8.6.5 Tensor Free Index Balance Requirement

The Tensor Free Index Balance Requirement ensures proper contraction and consistency in tensor algebra by balancing free indices across operations.

Tensor Free Index Balance Requirement is the condition that every additive term within a single tensor equation must expose exactly the same set of free indices, matching in name, matching in variance, and matching in count, so that no term contributes an index absent from the others and no term omits an index present elsewhere in the same equation.

Statement of the Requirement

Matching Free Indices Term by Term

For an equation combining several tensor expressions through addition or subtraction, the balance requirement demands that each term carry an identical roster of free indices, since a term with an extra or missing free index would represent an object of different rank than the others, making the sum meaningless as a single tensorial statement.

D_{i k} = A_{i k} + B_{i k} - C_{i k}

Every term here, on both sides of the equation, carries exactly the free indices (i) and (k), lower in variance, satisfying the balance requirement.

Matching Variance, Not Merely Matching Names

It is not sufficient for the same index letter to appear in every term; the balance requirement also demands that the variance match, since an index appearing as lower in one term and upper in another, even under the same letter, would indicate two structurally different objects being incorrectly combined.

P_{i} + Q^{i}

An expression of this form violates the balance requirement, since the first term carries a lower free index while the second carries an upper free index of the same name, and the two do not transform the same way under a change of basis.

Balance Requirement Across an Entire Equation

Both Sides Must Agree

The balance requirement applies not only among the terms on one side of an equation but across the equality itself: the free indices appearing on the left-hand side must exactly match, in name, variance, and count, the free indices appearing on the right-hand side.

W_{i} = g_{i j} U^{j}

The dummy index (j) is internal to the right-hand side and does not participate in the balance check, while the free index (i), lower in variance, is required to match precisely between the two sides.

Detecting Violations

A mismatch in the balance requirement, such as a free index present on one side but absent on the other, or present with opposite variance, signals either a typographical error or a genuine mathematical inconsistency in the proposed equation, and is one of the fastest checks available before attempting any deeper verification of a tensor identity.

Balance Requirement Under Common Operations

Preserved Under Addition, Scalar Multiplication, and Contraction

Operations such as adding two tensors of the same type, multiplying by a scalar, or contracting against another tensor using a properly paired dummy index, all preserve the balance requirement automatically, since none of these operations alters the set of free indices already present.

Broken by Careless Manipulation

The balance requirement is most often broken by manipulation errors, such as forgetting to include a free index in a newly introduced term, accidentally repeating an index that was meant to remain free, or transcribing an index with the wrong variance during a lengthy derivation.

Practical Verification

Verifying the free index balance requirement is a simple but essential first step whenever reading, deriving, or transcribing a tensor equation: listing the free indices, with their variance, present in each term of the equation and confirming that every list is identical catches the majority of notational errors before they propagate into an incorrect result.