11.19.5 Tensor Variance Convention Consistency Requirement

Ensuring tensor variance conventions are consistent across operations is essential for accurate mathematical modeling and clear notation in algebraic structures.

Tensor Variance Convention Consistency Requirement is the demand that every index appearing in a tensor equation obey the variance convention uniformly throughout the entire expression, so that free indices match in position on every term and repeated indices always appear once upper and once lower, without exception anywhere in the equation.

Foundational Setting

Consistency as a Precondition for Meaning

An equation involving indexed quantities is only interpretable as a tensor equation if it satisfies the consistency requirement. Without this uniform adherence to the convention, the individual terms of an equation could transform differently from one another under a change of basis, making any claimed equality between them meaningless.

The Two Facets of Consistency

The requirement has two distinct facets: consistency of free indices, which must occupy matching positions across every term, and consistency of repeated indices, which must always appear in exactly one upper and one lower position within any single term.

Free Index Consistency

Same Position Across All Terms

If a free index $i$ appears as an upper index in one term of an equation, it must appear as an upper index in every other term as well. Consider an equation of the form:

v^{i} = u^{i} + w^{i}

Here the free index $i$ is upper in every term, satisfying the requirement. An expression pairing $v^{i}$ with $u_{i}$ as if they could be added directly would violate it, since the two terms would transform by different, unmatched laws.

Why Violations Undermine Invariance

If free index positions were mismatched across terms, applying a basis change would multiply one term by the basis-change matrix and another by its inverse, so the two sides of the equation would generally cease to agree numerically after the transformation, breaking the equation's status as a valid, basis-independent statement.

Repeated Index Consistency

Exactly One Upper and One Lower Occurrence

Whenever an index letter is repeated within a single term, the consistency requirement demands that one occurrence be upper and the other lower, triggering the implicit summation and guaranteeing an invariant result:

\sum_{i} ω_{i} v^{i}

Disallowed Repetition Patterns

A term in which the same letter appears twice as an upper index, or twice as a lower index, does not satisfy the consistency requirement and is generally regarded as an ill-formed expression, since no legitimate contraction or invariance argument applies to such a repetition.

Visual Overview

Diagram of the Consistency Checks

Consistency Requirement in Composite Expressions

Checking Products and Sums Together

In an equation combining tensor products, contractions, and sums, the consistency requirement applies to the entire assembled expression: every free index arising after all contractions have been carried out must still match in position across every additive term, and every summed index within any individual product must still appear once upper and once lower.

A Practical Diagnostic

Because the requirement is purely structural, checking it does not require evaluating any numerical components, only inspecting the pattern of index letters and their vertical positions, making it a fast preliminary diagnostic for catching malformed tensor equations before any detailed calculation is attempted.

Consequences for Physical and Geometric Laws

Guaranteeing Coordinate Independence

A physical or geometric law expressed as a tensor equation that satisfies the consistency requirement is guaranteed to hold in every coordinate system once it is verified in one, since consistent index structure is precisely what ensures both sides of the equation transform identically under any change of basis.

Immediate Rejection of Malformed Candidates

Conversely, any proposed law that fails the consistency requirement cannot be a legitimate tensor equation regardless of how plausible it appears numerically in one particular coordinate system, since its two sides would not remain equal after a change of basis.

Summary of Key Traits

Defining Characteristics

Free indices must occupy the same vertical position on every term of a valid tensor equation.
Repeated indices within a single term must appear exactly once upper and once lower.
The requirement can be checked purely structurally, without evaluating numerical components.
Satisfying the requirement guarantees an equation's validity is preserved under any change of basis or coordinate system.