8.5 Tensor Lower Index Notation Role

Tensor Lower Index Notation Role is the function that a subscript index plays within tensor index notation: marking a slot as covariant, dictating that slot's transformation law under a change of coordinates, and identifying which of a tensor's several argument positions accepts a vector rather than a covector. It is the mirror-image counterpart of the upper index role, and the two together account for the complete range of behaviors a tensor's slots can exhibit.

The Transformation Duty

Carrying the Jacobian Directly

The primary duty of a lower index is to signal that its slot transforms using the Jacobian matrix directly, rather than its inverse; this is what makes a quantity like a covector's components ωᵢ covariant, meaning they vary in the same direction, in a specific technical sense, as the coordinate basis itself.

Compensating for Basis Change

This transformation duty exists for the same reason as its upper-index counterpart: to keep the underlying, basis-independent covector object unchanged as the coordinate system describing it is altered, with the lower index's transformation role serving as the compensating mechanism for covariant quantities specifically.

The Argument-Slot Duty

Accepting a Vector

In the interpretation of a tensor as a multilinear map, a slot marked with a lower index in the tensor's type designation (p, q) is one of the q slots that accepts an ordinary vector as its argument; this is the structural, argument-accepting role that lower indices play once a tensor is understood in its most basis-independent formulation.

Distinguishing Slots by Their Accepted Input

A (1, 2) tensor accepts one covector and two vectors, in that order, and it is precisely the lower indices in its symbolic representation T^{i}_{jk} that mark which two of its three total slots are the vector-accepting ones, leaving the upper-indexed slot as the covector-accepting one.

Diagram of the Lower Index Role

The Lower Index Role in Common Objects

Covectors and Basis Vectors as the Archetype

A covector's components ωᵢ are the clearest illustration of the lower index role: they transform covariantly, and the covector itself, viewed as a multilinear map, accepts a single vector as input to produce a scalar, matching exactly the single lower index it carries. The ordinary coordinate basis vectors eᵢ are likewise lower-indexed, since they combine with contravariant vector components under contraction to reconstitute the vector itself.

The Metric and Index Lowering

The metric tensor g_{ij}, carrying two lower indices, plays this role twice over: each of its two slots transforms covariantly, and it is this tensor, contracted against a vector, that performs the operation of lowering an index, converting a contravariant slot into a covariant one on some other tensor.

Consequences of the Lower Index Role

Determines Compatibility in Contraction

Because contraction pairs one upper index with one lower index, the lower index role directly determines which pairings are legitimate: a slot marked lower can only be validly contracted against a slot marked upper, never against another lower slot, since only an upper-lower pairing produces the Jacobian cancellation that makes the resulting contraction basis-independent.

Shapes How a Formula Is Read

Recognizing the lower index role immediately tells a reader that the corresponding quantity behaves like a covector rather than a vector under a change of coordinates, which in turn informs how that quantity is expected to combine with other tensors in a larger expression, well before any explicit computation with the formula is carried out.

Relationship to the Upper Index's Complementary Role

Mirror-Image Duties

The upper index plays an exactly complementary pair of duties — contravariant transformation and covector-accepting argument slots — so that the lower and upper index roles together account for the complete range of behaviors a tensor's slots can exhibit; understanding either role in isolation is aided by seeing it as one half of this matched, mirror-image pair.

A Foundational Piece of the Overall Notation System

The lower index role, together with its upper counterpart, forms the foundational distinction on which the entire apparatus of tensor index notation — summation, contraction, raising and lowering, symmetry — is built, since none of these operations can be correctly specified without first knowing which slots are lower and which are upper.