8.20.4 Tensor Index Range Expansion

Tensor Index Range Expansion is the step, preliminary to and underlying every other form of index expansion, of making explicit the concrete set of values — such as 1 through n, or 0 through n-1, or a labeled coordinate list like x, y, z — that a given index is understood to range over before any free-index equation or implicit summation can be written out in full. It is the operation of resolving the unstated convention that fixes how many values an index takes and which values those are, a fact that index notation itself does not display but that every subsequent expansion silently depends on.

Why the Range Must Be Fixed First

Index Notation Does Not State the Range Directly

An expression such as A^i_j gives no indication, by itself, of what values i and j may take; that information comes entirely from context, typically the stated or assumed dimension of the vector space on which the tensor is defined. Index range expansion supplies this missing piece explicitly, for example fixing that i, j ∈ {1, 2, 3} for a tensor defined on three-dimensional space, before any further expansion of free or dummy indices can proceed.

Range Expansion as a Prerequisite

Both free-index expansion and implicit-sum expansion require the index range as an input: free-index expansion needs to know how many separate equations to produce, and implicit-sum expansion needs to know how many terms its summation sign, once restored, actually contains. Index range expansion is therefore not typically performed as a separate, standalone written step but is the fact that must be settled — explicitly or by clear convention — before the other expansions can be carried out concretely rather than left in symbolic summation form.

Sources of the Range

Dimension of the Vector Space

In the most common setting, every index attached to a tensor built from a single vector space V and its dual ranges over the same set, of size equal to dim(V). Index range expansion in this setting reduces to a single fact — the dimension n — applied uniformly to every index of every tensor under discussion.

Coordinate-Specific or Labeled Ranges

In applied settings, the range is sometimes given as a list of named coordinates rather than bare integers, such as i ∈ {x, y, z} for three-dimensional Cartesian tensors or μ ∈ {t, x, y, z} for four-dimensional spacetime tensors with a distinguished time coordinate. Index range expansion in these cases must record the specific labels, not merely a count, since subsequent expanded equations are written using those labels as the subscripts or superscripts of individual components.

Restricted or Non-Uniform Ranges

Some conventions restrict certain indices to a subset of the full range, for instance letting Greek indices run over all spacetime coordinates while Latin indices are restricted to only the spatial coordinates within the same expression. Index range expansion must capture this distinction correctly, since expanding a Latin-indexed sum over the full spacetime range, or a Greek-indexed sum over only the spatial range, would produce an incorrect number of terms.

Effect on the Size of an Expansion

Range Determines Term and Equation Counts

Once the range of each index is fixed, the size of every subsequent expansion follows directly: a free index with range size n produces n separate equations, and a dummy index with range size n contributes a factor of n to the number of terms in any sum it appears in. Changing the assumed range — for example expanding a computation first in two dimensions for a simple check and then in four dimensions for the physically relevant case — changes every one of these counts proportionally, without changing the symbolic index expression itself.

where the product runs over each free index k and n_k is that index's range size, reducing to a single power of n when all indices share a common range.

Diagram of Range Expansion Preceding Other Expansions

Interaction With Dimension-Dependent Identities

Range Expansion Exposes Dimension Dependence

Certain tensor identities hold only for specific ranges, or take a different explicit form depending on the range chosen; for instance the fully contracted identity δ^i_i = n evaluates to a different concrete number for each different range size n. Index range expansion is what turns such a dimension-dependent symbolic identity into a specific numerical statement, and comparing the expansions obtained under two different assumed ranges is a standard way to confirm that a claimed identity's dependence on dimension has been stated correctly.

Necessary Step Before Numerical Computation

Any concrete numerical or symbolic evaluation of a tensor expression requires the index range to have been fixed, since a computer routine or a hand calculation must iterate over a definite, finite set of values for every index; index range expansion is thus the conceptual step that licenses treating an abstract index expression as a finite, computable object in the first place.