6.24.3 Tensor Lower Index Notation

Tensor Lower Index Notation is the convention of writing a covariant index as a subscript attached to the tensor symbol, as in Tⱼ or Tᵢⱼ, so that the lowered position of the letter signals that the corresponding component transforms using the change-of-basis matrix directly, in the same direction as the basis vectors themselves, rather than with its inverse. This is the mirror-image convention to upper index notation, and together the two conventions give every tensor index an unambiguous, position-encoded transformation rule.

What Lowering an Index Signals

Covariant Transformation Behavior

A lower index attached to a tensor component indicates that, under a change of basis with transition matrix A (so e′ᵢ = Σₖ Aₖᵢ eₖ), that component transforms directly with A:

This is the defining behavior of a covariant quantity: it "co-varies" with the basis, changing in the same direction the basis vectors change, unlike a contravariant quantity, which changes oppositely to compensate and keep the underlying vector fixed.

Association With the Dual Space

A lower index marks a component as belonging to an expansion involving the dual space V*: writing a covector φ = Σᵢ φᵢεⁱ in terms of the dual basis {εⁱ}, the coefficients φᵢ carry lower indices because they arise from evaluating φ on the basis vectors of V, φᵢ = φ(eᵢ), and this evaluation is what forces the covariant transformation law: as the basis {eᵢ} changes, the values φ(eᵢ) change in the same direction, with no compensating inverse needed.

Notational Rules Governing Lower Indices

Position Consistency Under Ordinary Manipulation

As with upper indices, a lower index remains lower throughout a calculation unless it is explicitly acted upon by an operation designed to change variance, such as raising it with the inverse metric; arithmetic operations like addition or scalar multiplication never change an index's position, since they do not alter the transformation law of the quantity involved.

Distinguishing Subscripts as Indices From Subscripts as Labels

Not every subscript in mathematical writing denotes a tensor index; a subscript can also label a sequence member, a component of an unrelated list, or a fixed numeral identifying a specific object (as in e₁, e₂, e₃ for basis vectors, where the subscript picks out which basis vector, rather than indicating covariant transformation of a component). Context and the surrounding discussion of tensor transformation determine whether a given subscript is functioning as a covariant tensor index or as a mere label.

Diagram of Index Lowering and Its Meaning

Where Lower Indices Appear Throughout Tensor Algebra

Covector Components and Covariant Tensor Slots

The coordinates of a linear functional, φᵢ, and every covariant slot of a higher-order tensor, T_{j₁...j_q}, use lower index notation; a fully covariant tensor of type (0, q) has only lower indices and no upper ones, the case relevant to bilinear and multilinear forms including the metric tensor.

Lowered Indices Produced by the Metric

When a metric tensor g is available, it can lower an upper index into a lower one via vᵢ = Σⱼ gᵢⱼvʲ, converting a contravariant quantity into a covariant one; the resulting lower index on vᵢ correctly signals that this lowered quantity now transforms covariantly, which is precisely why the metric is described as providing a canonical identification between V and V* — it supplies the exact transformation-compatible map needed to move consistently between upper and lower index notation.