13.12 Tensor Covariant Contravariant Contraction Case

Tensor Covariant Contravariant Contraction Case is the general requirement, underlying every valid tensor contraction, that the two indices selected for summation must consist of exactly one contravariant index and one covariant index, regardless of whether those indices belong to the same tensor or to two separate tensors. It identifies the structural condition common to trace contraction, matrix multiplication, inner products, and vector covector pairing alike, serving as the unifying rule from which all of these more specific named cases are derived.

Conceptual Basis

Variance as the Determining Factor

Every tensor index carries a variance, either contravariant, transforming via the Jacobian of a change of basis, or covariant, transforming via its inverse. The covariant contravariant contraction case identifies that only a pairing of one index of each type produces the cancellation of transformation factors necessary for a basis-independent result.

Why Same-Variance Pairing Fails

Attempting to sum over two contravariant indices or two covariant indices does not yield a basis-independent quantity, since both would transform by the same type of factor rather than by inverse factors, so no cancellation occurs and the resulting sum depends on the arbitrary choice of basis. The covariant contravariant requirement exists precisely to exclude such ill-defined pairings.

Applicability Across Single and Multiple Tensors

The covariant contravariant contraction case applies equally whether the two indices being paired belong to the same tensor, as in trace contraction, or to two separate tensors, as in matrix multiplication or vector covector pairing, since the underlying variance requirement is identical in both settings.

Formal Description

General Statement

For any two index slots designated for contraction, one carrying an upper position $$i$$ and the other a lower position $$j$$, the covariant contravariant contraction case requires setting these positions equal and summing:

where $$A^i$$ supplies the contravariant slot and $$B_i$$ supplies the covariant slot, whether these belong to the same or different tensors.

Transformation Cancellation

Under a change of basis with Jacobian $$J_i^i$$ and inverse $$J_{i\prime }^i$$, the contracted sum transforms as:

and because the two Jacobian factors combine to the identity, this reduces to $$A^i{B}_i$$, confirming basis independence follows precisely from the covariant contravariant pairing.

Exclusion of Mismatched Pairings

A proposed contraction pairing two contravariant indices, such as $$A^i{B}^i$$ without an intervening metric, does not satisfy the covariant contravariant contraction case and does not produce a basis-independent result, since both factors would transform identically rather than inversely.

Manifestations in Named Contraction Cases

Trace Contraction

In trace contraction, the covariant contravariant requirement is satisfied by pairing the single upper and single lower index belonging to the same mixed tensor, reducing a rank-two object to a scalar.

Matrix Multiplication

In matrix multiplication, the requirement is satisfied by pairing the covariant index of the first factor with the contravariant index of the second factor, linking the two tensors through their shared intermediate index.

Vector Covector Pairing

In the simplest possible instance, the requirement is satisfied directly by the natural opposite variances of a vector and a covector, requiring no auxiliary metric tensor to establish the necessary pairing.

Inner Products via a Metric

When two indices of the same variance must be paired, as with two vectors, the covariant contravariant contraction case is achieved indirectly by first invoking a metric tensor to convert one index to the opposite variance before the direct contraction is applied.

Practical Significance

A Single Rule Governing Diverse Operations

Recognizing the covariant contravariant contraction case as the common thread beneath trace, matrix multiplication, inner product, and vector covector operations clarifies that these are not independent rules but specific applications of one underlying requirement.

Diagnostic Use in Verifying Expressions

Checking that every contracted pair in a tensor expression consists of one upper and one lower index is a standard and necessary step in verifying that the expression is well formed and will yield a basis-independent result.

Foundation for Constructing New Contractions

Understanding the covariant contravariant contraction case as the essential requirement allows new, unnamed contractions to be constructed correctly by ensuring any newly proposed pairing of indices, however unfamiliar the specific tensors involved, still respects this same variance-matching rule.