6.16.2 Tensor Zero Two Component Index Pair

The Tensor Zero Two Component Index Pair denotes a specific arrangement of indices in tensor algebra, crucial for defining tensor components in multi-dimensional spaces.

Tensor Zero Two Component Index Pair is the pair of numerical labels i and j attached to each entry T_{ij} of a type (0,2) tensor once a basis has been fixed, identifying that entry's position within the full array of covariant components and providing the concrete, basis-dependent numerical description that complements the abstract, basis-independent tensor itself. This index pair is the same kind of bookkeeping device used for the type (2,0) classification, but placed entirely in the lower position, and understanding its behavior clarifies how symmetric and antisymmetric patterns, diagonal structure, and dimension counts are read directly off a covariant component array such as the metric.

Counting and Organizing the Components

Total Number of Components

If the underlying vector space V has dimension n, then each of the two lower indices i and j independently ranges over n values, giving:

number of components = n^{2}

exactly matching the component count of the type (2,0) classification, even though the two classifications transform oppositely under a change of basis.

Arranging the Pair into a Square Array

The component index pair is naturally arranged into a square array with n rows and n columns, the first index i selecting the row and the second index j selecting the column:

T = (\begin{matrix} T_{11} & T_{12} \\ T_{21} & T_{22} \end{matrix})

for n = 2. This array is the concrete presentation used, for instance, when a metric tensor is displayed as a matrix of inner products between basis vectors, T_{ij} = g(e_i, e_j).

Behavior of the Index Pair Under Relabeling and Transformation

Free Indices Versus Summed Indices

In the expression T_{ij} on its own, both i and j are free indices, ranging independently to enumerate every component. When contracted with vectors, as in T_{ij} v^i w^j, both i and j become summed dummy indices, since each is repeated once as a lower index on T and once as an upper index on a vector, leaving no free indices and producing a pure scalar.

Transformation of the Pair as a Unit

Under a change of basis with transition matrix A, the entire component index pair transforms together, each index picking up its own factor of A:

T_{' i j} = A_{i}^{k} A_{j}^{l} T_{k l}

so that, as with the type (2,0) case, each new component is generally a mixture of all n² old components rather than a single relabeled entry.

Symmetry Patterns Detected Through the Index Pair

Diagonal and Off-Diagonal Entries

Within the component index pair, entries with i = j lie on the diagonal, such as T_{11} and T_{22}, while entries with i ≠ j lie off the diagonal. A tensor is symmetric when swapping the pair leaves each component unchanged, T_{ij} = T_{ji}, which is the condition satisfied by every metric tensor, and antisymmetric when the swap negates the component, T_{ij} = -T_{ji}, forcing all diagonal entries to vanish.

Independent Components Under Symmetry

For a symmetric type (0,2) tensor, the component index pair is effectively unordered, reducing the count of independent components from n² to n(n+1)/2; for a metric in a three-dimensional space, this means only six independent numbers, T_{11}, T_{22}, T_{33}, T_{12}, T_{13}, T_{23}, are needed rather than nine. For an antisymmetric tensor, the count reduces further to n(n-1)/2.

Visualizing the Index Grid

Distinguishing the Index Pair from a Generic Array of Numbers

What the Index Pair Alone Does Not Guarantee

A square grid of n² numbers labeled with two subscripts is not automatically a type (0,2) tensor; the defining property is that the labeled entries transform according to the covariant rule under every change of basis. This transformation requirement is what distinguishes a genuine tensor such as a metric from a table of numbers that merely happens to be indexed by two subscripts without an associated transformation law.

Role of the Index Pair in Computation

Once the transformation law has been verified, the component index pair becomes the operational language for every computation involving the tensor: computing inner products via T_{ij} v^i w^j, checking symmetry via direct comparison of T_{ij} and T_{ji}, and constructing an inverse via the matrix inverse of the array T_{ij} all proceed by direct manipulation of the indices i and j.