10.20 Tensor Transformation Consistency Check

Tensor Transformation Consistency Check is the collection of practical verification procedures applied after carrying out a tensor coordinate transformation to confirm that the computed result is correct, relying on properties that any genuine tensorial transformation must satisfy, such as invariance of contracted scalars, round-trip recoverability, and agreement across alternative computational routes.

Invariance of Contracted Scalars

The Basic Contraction Test

The most direct consistency check forms any full contraction available in the transformed tensor expression, typically pairing an upper index with a lower index, and verifies that the resulting scalar is numerically identical whether computed from the original components or the newly transformed ones:

A mismatch here immediately signals an error somewhere in the transformation, either in the Jacobian entries used or in the assignment of forward versus inverse factors to the wrong index type.

Checking Multiple Independent Contractions

When a tensor admits more than one possible contraction, checking each one independently strengthens the consistency check, since an error affecting only one particular index pairing might otherwise go undetected if only a single contraction were verified.

Round-Trip Recovery

Transforming Back to the Source Chart

A second consistency check applies the reverse transformation, using the inverse Jacobian in place of the forward Jacobian and vice versa, to the newly computed components, and confirms that the original components are recovered exactly:

This check relies directly on the Jacobian product identity and confirms that the forward and inverse matrices used in the computation are genuinely consistent with one another, catching errors such as an incorrectly computed matrix inverse.

Sensitivity to Small Numerical Errors

Because round-trip recovery involves two successive matrix multiplications, small numerical errors introduced in computing the Jacobian entries can accumulate and appear amplified in this check, so an exact match is expected only when the Jacobian entries themselves were computed to sufficient precision or in exact symbolic form.

Agreement Across Alternative Routes

Direct Versus Composed Transformation

When an intermediate coordinate chart is available, transforming a tensor directly from the source chart to the target chart and, separately, transforming it through the intermediate chart in two steps, provides a third consistency check: agreement between the two routes confirms the correctness of both the direct Jacobian relation and the two intermediate Jacobian relations used.

Isolating the Source of an Error

If the two routes disagree, comparing intermediate results computed along the two-step route against independently verified values for the intermediate chart helps isolate which specific Jacobian relation, rather than the overall transformation process, contains the underlying error.

Diagram of the Consistency Checks

Three Independent Verification Paths

Checking Symmetry and Structural Properties

Preservation of Index Symmetry

If the original tensor is known to be symmetric or antisymmetric in a pair of indices, confirming that the newly transformed components retain the identical symmetry or antisymmetry pattern provides an additional structural consistency check, since this preservation is guaranteed automatically by the tensorial rule and its absence would indicate a computational error.

Dimensional and Unit Sanity Checks

Where the tensor components carry physical units, checking that the transformed components carry dimensionally consistent units, correctly adjusted for any change of coordinate scale implied by the transformation, offers a further practical sanity check independent of the purely mathematical verifications described above.

When Consistency Checks Fail

Distinguishing Genuine Errors From Non-Tensorial Quantities

A persistent, systematic failure of the contraction invariance check, one that cannot be traced to an arithmetic mistake in the Jacobian entries, may indicate that the quantity being transformed does not actually obey the tensorial transformation rule at all, exhibiting instead the non-tensorial transformation pattern, in which case the consistency check has correctly revealed a mismatch between the assumed and actual transformation behavior of the quantity.