8.17.3 Tensor Index Position Matching

Tensor Index Position Matching is the specific layer of the index matching rule requiring that a free index sharing the same letter across two terms of a tensor equation must also occupy the same vertical position — upper or lower — in each of those terms, since matching letters alone, without matching position, does not guarantee that the two terms transform by the same law under a change of coordinates. It isolates the positional aspect of matching from the separate question of letter identity, addressing situations where the correct letter is used throughout an equation but its position is inconsistently assigned.

Statement of the Requirement

Same Letter, Different Position, Still a Failure

An equation such as

uses the identical letter $i$ on both sides, satisfying letter identity, yet fails position matching because $i$ is upper on the left and lower on the right. Despite the superficial similarity of the two sides, position matching identifies this as an invalid tensor equation, since the letter alone is not sufficient evidence that both sides refer to compatible geometric objects.

Position Must Agree Term by Term, Not Just Overall

In an equation with several terms, position matching requires that every individual term carry the shared free index in the identical position, not merely that the correct position appear somewhere across the collection of terms. In $A^{i} + B^{i}{}{j}C^{j} = D{i}$, the free index $i$ is upper in the first two terms but lower on the right-hand side, and the equation fails position matching even though $i$ appears in an upper position in a majority of its occurrences.

Why Position Carries Independent Significance

Position Encodes the Transformation Law

Because the upper and lower positions signal two distinct, mutually inverse transformation laws under a change of coordinates, two terms sharing a letter but differing in position are asserted, by their very notation, to transform in opposite ways. An equation equating such terms would need to hold true in every coordinate system despite the two sides responding oppositely to any change of coordinates — a condition that fails except in the trivial case where both sides vanish identically.

Distinguishing Position Mismatch From Letter Mismatch

A mismatch in letter (as in $A^{k} = B^{i}$) indicates that no relationship between the two occurrences has been established at all, since nothing links two entirely different symbols. A mismatch in position with a shared letter (as in $A^{i} = B_{i}$) is a more specific error: it indicates that the same conceptual slot has been referenced correctly by name but incorrectly with respect to how it must transform, revealing that one of the two terms has likely undergone an unaccounted-for change in transformation type, such as an implicit and unacknowledged raising or lowering.

Position Matching in the Presence of a Metric

Legitimate Resolution via Explicit Raising or Lowering

When two terms that should represent the same underlying free index are found in different positions, the discrepancy is often resolved — rather than treated as an unrecoverable error — by explicitly inserting the metric tensor to raise or lower one of the two occurrences into agreement with the other. Given $A^{i} = B_{i}$, position matching can be restored by rewriting the right-hand side as $A^{i} = g^{ij}B_{j}$, in which the previously lower free index $j$ has been raised to the required upper position $i$ using the metric.

Position Matching Is Not Automatically Waived by the Existence of a Metric

Even in a space equipped with a metric, where raising and lowering are always available, position matching remains a strict requirement on the equation as written; the existence of a metric provides a tool for repairing a position mismatch, but it does not mean position can be ignored or treated as interchangeable without that explicit repair being carried out and displayed in the notation.

Position Matching for Multiple Free Indices

Each Free Index Checked Independently

When an equation carries several free indices, position matching is verified separately for each one: one free index might correctly match in position across all terms while a different free index in the same equation fails to match, and each such discrepancy must be identified and addressed on its own rather than assuming that fixing one automatically resolves another.

Position Matching Within Staggered Mixed Indices

For tensors with several indices in a specific left-to-right order, such as $T^{i}{}{j}{}^{k}$, position matching must additionally respect which slot within that ordering each index occupies, since a term with the same letters but a different arrangement of upper and lower slots, such as $T{j}{}^{ik}$, represents a structurally different object even if, letter for letter, the same three symbols are present.

Practical Verification

A Targeted Check Once Letters Are Confirmed

Position matching is most efficiently checked as a second pass after letter identity has already been confirmed: having established that the same letters recur across the terms of an equation, the remaining task is simply to verify, for each shared letter, that its vertical placement — superscript or subscript — is identical in every term where it appears as a free index.

Role Within the Index Matching Rule

Position matching occupies the layer of the index matching rule between the coarse type-count check and the full letter-and-position correspondence required of a completely valid equation: it isolates and enforces the specifically positional component of that correspondence, ensuring that even when the correct letters have been used throughout an expression, the transformation behavior those letters are meant to signal has not been silently altered or overlooked in some of the terms being combined.