8.5.1 Tensor Lower Index Position Role

Tensor Lower Index Position Role is the specific function played by the horizontal ordering among a tensor's several subscript indices: the left-to-right sequence in which lower indices are written, which determines which vector-accepting argument slot, among the several such slots a tensor may have, each particular lower index refers to. It narrows the general lower index role down to the question of order among indices that already share the lower, covariant position.

Order Among Multiple Lower Indices

Each Position Names a Distinct Argument Slot

A tensor of type (p, q) with q ≥ 2 has more than one covariant argument slot, and the horizontal order of its subscript indices in an expression such as T_{ijk} records which slot each of i, j, and k is understood to fill: i names the first vector-accepting slot, j the second, k the third, and reordering them generally refers to a different assignment of arguments to slots.

in general, since the left side feeds its first slot the value assigned to i and its second slot the value assigned to j, while the right side does the reverse.

Position Is Independent of Which Letters Are Used

The role played by position among lower indices is entirely about order, not about which specific letters are chosen to fill each position; renaming the indices in T_{ijk} to T_{abc}, preserving their relative order, refers to exactly the same slot assignment, whereas permuting the letters while keeping the same names, as in T_{jik}, changes which slot each named index refers to.

Symmetry as a Statement About Lower Index Position

Symmetric Pairs of Lower Indices

When a tensor happens to be symmetric in a pair of its lower indices, T_{ij} = T_{ji}, this is a substantive claim specifically about the lower index position role: it asserts that, for this particular tensor, the assignment of arguments to its first and second covariant slots may be swapped without changing the value, a property that does not hold for a generic tensor and must be established or assumed explicitly. The metric tensor g_{ij} = g_{ji} and the stress tensor σᵢⱼ = σⱼᵢ are standard examples of this symmetry holding in practice.

Antisymmetric Pairs

Conversely, T_{ij} = -T_{ji} asserts that swapping the position of a pair of lower indices flips the sign of the result, as occurs with the electromagnetic field-strength tensor's lower-index form; both symmetric and antisymmetric statements are only meaningful once the lower index position role is understood, since they are precisely claims about what happens under reordering.

Diagram of Lower Index Position

Lower Index Position in Staggered Notation

Recording Original Position After Lowering

When one index of a mixed tensor is lowered from an upper to a lower position, staggered notation, such as T_{i}{}^{j}{}_{k}, is used to preserve a record of the original horizontal position that the newly lowered index occupied among all the tensor's slots before the lowering operation, ensuring that the lowered index is not mistaken for occupying a different original slot than it actually did.

Why This Matters Specifically for Lower Positions

Because lowering an index moves it into lower position without necessarily placing it at the leftmost or rightmost lower slot, the position role for lower indices must account for indices that arrived at their lower status through lowering, not only those that were lower from the tensor's original definition; staggered notation is the standard device for keeping this positional history unambiguous.

Practical Consequences

Contraction Order in a Chain

When several lower indices participate in different contractions within a longer expression, their relative horizontal position, together with the corresponding upper indices they are paired against, determines the order in which the associated operations combine, mirroring the non-commutativity of the operations being represented, such as matrix or operator composition.

Verifying Claimed Symmetries

Any claimed symmetry or antisymmetry among a tensor's lower indices should be checked specifically against the positional role: verifying that swapping the horizontal position of the claimed pair, while leaving all other indices fixed, produces the stated relationship, rather than assuming the relationship holds merely because the indices in question happen to share the same vertical, lower placement.