11.20.5 Tensor Variance Verification Coordinate Consistency

Tensor Variance Verification Coordinate Consistency is the stage of the verification procedure that confirms a candidate tensor's transformation behavior holds correctly not merely for one arbitrary coordinate change but across an entire sequence or network of coordinate changes, checking that composing several individual transformations produces the same overall result as computing the direct transformation in one step.

Foundational Setting

Beyond a Single Transformation

An object might appear to satisfy the correct transformation law under one particular coordinate change yet fail to do so consistently when a different, or a chained, coordinate change is tested. Coordinate consistency verification closes this gap by testing multiple related transformations together rather than relying on a single instance.

The Composition Requirement

The check rests on the chain rule property that composing two coordinate changes should yield a combined Jacobian equal to the product of the individual Jacobians:

Procedure for the Coordinate Consistency Check

Step One: Choose an Intermediate Coordinate System

Select coordinates $$x^i$$, an intermediate system $${\hat{x}}^i$$, and a final system $${\tilde{x}}^i$$, so that the transformation from the first to the third can be carried out either directly or by passing through the second.

Step Two: Transform in Two Different Ways

Compute the candidate tensor's components in the final coordinate system twice: once by applying the transformation law directly from the first system to the third, and once by applying it first from the first system to the intermediate system, and then separately from the intermediate system to the third.

Step Three: Confirm Agreement

Verify that both routes produce identical numerical components:

Why Agreement Confirms Genuine Tensor Behavior

Algebraic Guarantee for True Tensors

If the candidate object genuinely satisfies the tensor transformation law with a well-defined Jacobian factor, the composition property of Jacobians guarantees the two computed routes agree automatically, since the matrix product structure of the transformation law respects the chain rule by construction.

What Disagreement Would Indicate

If the two routes yield different results, this indicates that the transformation being applied is not a genuine, well-defined tensor law but instead depends on some extraneous feature of the specific coordinate path chosen, a hallmark of objects that fail to be true tensors despite superficially tensor-like notation.

Visual Overview

Diagram of the Two-Route Comparison

Extending the Check Across Coordinate Patches

Overlap Consistency on Manifolds

On a manifold covered by several overlapping coordinate charts, coordinate consistency verification also confirms that a candidate tensor's components, transformed between any two charts through their region of overlap, agree with a direct comparison whenever a third chart's overlap region provides an alternative path, matching the composition check across an entire network of charts rather than a single pair.

Practical Importance for Global Objects

This broader form of the check is what justifies treating a tensor field as a single, globally well-defined object across an entire manifold rather than as a disconnected collection of locally defined quantities that happen to agree by coincidence in each individual patch.

Summary of Key Traits

Defining Characteristics

• Coordinate consistency verification tests a candidate tensor's transformation law across multiple or chained coordinate changes, not just one.
• Genuine tensors automatically satisfy this check due to the composition property of the Jacobian matrices governing their transformation law.
• A discrepancy between direct and intermediate-routed transformations reveals that the candidate is not a well-defined tensor.
• The check extends to networks of overlapping coordinate charts, underpinning the treatment of tensor fields as globally consistent objects.