11.13 Tensor Covariant Slot Behavior

Tensor Covariant Slot Behavior is the characteristic pattern by which each lower index position, or slot, of a tensor acts as an input for a contravariant vector when the tensor is regarded as a multilinear map, so that filling a covariant slot with a vector, rather than a covector, is what produces a well-defined scalar or lower-rank tensor output.

Definition and Basic Setting

Slots as Argument Positions

A tensor of a given type can be understood as a multilinear function accepting a fixed number of vector and covector arguments, and each lower index of the tensor corresponds to one argument position, or slot, that specifically expects a contravariant vector to be inserted into it.

Contrast With Upper Index Slots

Whereas a lower index slot expects a contravariant vector as its filling argument, an upper index slot expects a covariant covector instead, so the two kinds of slots impose opposite requirements on what type of object may be validly inserted into them.

Linearity Within Each Covariant Slot

Linearity in the Inserted Vector

Filling a covariant slot with a vector produces a result that is linear in that vector, meaning that inserting a sum of two vectors gives the sum of the individual results, and inserting a scaled vector gives the correspondingly scaled result, which is the defining multilinearity property of the tensor in that slot.

Independence of Behavior Across Different Slots

When a tensor has several lower index slots, the linearity behavior of each slot is independent of the others, meaning that a vector inserted into one covariant slot can be varied while the vectors inserted into the remaining slots are held fixed, without disturbing the linearity property established in that other slot.

Consequences of Slot Behavior

Consistency With the Covariant Transformation Law

The requirement that a covariant slot accepts a contravariant vector is precisely what makes the covariant transformation law, using the inverse Jacobian factor, produce a coordinate-independent scalar when the tensor is fully evaluated on vectors, since it is the pairing invariance between covariant and contravariant objects that guarantees this.

Basis for the Metric's Two Covariant Slots

The covariant metric tensor, carrying two lower indices, exhibits covariant slot behavior in both of its argument positions, accepting a vector in each slot, which is exactly the structure needed for it to function as a bilinear form on pairs of vectors.

Role Within Tensor Algebras

Foundation for Interpreting Tensors as Multilinear Maps

Covariant slot behavior, together with the corresponding contravariant slot behavior for upper indices, provides the conceptual basis for interpreting a tensor of any type as a multilinear map with a fixed number of vector and covector argument slots, rather than as merely an indexed array of numbers.

Guiding Correct Contraction of Indices

Understanding covariant slot behavior clarifies why a lower index of a tensor must be contracted specifically with an upper index of another tensor, and not with another lower index, since only an upper index supplies the contravariant vector that a covariant slot is structured to accept.