms such as inner products, distinguishing them from tensors whose slots mix variance types or share the opposite variance.

Anatomy of the Slot Pair

Each Slot as an Independent Input Channel

A type (0,2) tensor T with components T_{ij} can be understood as a function of two arguments drawn from V, where the first slot is filled by a vector v and the second slot is filled by a vector w:

T (v, w) = T_{i j} v^{i} w^{j}

Each slot contracts with exactly one vector, and linearity in each argument separately makes T a bilinear map on V × V. The two slots are distinguished by position, first or second, independently of whether the numerical values produced turn out to be symmetric between the two arguments.

Both Slots Share the Same Transformation Type

Both slots of the covariant pair transform identically under a change of basis, each governed by the transition matrix A rather than its inverse:

T_{' i j} = A_{i}^{k} A_{j}^{l} T_{k l}

This shared transformation behavior justifies treating the two indices as a genuine pair, since exchanging the two slots yields another type (0,2) tensor obeying exactly the same transformation law, unlike a mixed slot pair where the two positions are governed by different matrices.

Symmetric and Antisymmetric Decomposition of the Pair

Splitting the Slot Pair

Because both slots transform identically, the tensor decomposes uniquely into a part symmetric under exchange of the two slots and a part antisymmetric under that exchange:

T_{i j} = T_{(i j)} + T_{[i j]}

with the symmetric part given by:

T_{(i j)} = \frac{T_{i j} + T_{j i}}{2}

This decomposition is basis-independent precisely because both slots belong to the same variance type; the metric tensor, being symmetric by definition, occupies the extreme case in which the antisymmetric part vanishes entirely.

Interpretation of Each Piece

The symmetric part behaves like a symmetric bilinear form on V, the natural setting for quadratic forms and inner products, while the antisymmetric part behaves like an alternating bilinear form on V, the setting used for constructions such as the curl-like pairing that appears in the study of differential two-forms.

Diagram of the Slot Pair

The Slot Pair in Relation to Tensor Products

Building the Pair from Elementary Tensors

An elementary tensor φ ⊗ ψ realizes the slot pair concretely: the first slot corresponds to the covector φ, the second to the covector ψ, and the components T_{ij} = φ_i ψ_j display exactly how each slot inherits its transformation behavior from the covector occupying it. General type (0,2) tensors are sums of such elementary pairs.

Contraction Requires an Additional Structure

Because both slots are covariant, they cannot be contracted directly against each other; summing i and j in T_{ij} is not intrinsically meaningful without an auxiliary contravariant object such as the inverse metric g^{ij} to raise one slot first, for instance forming g^{ij} T_{ij}. This mirrors, in reverse, the same limitation seen in the type (2,0) slot pair, and it is precisely why raising and lowering operations via the metric are the standard tool for producing scalars from purely covariant or purely contravariant tensors, in contrast to the mixed (1,1) slot pair, which admits a direct, metric-free contraction.