8.5.3 Tensor Lower Index Component Reference

Tensor Lower Index Component Reference is the act of pointing to one specific numerical entry among a tensor's covariant components by fixing a subscript index to a particular value, as in referring to ω₂ from the general symbolic component ωᵢ. It is the covariant-side counterpart of component reference in general, narrowed specifically to the case where the fixed index occupies the lower, subscript position.

Fixing a Lower Index to a Value

From Symbolic Subscript to a Single Number

Writing ωᵢ symbolically describes the entire family of covariant components of a covector, one for each value the index i may take; substituting a specific value, such as 2, produces ω₂, a reference to exactly one number in that family, with no ambiguity about which entry is meant.

Distinguishing a Lower Reference from an Upper One

A reference such as ω₂ picks out a covariant component directly, while a reference such as ω^{2}, obtained instead from a contravariant version of the same underlying covector via raising with the inverse metric, generally refers to a numerically different value; the vertical position of the fixed index in a component reference therefore carries meaning just as much as it does in the symbolic, unfixed form.

Basis Dependence of a Lower Index Reference

The Value Is Tied to the Chosen Basis

Because covariant components are defined relative to a chosen basis, a specific lower index component reference such as ω₂ = 5 is only meaningful once the basis in which it was computed is specified; the same underlying covector generally has a different second covariant component in a different basis, so the reference alone, without its associated basis, does not carry complete information.

Recomputing a Reference After a Change of Basis

Under a change of coordinates, the value referred to by a fixed lower index reference is recomputed according to the covariant transformation law; ω₂ in one coordinate system and ω₂′ in another are generally different numbers, both legitimately referring to "the second covariant component" of the same covector, but relative to their own respective bases.

Diagram of a Lower Index Component Reference

Lower Index References in Practice

Numerical Lookup in Computational Settings

In software representing a covector's covariant components as an array, a lower index component reference corresponds directly to indexing into that array at the position matching the fixed value, with the array's own indexing convention (commonly starting at 0) matched against whatever convention the surrounding index set assumes.

Citing Specific Physical Quantities

A physical statement referencing one particular covariant component, such as citing one component of a metric tensor or a gradient at a point, is a lower index component reference in practice, and correctly interpreting it requires knowing which coordinate system, and hence which covariant basis, the cited numerical value is expressed relative to.

Relationship to Symbolic Lower Index Roles

A Reference Is Not Itself a General Statement

A specific lower index component reference, having had its index fixed, no longer participates in the free-index balancing or dummy-index summation rules that govern symbolic expressions; those rules apply to the general, unfixed form ωᵢ from which the reference was derived, not to the single fixed number ω₂ itself.

Reconstructing the General Form from Several References

Enumerating every possible lower index component reference for a given tensor, one for each value in its index set, recovers the complete symbolic component in general form; conversely, the symbolic form is best understood as a compact way of describing this entire collection of individual references all at once.