6.9.2 Tensor Contravariant Index Position

Tensor Contravariant Index Position is the specific place, among all the upper indices carried by a tensor's components, that a given contravariant index occupies relative to the others, this placement determining which one-form argument slot the index labels and how the index behaves under operations that treat the different upper slots asymmetrically. Two contravariant indices belonging to the same tensor are distinguished from one another not by any difference in their transformation law, since every upper index transforms by the same direct-Jacobian rule, but by their position in the ordered sequence of upper slots, and this position is precisely the information tracked by the contravariant index position.

What Index Position Records

Position as an Ordering Label

A tensor with several upper indices requires some means of telling its indices apart beyond their shared transformation behavior, since all contravariant indices obey the identical rule under a change of coordinates. The device used for this is position: the indices are written in a fixed order in the component notation, and the position of an index in that order, first, second, third, and so on, is what identifies which argument slot it corresponds to and what distinguishes it from every other upper index on the same tensor.

In this expression the index in the first position is conventionally called a, the index in the second position b, and the index in the third position c, and this positional assignment is exactly what is meant by the covariant index position for the contravariant case, applied here to upper indices.

Position Versus Identity of Value

The letter chosen to name an index, and the numerical value it takes when a particular dual basis one-form is substituted, are both secondary to its position: renaming every occurrence of an index consistently leaves the tensor's meaning unchanged, but moving an index from one position to another, without also permuting the corresponding slot of the underlying multilinear map, generally changes which tensor is being described. Position is the invariant structural fact, while the letter used is only a notational convenience.

Position Sensitivity and Its Consequences

General Tensors Are Position-Sensitive

For a tensor without any special symmetry, exchanging the one-forms fed into two upper slots that occupy different positions produces, in general, a different scalar. Correspondingly, exchanging the position labels of two upper indices in the component notation produces, in general, a genuinely different array of numbers, not merely a relabeling of the same one. This position sensitivity is the generic case, and it is the reason the ordered sequence of upper indices must be tracked explicitly.

Symmetric and Antisymmetric Exceptions

Tensors that are symmetric in a given pair of upper positions satisfy equality of components under exchange of those two specific positions, while tensors antisymmetric in a pair of upper positions acquire a sign change under that exchange. These properties are always stated relative to particular positions: a tensor may be symmetric in its first and second upper positions while showing no special relation between its first and third, so position sensitivity or its absence must be specified position pair by position pair.

Position and the Placement of Contracted Indices

When an upper index at a given position is contracted against a lower index of another tensor, only the one-form slot at that particular position is saturated; the remaining upper positions retain their open, unsaturated status. Contraction therefore acts on contravariant index position selectively, and the resulting tensor's remaining upper indices inherit the relative ordering of whichever positions were not involved, with the contracted position simply removed from the sequence.

Position Under Structural Operations

Position Under Change of Basis

A change of coordinates applies the same direct-Jacobian transformation factor to every upper index regardless of position, but it applies a separate copy of that factor to each position independently, with the dummy index summed in that copy matching only the original index that occupied the same position. Position is thus preserved by a coordinate transformation: the index in the first position remains, after transformation, associated with the first slot.

Position Under Tensor Product Formation

Forming a tensor product of two contravariant tensors places all of the upper positions belonging to the first factor before all of the upper positions belonging to the second factor, according to a fixed convention, so that the position of every index in the resulting tensor is determined jointly by its original position within its own factor and by which factor it came from.

Explicit Permutation of Position

An operator that deliberately exchanges the one-forms assigned to two chosen upper positions, without altering anything else about the tensor, produces a new tensor whose components are obtained from the original by swapping the two corresponding position labels throughout. Such an explicit permutation is the standard tool used to define symmetrization and antisymmetrization on contravariant indices: the symmetric part in two upper positions is built by averaging the original with its position-swapped version, and the antisymmetric part is built by taking half the difference between the two.