6.15.1 Tensor Two Zero Contravariant Slot Pair

A tensor with two zero contravariant slots pairs indices, defining a bilinear form in tensor algebra.

Tensor Two Zero Contravariant Slot Pair is the pair of index positions occupied by the two upper indices of a type (2,0) tensor, understood as two independent "slots" each capable of accepting a covector from the dual space V* and each transforming under a change of basis according to the same contravariant rule that governs ordinary vector components. This slot pair is the structural core of the type (2,0) classification, and examining how the two slots behave individually, together, and under symmetry operations reveals why type (2,0) tensors function as bilinear maps on the dual space rather than as operators or as forms acting directly on vectors.

Anatomy of the Slot Pair

Each Slot as an Independent Input Channel

A type (2,0) 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 covector φ and the second slot is filled by a covector ψ:

T (φ, ψ) = T^{i j} φ_{i} ψ_{j}

Each slot contracts with exactly one covector, and because the tensor is linear in each argument separately, the map T is bilinear on V* × V*. The two slots are distinguished by their position, first or second, even when the tensor is later found to be symmetric in the values it produces.

Both Slots Share the Same Transformation Type

Unlike a mixed tensor, where one slot transforms with the transition matrix A and the other with its inverse B, both slots of the contravariant pair transform identically, each governed by B = A^{-1}:

T^{' i j} = B_{k}^{i} B_{l}^{j} T^{k l}

This shared transformation behavior is what justifies calling the two indices a "pair" rather than treating them as structurally distinct roles, since permuting the two slots produces another valid type (2,0) tensor governed by the exact same transformation law.

Symmetric and Antisymmetric Decomposition of the Pair

Splitting the Slot Pair

Because both slots of the pair transform the same way, the tensor can always be decomposed 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}

and the antisymmetric part given by the corresponding difference divided by two. This decomposition is only well-defined and basis-independent because both slots in the pair belong to the same transformation type; attempting the same operation on a mixed (1,1) tensor would not produce basis-independent symmetric and antisymmetric parts, since swapping an upper index with a lower index is not meaningful.

Interpretation of Each Piece

The symmetric part behaves like a symmetric bilinear form evaluated on covectors and is the natural object for constructing quadratic expressions, while the antisymmetric part behaves like an alternating bilinear form and is the natural object underlying constructions such as bivectors, which represent oriented plane elements spanned by pairs of directions.

Diagram of the Slot Pair

The Slot Pair in Relation to Tensor Products

Building the Pair from Elementary Tensors

An elementary tensor v ⊗ w realizes the slot pair concretely: the first slot corresponds to the vector v, the second slot corresponds to the vector w, and the components T^{ij} = v^i w^j show explicitly how each slot inherits its transformation behavior from the vector occupying it. General type (2,0) tensors are sums of such elementary pairs and need not factor into a single product of two vectors.

Contraction Requires an Additional Structure

Because both slots are contravariant, they cannot be contracted directly against each other the way an upper and a lower index can; summing i and j in T^{ij} is not a valid tensor operation on its own. Producing a scalar from a type (2,0) tensor's slot pair requires an auxiliary covariant object, such as a metric g_{ij}, to lower one of the two contravariant slots before contraction, for instance forming g_{ij} T^{ij}. This dependence on an external structure to contract the pair is a defining limitation that distinguishes type (2,0) tensors from type (1,1) tensors, whose mixed slot pair already admits a natural, metric-free contraction through the trace.