8.13.2 Tensor Index Lower Position

Tensor Index Lower Position denotes the placement of indices beneath a tensor symbol, indicating contraction with covariant components in tensor algebra.

Tensor Index Lower Position is the placement of a tensor index as a subscript, written beneath and to the right of the base symbol, used to denote a covariant component or covariant slot of a tensor. An index written in the lower position, such as the $i$ in $A_i$, transforms under a change of basis in the same way that basis covectors transform, and it is this transformation behavior — not merely the visual placement — that the lower position is meant to signal. The position of an index is therefore not a stylistic choice but a load-bearing piece of notation: it fixes how the labeled quantity behaves algebraically and geometrically.

Meaning of the Lower Position

Covariant Transformation Behavior

A quantity carrying an index in the lower position transforms according to the same rule used to transform the basis vectors of the space. Under a change of coordinates or basis described by a transformation matrix, the components of a covariant object combine with the matrix itself, rather than with its inverse:

A_{i′}^{} = \frac{\partial x^{j}}{\partial x^{i′}} A_{j}

This transformation law is the defining property of a lower-position index: whatever combination of partial derivatives or transformation coefficients is used to convert old basis vectors into new ones is applied directly, without inversion, to every index written in the lower position.

Association With Covectors and One-Forms

The lower position is conventionally reserved for the components of covectors, also called one-forms or dual vectors. A covector $\omega$ expressed in a coordinate basis is written with all of its indices lowered, $\omega_i$, reflecting that covectors are elements of the dual space, whose natural pairing with ordinary (contravariant) vectors produces a scalar without requiring a metric.

Lower Position in Higher-Rank Tensors

Fully Covariant Tensors

A tensor with every index in the lower position, such as $T_{ij}$ or $T_{ijk}$, is called a fully covariant tensor of the corresponding rank. Each lower index transforms independently according to the covariant law, so that a rank-2 fully covariant tensor obeys

T_{i′ j′} = \frac{\partial x^{k}}{\partial x^{i′}} \frac{\partial x^{l}}{\partial x^{j′}} T_{k l}

Mixed Tensors With Lower Indices

In a mixed tensor, lower and upper positions coexist on the same symbol, and each index transforms strictly according to its own position regardless of the others present. A mixed tensor $T^{i}{}_{j}$ has its upper index $i$ transforming contravariantly and its lower index $j$ transforming covariantly, independently of one another within the same expression.

Producing a Lower-Position Index From an Upper One

Index Lowering via the Metric Tensor

An index that begins in the upper position can be converted into a lower-position index by contracting it with the covariant metric tensor $g_{ij}$:

A_{i} = g_{ij} A^{j}

This operation, called lowering the index, produces a covariant component from a contravariant one and is only well-defined once a metric has been fixed on the space. In a purely affine setting, with no metric, an upper index cannot be converted into a lower one, and the two positions remain permanently distinct kinds of objects.

Symmetry of the Lowering Operation

Because the metric tensor is itself symmetric, $g_{ij} = g_{ji}$, the lowering operation does not depend on which of the metric's two indices is contracted against the tensor's upper index, so the resulting lower-position component is unambiguous.

Distinguishing the Lower Position From Ordinary Subscripts

Tensor Indices Versus Labeling Subscripts

Not every subscript attached to a symbol in mathematics denotes a lower tensor index; subscripts are also used to label discrete objects in a sequence, such as $a_1, a_2, a_3$ for the entries of an ordinary list, without any implication of covariant transformation behavior. Within tensor algebra specifically, however, a subscript on a tensor symbol is understood to carry the full covariant transformation meaning described above, and this convention is assumed whenever the surrounding context has established that the symbol denotes a tensor.

Role Within Index Position Notation

Together with the upper (contravariant) position, the lower position forms the complete vocabulary of tensor index placement: every index on every tensor must occupy one of these two positions, and the total pattern of upper and lower positions on a symbol determines its type $(p,q)$, where $p$ counts the upper indices and $q$ counts the lower ones. The lower position is therefore not merely a graphical convention but one half of the classification system that governs how tensors combine, contract, and transform throughout tensor algebra.