8.22 Tensor Indexed Equation Notation

Tensor Indexed Equation Notation is the overall convention by which a full equality between two tensor expressions is written using index notation, combining an equals sign with the rules governing free indices, dummy indices, and term structure so that the resulting statement is unambiguous, well-formed, and correctly indicates which components of one side correspond to which components of the other. It is the framework within which individual index expressions — vectors, contractions, sums of terms — are assembled into complete assertions of equality, and it is this framework that determines whether such an assertion is even grammatically meaningful before any question of its truth arises.

The Two Requirements of a Well-Formed Equation

Matching Free Index Sets on Both Sides

The first requirement of indexed equation notation is that the free index set of the left-hand side must equal the free index set of the right-hand side, letter for letter and variance for variance, since each side of the equation must represent the same tensor type and the same specific set of surviving component labels. An equation such as v^i = A^i_j w^j is well-formed under this requirement because both sides share the single free index i in the upper position.

Internally Consistent Term Structure Within Each Side

The second requirement is that every additive term within each side must itself be internally consistent: any repeated letter within a single term must appear exactly twice with opposite variance (making it a dummy index for that term), and every term added within one side must reduce, after its own contractions, to the same free index set as every other term in that sum. These two requirements together, applied consistently, define what it means for a string of symbols to be a legitimate tensor equation rather than a meaningless juxtaposition of indices.

Reading an Indexed Equation as a Family of Scalar Equations

One Equals Sign, Many Underlying Equalities

A single indexed equation with a nonempty free index set is understood as an abbreviation for a whole family of ordinary scalar equalities, one for each combination of values the free indices can take; the notation's compactness comes precisely from writing this entire family with a single equals sign rather than listing every instance separately. The equation v^i = A^i_j w^j in three dimensions stands for exactly three separate scalar equalities, one for v¹, one for v², and one for v³.

The Implicit Universal Quantification

Indexed equation notation carries an implicit universal quantifier over its free indices: the equation is asserted to hold for every value each free index can independently take, not merely for some particular value, which is why establishing an indexed equation typically requires an argument valid for a generic, unspecified value of each free index rather than a check at one specific value alone.

Combining Equations Under the Notation

Adding, Subtracting, and Multiplying Indexed Equations

Two indexed equations sharing the same free index set can be added or subtracted, term by term, to produce a new valid indexed equation with that same free index set; an indexed equation may also be multiplied through by an additional tensor factor carrying its own, previously unused indices, which then simply appear as additional free (or, if contracted against something already present, dummy) indices in the resulting equation. These operations mirror ordinary algebraic manipulation of scalar equations, applied uniformly across the entire implicit family of component equalities at once.

Substitution of One Expression for Another

A term or an entire side of an indexed equation may be replaced by an equivalent expression established elsewhere, provided the substituted expression carries exactly the same free indices, with the same variance, as what it replaces; this substitution rule is what allows chains of indexed equations, each derived from the previous one, to be assembled into longer tensor derivations while preserving well-formedness at every step.

Diagram of an Indexed Equation Unpacking Into Its Family

Distinguishing Notation From Content

The Notation Does Not Assert Truth, Only Grammaticality

Indexed equation notation is a purely syntactic framework: an expression can be perfectly well-formed under its rules — matching free index sets, consistent term structure — while still being a false statement about the tensors involved. Well-formedness under the notation is a necessary precondition for an equation to be meaningfully checked for truth, but it is not itself a guarantee of truth, in the same way that a grammatically correct sentence in ordinary language need not be a true one.

The Standard Vehicle for Stating Tensor Identities

Because it compresses an entire family of component-wise equalities into a single line while remaining precise about variance and summation, indexed equation notation is the standard form in which tensor identities — from elementary ones such as the symmetry of the metric to advanced ones such as the Bianchi identities of curvature — are stated, derived, and communicated throughout the subject.