8.14.5 Tensor Index Range Convention

Tensor Index Range Convention is the set of customary, discipline-specific rules by which authors fix the starting point, alphabet, and default extent of index ranges before any tensor expression is written, so that the numerical meaning of an index range does not need to be restated in every formula. Unlike the dimension dependence of a range, which is a mathematical fact about the ambient space, the range convention is a matter of agreed practice: two authors working in identical dimensions can adopt different conventions and thereby assign different starting values or alphabets to what is otherwise the same underlying range.

Starting-Value Conventions

One-Based Indexing

The most common convention across pure mathematics and much of physics is to begin index ranges at $1$, so that an index in an $n$-dimensional space runs over

This one-based convention aligns naturally with the ordinal counting of basis vectors: the first basis vector is $e_1$, the second is $e_2$, and so on.

Zero-Based Indexing

In relativistic physics, a competing and equally standard convention starts spacetime indices at $0$, reserving that value for the time coordinate and running the remaining values over the spatial directions:

The choice of zero-based indexing here is not arbitrary: it reflects the convention of labeling the time coordinate $x^{0}$, distinguishing it structurally from the spatial coordinates $x^{1}, \dots, x^{n-1}$.

Alphabet-Based Conventions

Latin Indices for Spatial or General Ranges

A widespread convention uses lowercase Latin letters — $i, j, k, l, \dots$ — for indices ranging over purely spatial dimensions or over a general $n$-dimensional space without a distinguished time-like direction. This alphabet choice by itself signals to a reader, without further comment, that the governing range is expected to be the ordinary spatial or abstract dimension of the problem.

Greek Indices for Spacetime Ranges

A parallel and equally entrenched convention reserves lowercase Greek letters — $\mu, \nu, \rho, \sigma, \dots$ — specifically for indices ranging over the full set of spacetime coordinates, including the time-like direction. Under this convention, encountering a Greek letter as an index is itself sufficient to communicate that its range includes the $0$ (or time) value, while a Latin letter in the same document is understood to range over spatial values only.

Consequences of Mixing Alphabets Within One Discussion

Because Latin and Greek indices are conventionally assigned different ranges, a single expression that mixes both alphabets, such as $A^{\mu} = (A^{0}, A^{i})$, is a compact and standard way to split a full spacetime tensor into its time and spatial parts without restating any range explicitly; the alphabet convention alone carries this information.

Declaring and Overriding the Convention

Explicit Statement at the Outset of a Discussion

Because index range conventions vary between fields and even between individual texts, an author typically states the adopted convention once, near the beginning of a derivation or document — for example, declaring that "Latin indices run from 1 to 3" and "Greek indices run from 0 to 3" — after which every subsequent tensor expression in that document is interpreted according to the declared convention without repetition.

Local Overrides

An author may locally suspend the ambient range convention for a specific index by attaching an explicit range annotation to that index alone, temporarily overriding the inherited starting value or alphabet-based range for the scope of that particular expression, while leaving the general convention intact for every other index in the surrounding discussion.

Convention Versus Dimension Dependence

Two Independent Sources of Range Information

The numerical size of an index range is fixed by the dimension of the space, an objective mathematical fact independent of notation. The starting value and the alphabet used to write the index are matters of convention, chosen for readability and disambiguation rather than dictated by the mathematics itself. A four-dimensional spacetime has exactly four values for any index regardless of convention, but whether those four values are labeled ${0,1,2,3}$ with a Greek letter or ${1,2,3,4}$ with a Latin letter is a question of adopted practice, not of dimension.

Role Within Index Range Notation

The range convention supplies the human-readable scaffolding around the purely mathematical fact of dimension dependence: it fixes where counting begins and which alphabet signals which kind of range, so that readers within a given field can infer the intended range of an index from its letter and typical usage alone, without needing an explicit range clause attached to every single tensor expression they encounter.