3.14.5 Tensor Covector Pairing Invariance

Tensor covector pairing invariance ensures pairings stay unchanged under coordinate transformations, crucial in differential geometry and physics.

Tensor Covector Pairing Invariance is the property that the scalar value produced by pairing a covector with a vector, φ(v), remains exactly the same number regardless of which basis is used to compute it, even though the individual components of φ and v change from one basis to another. This invariance is the organizing principle behind the specific transformation rules assigned to covector and vector components: those rules are not arbitrary but are precisely the ones required to keep the pairing invariant.

Statement of the Invariance

The Basic Claim

For a covector φ and vector v, and any two bases related by a change-of-basis matrix A, the scalar computed from components in the old basis equals the scalar computed from components in the new basis:

\sum_{i = 1}^{n} φ_{i} v^{i} = \sum_{i = 1}^{n} φ_{i}^{'} v^{i'}

Both sides equal φ(v), computed the abstract, basis-free way; the equation asserts that computing this same number through coordinates gives an identical result no matter which coordinate system is used.

Why This Is Not Automatic

Invariance is not a generic property of arbitrary pairs of number lists; it depends critically on φ_i and v^i transforming by mutually inverse-type matrices. If both were to transform by the same matrix A, or if either transformed by an unrelated rule, the sum Σ φ_i v^i would generally take different numerical values in different bases, and the pairing would fail to represent any basis-independent quantity.

Proof of Invariance

Substituting Both Transformation Laws

Starting from the sum computed with new components and substituting the transformation laws φ'_i = A^j_i φ_j and v'^i = (A^{-1})^i_k v^k:

\sum_{i = 1}^{n} φ_{i}^{'} v^{i'} = \sum_{i = 1}^{n} (\sum_{j = 1}^{n} A_{i}^{j} φ_{j}) (\sum_{k = 1}^{n} {(A^{- 1})}_{i}^{k} v^{k})

Collapsing the Sum Over the New Index

Rearranging the order of summation to sum over i first isolates a product of A and A^{-1} entries:

\sum_{j = 1}^{n} \sum_{k = 1}^{n} (\sum_{i = 1}^{n} A_{i}^{j} {(A^{- 1})}_{i}^{k}) φ_{j} v^{k}

The bracketed inner sum is the (j, k) entry of the product A^T (A^{-1})^T, or equivalently reduces via the standard matrix inverse identity to δ^k_j. Substituting this delta collapses the double sum to a single sum over the shared index:

\sum_{j = 1}^{n} φ_{j} v^{j}

which is exactly the original sum computed with the old components, completing the proof.

Invariance as the Defining Test for Tensor Objects

A Necessary Condition for Being a Genuine Pairing

Invariance of the pairing is often used as the criterion by which a proposed transformation rule for some array of numbers is judged to define a genuine covector or vector, rather than an arbitrary array that happens to carry indices. Any assignment of components across bases that fails to preserve Σ φ_i v^i cannot represent a basis-independent scalar pairing, and so cannot correspond to an actual covector paired with an actual vector.

Generalization to Full Contractions of Tensors

The same style of argument extends beyond a single covector and vector to full contractions of general tensors, where every upper index of one factor is paired against a matching lower index of another. In each such contraction, the A factors contributed by lower indices and the A^{-1} factors contributed by upper indices cancel in matching pairs, leaving an invariant scalar. Covector pairing invariance is the simplest, foundational instance of this broader cancellation phenomenon that underlies tensor contraction in general.

Geometric Interpretation

The Pairing as a Basis-Free Number

Because φ(v) does not depend on the basis used to compute it, it can be regarded as a genuine, intrinsic property of the pair (φ, v), existing independently of any coordinate system. This is consistent with the original definition of φ as a function on V, which never referenced coordinates in the first place; invariance simply confirms that the coordinate computation faithfully reproduces this coordinate-free function value.

Physical Analogy

In settings where covectors and vectors represent physical quantities, such as a gradient covector and a displacement vector, the invariance of their pairing corresponds to the physical requirement that a quantity such as a change in a scalar field along a displacement should not depend on the arbitrary choice of coordinate axes used by an observer to describe the situation.

Diagrammatic Summary

The diagram shows two separate computations, one performed with old-basis components and one with new-basis components, converging on the same single scalar output, illustrating that the pairing invariance guarantees agreement regardless of the path taken to reach it.