8.4.3 Tensor Upper Index Component Reference

Tensor Upper Index Component Reference is the act of pointing to one specific numerical entry among a tensor's contravariant components by fixing a superscript index to a particular value, as in referring to v^{2} from the general symbolic component v^{i}. It is the contravariant-side counterpart of component reference in general, narrowed specifically to the case where the fixed index occupies the upper, superscript position.

Fixing an Upper Index to a Value

From Symbolic Superscript to a Single Number

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

Distinguishing an Upper Reference from a Lower One

A reference such as v^{2} picks out a contravariant component, obtained without use of the metric, while a reference such as v_{2}, obtained instead from a covariant version of the same underlying vector via lowering with the 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 an Upper Index Reference

The Value Is Tied to the Chosen Basis

Because contravariant components are defined relative to a chosen basis, a specific upper index component reference such as v^{2} = 5 is only meaningful once the basis in which it was computed is specified; the same underlying vector generally has a different second contravariant 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 upper index reference is recomputed according to the contravariant transformation law; v^{2} in one coordinate system and v^{2′} in another are generally different numbers, both legitimately referring to "the second contravariant component" of the same vector, but relative to their own respective bases.

Diagram of an Upper Index Component Reference

Upper Index References in Practice

Numerical Lookup in Computational Settings

In software representing a vector's contravariant components as an array, an upper 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 contravariant component, such as citing the radial component of a velocity field at a point, is an upper index component reference in practice, and correctly interpreting it requires knowing which coordinate system, and hence which contravariant basis, the cited numerical value is expressed relative to.

Relationship to Symbolic Upper Index Roles

A Reference Is Not Itself a General Statement

A specific upper 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 v^{i} from which the reference was derived, not to the single fixed number v^{2} itself.

Reconstructing the General Form from Several References

Enumerating every possible upper 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.