13.11.1 Tensor Vector Covector Contraction Pair

Tensor Vector Covector Contraction Pair is the specific combination of one vector and one covector selected to be contracted together, considered as the pair of participating objects rather than as the shared index linking them or the scalar they produce. It identifies which vector and which covector are being brought together for contraction, a distinction that matters once several vectors and covectors are available and a specific pairing among them must be designated.

Conceptual Basis

The Pair as the Operands of Contraction

Contraction always requires two participating objects when performed across separate tensors rather than within a single tensor, and the vector covector contraction pair names exactly these two objects, one contravariant and one covariant, that are selected for a given contraction operation.

Selection Among Multiple Candidates

In a setting where several vectors and several covectors are simultaneously available, such as within a larger expression or computation, identifying the vector covector contraction pair is the act of specifying which particular vector is to be contracted with which particular covector, since arbitrary combinations are not implied automatically.

Pair as a Prerequisite for the Summed Index

Before a summed index can be assigned to link a vector and covector, the pair itself must first be identified: the contraction pair determines which two objects are involved, and only afterward does the shared index specify how their components are to be matched and summed.

Formal Description

Naming the Pair

Given several vectors $$u^i,v^i$$ and covectors $${\omega }_i,{\eta }_i$$, one valid vector covector contraction pair might designate $$u^i$$ together with $${\eta }_i$$, giving:

as distinct from the alternative pair formed by $$v^i$$ together with $${\omega }_i$$.

Dimensional Requirement on the Pair

For a vector covector contraction pair to be valid, the vector and covector selected must range over spaces of identical dimension, since only then does a well-defined shared index exist that can be summed to produce a scalar.

Distinctness From Unrelated Objects

Any vector and covector not designated as part of the same contraction pair remain independent of one another within the expression, retaining their own free indices and playing no role in that particular contraction, even if they happen to share the same dimension.

Properties

Uniqueness of Result Given a Fixed Pair

Once a specific vector covector contraction pair has been designated, the resulting scalar is fully determined, since no further choice remains beyond identifying which two objects are involved and summing over their shared index.

Multiplicity of Available Pairs

When multiple vectors and covectors of compatible dimension are present, several distinct vector covector contraction pairs may be formed from them, each yielding, in general, a different scalar result, since different pairs need not encode the same functional relationship.

Symmetry Considerations

The vector covector contraction pair is inherently asymmetric in role, since one member must be contravariant and the other covariant, meaning the notion of exchanging the two members of the pair does not apply in the same way it would for two objects of identical variance.

Practical Considerations

Explicit Identification in Complex Expressions

In expressions involving several vectors and covectors, explicitly naming the vector covector contraction pair intended for a given operation avoids ambiguity about which specific combination is meant, particularly when multiple valid pairings of compatible dimension exist.

Role in Constructing Bilinear Forms

Selecting different vector covector contraction pairs from a fixed collection of vectors and covectors is the underlying mechanism by which bilinear expressions, such as those appearing in physical or geometric formulas, are built up term by term from elementary contractions.

Foundation for General Tensor Pairing

The vector covector contraction pair represents the simplest case of the more general notion of selecting which tensors, among several available, are to participate in a given contraction, a consideration that persists in more elaborate settings involving higher-rank tensors and multiple simultaneous contractions.