11.13.2 Tensor Covariant Slot Lower Index Association

Tensor Covariant Slot Lower Index Association links indices to covariant components, defining how tensors transform in coordinate systems.

Tensor Covariant Slot Lower Index Association is the direct correspondence, fixed by notational convention, between each lower index appearing on a tensor's symbol and one specific covariant slot of that tensor viewed as a multilinear map, ensuring that the written position of an index unambiguously identifies which argument position expects a contravariant vector.

Definition and Notational Basis

The Correspondence Stated Precisely

Every distinct lower index written on a tensor's symbol is associated with exactly one covariant slot, and the order in which lower indices are written from left to right is taken, by convention, to match the order in which the corresponding vector arguments are supplied to the multilinear map.

T_{i j} (u, v) = T_{i j} u^{i} v^{j}

Distinguishing Index Labels From Slot Identity

Although two different letters may be used to label two different lower indices, the association is with the slot's position, not the specific letter chosen, so relabeling a lower index with a different letter does not change which covariant slot it identifies, provided its position among the indices is preserved.

Consequences of the Association

Enabling Unambiguous Partial Evaluation

Because each lower index is tied to a specific slot, a tensor with several lower indices can have just one of those slots filled with a vector while the others remain open, and the lower index association makes clear exactly which slot has been filled and which slots still await an argument.

T_{i j} u^{i} = S_{j}

Consistency Requirement in Symmetric and Antisymmetric Tensors

For tensors with special symmetry properties among their lower indices, the lower index association must still be respected when applying the symmetry, since exchanging the roles of two covariant slots is only meaningful once each slot's association with a specific index label has been correctly identified beforehand.

Interaction With Coordinate Transformation

Association Preserved Under a Change of Coordinates

When a tensor's components are transformed to a new coordinate system, the lower index association with its covariant slots is preserved exactly, since the covariant transformation law updates each index's numerical value while leaving unchanged which slot that index refers to.

Association Independent of the Specific Vector Supplied

The lower index association identifies a slot as a fixed position in the tensor's structure, entirely independent of which particular vector is eventually supplied to fill that slot, so the association remains stable even as different vectors are substituted into the same covariant slot across different calculations.

Role Within Tensor Algebras

Supporting Precise Communication of Tensor Structure

The lower index association provides the precise bookkeeping needed to communicate unambiguously which covariant slot of a tensor is being referenced in any given expression, which becomes essential once tensors possess more than one covariant slot and careful tracking is required.

Foundation for Defining Operations on Specific Slots

Operations such as contraction, symmetrization, or index raising that target one particular index of a tensor rely entirely on the lower index association to specify correctly which covariant slot is being acted upon, distinguishing it from any other covariant or contravariant slot the tensor may possess.