6.8.2 Tensor Covariant Index Position

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

What Index Position Records

Position as an Ordering Label

A tensor with several lower indices requires some means of telling its indices apart beyond their shared transformation behavior, since all covariant indices obey the identical rule under a change of coordinates. The device used for this is position: the indices are written in a fixed left-to-right 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 lower 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 of each one.

Position Versus Identity of Value

The letter chosen to name an index, and the numerical value it takes when a particular basis vector is substituted, are both secondary to its position: renaming every occurrence of an index consistently, or relabeling the dummy indices used in a summation, 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 therefore 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 vectors fed into two lower slots that occupy different positions produces, in general, a different scalar. Correspondingly, exchanging the position labels of two lower 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 lower indices must be tracked explicitly rather than treated as an unordered collection.

Symmetric and Antisymmetric Exceptions

Tensors that are symmetric in a given pair of positions satisfy equality of components under exchange of those two specific positions, while tensors antisymmetric in a pair of 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 lower 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 rather than asserted for the tensor as an undifferentiated whole.

Position and the Placement of Contracted Indices

When a lower index at a given position is contracted against an upper index of another tensor, only the vector slot at that particular position is saturated; the remaining lower positions retain their open, unsaturated status. Contraction therefore acts on covariant index position selectively, and the resulting tensor's remaining lower indices inherit the relative ordering of whichever positions were not involved in the contraction, 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 inverse-Jacobian transformation factor to every lower 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, and likewise for every other position.

Position Under Tensor Product Formation

Forming a tensor product of two covariant tensors places all of the lower positions belonging to the first factor before all of the lower 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, first or second, it came from.

Explicit Permutation of Position

An operator that deliberately exchanges the vectors assigned to two chosen 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: the symmetric part of a tensor in two 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.