9.8.3 Tensor Basis Tensor Index Address

Tensor Basis Tensor Index Address is the labeling of every slot belonging to a composite basis tensor by an index bearing both a numerical value and an upper-or-lower type, recording simultaneously which basis or dual-basis element occupies that slot and whether that slot accepts a vector or a covector argument; it extends the plain multi index address of a tensor product basis element by attaching, to each entry of the tuple, the type information needed to interpret the basis tensor correctly as a mixed object with a specific number of upper and lower slots.

What the Index Address Records

Numerical Value and Type Together

For a composite basis tensor such as e_i ⊗ e^j ⊗ e_k, the index address is not simply the tuple (i, j, k) of numerical values, but that tuple together with the type of each entry: i and k are recorded as lower-type indices, while j is recorded as an upper-type index, since each arises from a different kind of factor.

Fixing the Type of the Composite Tensor Overall

Reading the type of every entry in a basis tensor's index address in sequence gives directly the overall type of the composite tensor: the count of lower-type entries fixes how many covector slots it has, and the count of upper-type entries fixes how many vector slots it has.

Why Numerical Value Alone Is Insufficient

The Same Numeral Can Occupy Either Type of Slot

Because index values are typically drawn from the same range regardless of whether they label a primal or a dual basis element, the bare numeral in a tuple cannot by itself indicate whether a given slot is a vector or a covector slot; only the addition of type information to the index address resolves this ambiguity.

Type Information Determines Which Transformation Rule Applies

Whenever a basis tensor's components must be transformed under a change of coordinate basis, the index address's type information determines which of the two transformation rules — the one for upper indices or the one for lower indices — applies to each entry, so omitting type information would leave the correct transformation undetermined.

Index Address and the Ordered Tuple Role

Position Combines With Type to Fully Specify a Slot

The index address builds directly on the ordered tuple role of the tensor product basis: position within the tuple fixes which factor an entry belongs to, while the type attached to that entry fixes whether the factor is a primal or dual basis element, so that position and type together give a complete specification of every slot.

Reordering Requires Carrying Types Along

Whenever the slots of a basis tensor are permuted, whether to reflect symmetry considerations or a change of convention, the type attached to each entry of the index address must be carried along with its numerical value, since a permutation that moves values without their associated types would misassign vector slots as covector slots or the reverse.

Diagram of the Index Address

Consequences of a Well Formed Index Address

It Removes Ambiguity From Component Notation

A fully specified index address, giving both value and type for every slot, removes any ambiguity about how many upper and lower indices a component array carries and in what order, allowing an expression such as T^i_{jk} to be read unambiguously as having one upper and two lower slots in that specific sequence.

It Enables Correct Contraction and Raising or Lowering

Only once the type of each index is known from its address can operations such as contracting a specific upper index against a specific lower index, or raising a particular lower index using the metric, be carried out correctly, since these operations are defined in terms of index type rather than numerical value alone.