10.19.3 Tensor Non Tensorial Rule Violation

Tensor Non Tensorial Rule Violation is the specific mismatch that occurs when an indexed quantity's actual transformation law, derived directly from its own definition, fails to equal the transformation law predicted by the general tensorial transformation rule for an object carrying that same arrangement of upper and lower indices, establishing formally that the quantity is not a tensor.

Formally Establishing a Violation

The Comparison Test

A rule violation is demonstrated by deriving the true transformation law of a quantity from first principles, typically by differentiating its defining formula and applying the chain rule, and then comparing the result term by term against what the tensorial rule alone would predict for an object with the same index pattern:

If the actually derived transformation law contains this predicted expression plus any additional surviving term, a violation of the tensorial rule has been established for that quantity.

Sufficiency of a Single Counterexample

Establishing a rule violation requires only a single explicit coordinate transformation for which the actual and predicted transformation laws disagree; unlike proving that a quantity does obey the tensorial rule, which by the quotient rule requires checking every possible contraction, disproving tensorial status requires exhibiting just one transformation, or one contraction, where the predicted invariance fails.

Categorizing Violations by Their Origin

Violations From Genuine Second-Derivative Content

The most common source of violation, exemplified by the Christoffel symbols, arises because the quantity's own definition intrinsically involves how the coordinate basis itself varies from point to point, introducing second derivatives of the coordinate map into its transformation law that have no counterpart in the purely first-derivative-based tensorial rule.

Violations From Explicit Coordinate Reference

A second source of violation occurs when a quantity is defined using an explicit, non-covariant reference to a particular coordinate system, such as an ordinary partial derivative of a tensor component taken without any correction for the variation of the basis, producing a quantity whose transformation law depends on the specific coordinates used in its very definition rather than only on the underlying geometric content.

Diagram of a Detected Violation

Predicted Versus Actual Transformation

Consequences of a Confirmed Violation

Loss of Coordinate-Free Meaning

Once a rule violation is confirmed, no single component of the quantity, and no naive contraction built from it alone, can be assigned coordinate-independent geometric meaning, since its numerical value depends in part on the arbitrary coordinate system chosen, a direct consequence detailed as the non-tensorial coordinate dependence exhibited by such quantities.

Restriction on Permitted Uses

A quantity confirmed to violate the tensorial rule cannot be substituted freely into an otherwise valid tensor equation without care, since doing so risks producing an equation whose two sides transform differently under a change of basis, and any correct usage of such a quantity, such as its role inside a covariant derivative, must be arranged so that its non-tensorial extra term is explicitly cancelled by a matching term elsewhere in the expression.

Confirming the Absence of a Violation

When No Violation Is Found

If, after deriving the true transformation law and comparing it against the tensorial-rule prediction, no extra term survives for any coordinate transformation tested, and the quotient rule test also succeeds across every relevant contraction, the quantity is confirmed to be a genuine tensor, with no rule violation present.

Re-Examining Borderline Cases

Some quantities appear to violate the tensorial rule under one class of coordinate transformations, such as general curvilinear changes, while satisfying it under a more restricted class, such as changes of basis that are themselves linear, so establishing or ruling out a violation requires specifying clearly which class of admissible transformations is being tested, since a quantity's tensorial status can depend on this scope.