7.5.4 Tensor Component Lower Index Position

In tensor algebra, the lower index position of a component denotes its covariant nature, indicating contraction with other tensors through summation over repeated indices.

Tensor Component Lower Index Position is the placement of a covariant index as a subscript on a tensor's symbol, marking that index as one whose components transform using the change-of-basis matrix directly, in contrast to an upper index position, whose components transform using the inverse of that matrix.

Definition and Scope

Subscript as a Marker of Variance

A lower index position is written as a subscript, as in (T_i), and identifies a slot in the tensor's index structure associated with covariant transformation behavior. As with the upper index position, the placement is a notational device carrying a specific algebraic consequence rather than any indication of subtraction or division.

T_{i}

Origin in Covector Components

The lower index position originates from how a covector's components are defined relative to a dual basis: writing (\omega = \omega_i e^i), the coefficients (\omega_i) carry a lower index because they must vary in the same direction as a change in the basis vectors (e_i) of the underlying space, keeping the dual pairing between covectors and vectors consistent regardless of basis.

Structural Properties

Transformation Rule Attached to the Lower Position

Under a change of basis given by a matrix (A), with (e_i' = A_i^{\ k} e_k), a component in the lower index position transforms with the matrix itself, not its inverse:

ω_{'} = A_{i}^{k} ω_{k}

so that as the basis vectors are rescaled, the lower-index components scale in step with them, opposite to the compensating behavior of upper-index components.

Position Among Several Lower Indices

When a tensor carries multiple lower indices, their relative order within the lower-index group must be tracked independently unless a symmetry ties them together, since each lower slot may be contracted against a different upper index elsewhere, and permuting lower indices without a symmetry justification produces, in general, a different tensor.

Distinction From Upper Index Position

The lower index position is defined by contrast with the upper index position: only a lower index paired with an upper index, one of each variance, forms a valid contraction without additional structure, and a sum taken over two lower indices directly does not constitute a standard contraction unless a metric is first used to raise one of them.

g_{i j} g^{j k} = δ_{i}^{k}

Role Within Tensor Algebra

Determining Valid Contractions and Products

The lower index position is what a contraction rule seeks to pair with an available upper index; recognizing which slots in a tensor's structure carry the lower position is a prerequisite for determining, before any computation, which contractions between two tensors are legitimate.

Interaction With Metrics

In spaces equipped with a metric tensor, lower and upper index positions become interconvertible through the operations of raising and lowering, in which the metric and its inverse are contracted against a tensor to move an index from one position to the other, though the index's original position still determines how it transforms before any such conversion is applied.