11.20 Tensor Variance Verification Procedure

Tensor Variance Verification Procedure is the systematic set of checks applied to a candidate mathematical object or equation to confirm that it genuinely qualifies as a tensor of the claimed variance type, testing whether its components truly obey the covariant or contravariant transformation law under an explicit change of basis rather than merely resembling a tensor in notation alone.

Foundational Setting

Why Verification Is Necessary

Not every quantity written with indices is automatically a tensor. Some index-bearing objects, such as connection coefficients, fail to satisfy any consistent transformation law, while others transform correctly only under a restricted class of basis changes. The verification procedure exists to distinguish genuine tensors from such look-alikes.

The Core Question Being Tested

At its heart, the procedure asks a single question for a claimed contravariant object $$v^i$$: does an explicit change of basis produce components related to the original ones exactly according to the inverse-matrix transformation law, with no additional terms left over?

The Verification Steps

Step One: Introduce an Explicit Basis Change

Select a general, invertible matrix $$A$$ relating an old basis to a new basis:

Step Two: Compute the New Components Directly

Using the definition of the candidate object independent of any assumed transformation law, compute its components in the new basis directly from first principles, rather than by simply asserting the desired transformation rule.

Step Three: Compare to the Claimed Law

Check whether the directly computed new components match exactly what the claimed transformation law would predict:

If this equality holds for every admissible choice of $$A$$, the object passes verification as a genuine contravariant tensor of that rank.

Applying the Procedure to a Known Counterexample

Testing the Connection Coefficients

Applying the same three steps to a set of connection coefficients reveals that the directly computed new components differ from the prediction of either the covariant or contravariant law by an additional, inhomogeneous term:

The verification procedure correctly flags this object as failing to qualify as a tensor, despite its index notation resembling one.

Visual Overview of the Procedure

Flow of the Verification Steps

Extending the Procedure to Mixed and Higher-Rank Tensors

Verifying Each Index Separately

For a mixed tensor with several indices, the verification procedure is applied index by index: each upper index is checked against the inverse-matrix law, and each lower index is checked against the direct-matrix law, with the object confirmed as a tensor of its claimed type only if every index passes its respective check simultaneously.

Verifying Invariance of Contractions

As a complementary check, the procedure can be applied to a fully contracted expression by confirming that its value, computed directly in two different bases, agrees exactly, providing indirect confirmation that the individual factors involved transform with correctly matched, mutually inverse laws.

Summary of Key Traits

Defining Characteristics

• The procedure tests whether directly computed new-basis components match the components predicted by a claimed transformation law.
• Verification must hold for every admissible basis change, not merely a single special case, to confirm genuine tensor status.
• Objects such as connection coefficients fail verification due to an inhomogeneous extra term, distinguishing them from true tensors.
• For mixed and higher-rank objects, each index is verified independently, and invariance of full contractions serves as a complementary consistency check.