13.12.1 Tensor Covariant Contravariant Slot Pair

Tensor Covariant Contravariant Slot Pair is the specific pair of index positions, one contravariant and one covariant, identified as the two slots to be contracted together in a given tensor operation, considered as concrete positions within a tensor's index structure rather than as the abstract rule requiring such pairing. It names the actual slots singled out for summation, distinguishing this specific identification from the general variance-matching requirement that governs which slots are eligible to form such a pair in the first place.

Conceptual Basis

Slots as Positions, Not Just Types

Every tensor index occupies a specific position within the tensor's overall index structure, and a covariant contravariant slot pair identifies two such positions explicitly, one of upper type and one of lower type, that have been selected from among possibly several available candidates for contraction.

Distinguishing the Pair From the General Rule

The requirement that contraction must pair a contravariant index with a covariant index is a general structural rule applicable to any tensor. A covariant contravariant slot pair, by contrast, refers to one particular instance of this rule being applied to two named, located positions within a specific tensor or pair of tensors.

Necessity of Explicit Identification

When a tensor possesses more than one contravariant or more than one covariant index, several distinct covariant contravariant slot pairs may be available, and identifying which specific pair is intended for a given contraction is necessary to avoid ambiguity about which indices are actually being summed.

Formal Description

Locating a Slot Pair Within a Single Tensor

For a tensor $$T_{kl}^{ij}$$, one possible covariant contravariant slot pair consists of the position holding $$i$$ and the position holding $$k$$, yielding upon contraction:

while a different, equally valid slot pair could instead link the position holding $$j$$ with the position holding $$k$$.

Locating a Slot Pair Across Two Tensors

For two separate tensors $$A^i$$ and $$B_j$$, the covariant contravariant slot pair consists of the single contravariant position on $$A$$ and the single covariant position on $$B$$, with contraction proceeding by identifying $$i$$ and $$j$$ as the same summed label.

Dimensional Compatibility of the Located Pair

A covariant contravariant slot pair is valid for contraction only when the two identified positions range over spaces of identical dimension, since the pair must support a well-defined shared index before summation can be carried out.

Properties

Enumerability of Available Pairs

For a tensor with $$p$$ contravariant and $$q$$ covariant indices of compatible dimension, the number of distinct covariant contravariant slot pairs available for a single contraction is the product $$p\times q$$, each constituting a separately identifiable choice.

Result Dependent on the Specific Pair Chosen

Selecting a different covariant contravariant slot pair from the same tensor generally produces a different resulting object, since each pair sums over a distinct pair of positions and therefore preserves a different subset of directional information from the original tensor.

Disjointness Requirement for Multiple Simultaneous Pairs

When more than one covariant contravariant slot pair is contracted simultaneously within a larger expression, the pairs selected must be disjoint, meaning no single slot position may be claimed by more than one pair at the same time.

Practical Considerations

Explicit Notation for Unambiguous Reference

In tensors with several indices of the same variance, explicitly naming or indexing the covariant contravariant slot pair intended for a given contraction, rather than relying on position alone, helps ensure that the specific pairing being used is communicated without ambiguity.

Role in Symbolic Tensor Manipulation

Symbolic computer algebra systems handling tensor expressions must track covariant contravariant slot pairs explicitly as part of their internal representation, since correctly identifying which slots are linked is essential to correctly simplifying or evaluating any contraction involving tensors with multiple indices.

Foundation for Describing Contraction Sequences

When a computation involves several successive contractions, each step can be described precisely by naming the covariant contravariant slot pair consumed at that step, providing a clear and unambiguous record of how a complex tensor expression is reduced stage by stage.