11.20.3 Tensor Variance Verification Pairing Invariance

Tensor Variance Verification Pairing Invariance ensures consistent behavior under transformations, maintaining tensor properties across coordinate systems.

Tensor Variance Verification Pairing Invariance is the specific check within the verification procedure that confirms a full contraction between a claimed covariant object and a claimed contravariant object produces a genuinely basis-independent scalar, by explicitly computing the paired sum in two different bases and verifying that the two numerical results agree exactly.

Foundational Setting

Why This Check Targets Pairs

Individual index-position and transformation-factor checks confirm that a single object transforms correctly in isolation. Pairing invariance goes further by testing the joint behavior of two objects at once, since the entire justification for labeling one object covariant and another contravariant rests on their contraction producing an invariant, and this joint claim deserves its own explicit test.

The Quantity Under Test

Given a candidate covector $ω_{i}$ and a candidate vector $v^{i}$ , the pairing invariance check examines the contracted sum:

\sum_{i} ω_{i} v^{i}

Procedure for the Pairing Check

Step One: Compute in the Original Basis

Evaluate the contracted sum directly using the components of both objects as given in the original basis, producing a specific numerical value.

Step Two: Transform Both Objects Independently

Apply the claimed transformation laws separately to each object under an explicit basis change matrix $A$ , obtaining new-basis components ${\tilde{ω}}_{i}$ and ${\tilde{v}}^{i}$ for each.

Step Three: Recompute and Compare

Evaluate the contracted sum again using the new-basis components, and confirm that this second numerical value exactly matches the value obtained in the original basis:

\sum_{i} {\tilde{ω}}_{i} {\tilde{v}}^{i} = \sum_{i} ω_{i} v^{i}

Why Success Confirms the Claimed Pairing

The Algebraic Reason for Success

If both objects genuinely satisfy their claimed transformation laws, substituting these laws into the recomputed sum causes the matrix $A$ and its inverse $A^{- 1}$ to multiply together to the identity, algebraically forcing the two sums to agree:

\sum_{j} A_{i}^{j} {(A^{- 1})}_{j}^{i} = δ_{i}^{i}

What a Failure Reveals

If the two computed values disagree, the pairing invariance check reveals that at least one of the two objects does not actually obey its claimed transformation law, even if that object appeared to pass an isolated index-position check under a more limited set of test transformations.

Visual Overview

Diagram of the Pairing Check

Extending the Check to Higher-Rank Contractions

Multiple Simultaneous Pairings

For an expression involving several contracted index pairs at once, the pairing invariance check can be applied to the entire fully contracted scalar, confirming overall invariance, or it can be applied one contraction at a time by first performing all but one contraction and checking that the remaining paired sum alone is invariant, isolating which specific pairing might be responsible for a failure.

Confirming Mixed Tensor Traces

The same check applies directly to the trace of a mixed tensor, confirming that $T_{i}^{i}$ , summed over $i$ , evaluates to the same number in every basis tested, which is the mixed-tensor analogue of the vector-covector pairing check.

Summary of Key Traits

Defining Characteristics

Pairing invariance checks that a contracted sum between a claimed covariant and a claimed contravariant object gives the same numerical value in every tested basis.
Success is guaranteed algebraically whenever both objects genuinely satisfy their claimed, mutually inverse transformation laws.
A failure of the check reveals that at least one of the paired objects does not actually obey its claimed transformation law.
The check generalizes to multiple simultaneous contractions and to the trace of mixed tensors, allowing failures to be isolated to a specific pairing when needed.