1.2.57 Tensor Component Index Definition

Tensor Component Index Definition is the specification of what an individual index attached to a tensor component represents, how it is positioned as upper or lower, what range of values it takes, and how it participates in the summation conventions and transformation rules that govern tensor algebra. An index is a label, typically a single letter such as i, j, or k, attached to a tensor symbol as a superscript or subscript, and each index picks out one particular direction of the multidimensional array of numbers that constitutes the tensor's components.

The Role of an Index

Index as a Coordinate Selector

Each index attached to a tensor component selects one entry along one of the tensor's dimensions. A tensor of type (p, q) has p + q indices in total, and fixing a numerical value for every index picks out a single scalar entry from the full component array.

Here, i^1, ..., i^p are the upper indices and j_1, ..., j_q are the lower indices, each independently taking a value from 1 to n, where n is the dimension of the underlying vector space.

Index Range

An index ranges over the integers from 1 to n, where n equals the dimension of the vector space V on which the tensor is built. If the space has dimension 4, for instance, every index independently ranges over the values 1, 2, 3, 4, and the tensor's full component array has as many entries as n raised to the power of the total number of indices.

Upper Indices

Definition

An upper index, written as a superscript, marks a contravariant slot of the tensor. It corresponds to one factor of the vector space V in the tensor product defining the tensor's type.

Transformation Behavior

Under a change of basis governed by a matrix A, an upper index transforms using the matrix A itself, following the same rule used to convert old vector components into new ones.

Lower Indices

Definition

A lower index, written as a subscript, marks a covariant slot of the tensor. It corresponds to one factor of the dual space V* in the tensor product defining the tensor's type.

Transformation Behavior

Under the same change of basis, a lower index transforms using the inverse matrix A^{-1}, following the rule used to convert old dual-basis components into new ones.

Free Indices and Dummy Indices

Free Indices

A free index is an index that appears once in a given term and is not summed over. It labels which component of the resulting tensor equation is being described, and it must appear consistently, with the same letter in the same position, upper or lower, on every term of a valid tensor equation.

Dummy Indices

A dummy index, also called a summation index, is an index that appears exactly twice in a single term, once as an upper index and once as a lower index, signaling that the term is summed over all values of that index according to the Einstein summation convention.

The letter chosen for a dummy index carries no meaning outside the term in which it appears and may be freely renamed, provided the new letter does not collide with another index already in use in the same expression.

Index Repetition Rule

A valid tensor expression under the Einstein summation convention never repeats an index more than twice within a single term, and any repetition must pair one upper occurrence with one lower occurrence; repeating an index twice in the same position, both upper or both lower, is not a valid summation and signals an error in the expression.

Index Position and Meaning

Position Encodes Slot Identity

The order in which indices are written, left to right among the upper indices and left to right among the lower indices, encodes which argument slot of the multilinear map each index corresponds to. Swapping the order of two indices generally produces a different component unless the tensor possesses a symmetry that equates them.

Symmetric and Antisymmetric Index Behavior

When a tensor's components are unchanged under the exchange of two indices of the same type, the tensor is called symmetric in those indices. When the components change sign under such an exchange, the tensor is called antisymmetric, or skew-symmetric, in those indices.

Diagrammatic Summary

The diagram shows a tensor symbol T carrying upper indices, associated with contravariant slots, and lower indices, associated with covariant slots, each index independently ranging over the dimension of the underlying vector space.