8.9.1 Tensor Summation Index Range

Tensor Summation Index Range defines the scope of indices in tensor operations, specifying how summation occurs across repeated indices in mathematical expressions.

Tensor Summation Index Range is the complete set of admissible values that a summation index is understood to sweep through when the Einstein summation convention is applied, with that set determined entirely by the dimension of the space in which the tensor expression is defined, even though the range itself is never written explicitly in the notation.

Determining the Range

Range Fixed by Ambient Dimension

The range of a summation index is set by the dimension of the vector space, manifold, or coordinate system underlying the expression: in an (n)-dimensional space, a summation index conventionally runs over the integer values from (1) to (n), or from (0) to (n - 1) depending on the indexing convention adopted for that context.

A_{i} B^{i} = \sum_{i = 1}^{n} A_{i} B^{i}

Range Implicit, Not Written

Unlike an explicit summation, which typically states its lower and upper bounds directly beneath and above the summation symbol, the summation index range under the Einstein convention is left entirely unstated in the expression itself and must be inferred from the surrounding context, such as the stated dimension of the space or the number of spatial coordinates being used.

Common Range Conventions

Spatial Indices

In ordinary three-dimensional space, a summation index associated with spatial coordinates conventionally ranges over three values, corresponding to the three independent spatial directions, typically labeled with Latin letters such as (i), (j), or (k).

A_{i} B^{i} = A_{1} B^{1} + A_{2} B^{2} + A_{3} B^{3}

Spacetime Indices

In contexts involving relativistic spacetime, a summation index associated with spacetime coordinates conventionally ranges over four values, one temporal and three spatial, typically labeled with Greek letters such as (\mu) or (\nu), a convention that visually distinguishes a four-valued range from a three-valued spatial range.

A_{μ} B^{μ} = \sum_{μ = 0}^{3} A_{μ} B^{μ}

Consistency of Range Across an Expression

Same Index, Same Range Throughout

Whenever a given summation index appears within an expression, its range must remain the same dimension throughout that expression; it is not permissible for the same index letter to silently switch between a three-valued spatial range and a four-valued spacetime range within a single coherent calculation.

Range Determines Family Size

The range of a summation index also determines how many individual scalar products are being added together in the implied sum, so a larger range corresponds to a longer, though still implicit, chain of additions collapsed into the single repeated-index expression.

Practical Illustration

Correctly identifying the summation index range for any given expression, by recognizing the dimension of the space implied by context or by the letter convention used for the index, is essential for evaluating a tensor expression numerically, since the compact notation itself provides no explicit statement of how many terms the implied sum actually contains.