11.14 Tensor Contravariant Slot Behavior

Tensor Contravariant Slot Behavior is the characteristic pattern by which each upper index position, or slot, of a tensor acts as an input for a covariant covector when the tensor is regarded as a multilinear map, so that filling a contravariant slot with a covector, rather than a vector, 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 upper index of the tensor corresponds to one argument position, or slot, that specifically expects a covariant covector to be inserted into it.

Contrast With Lower Index Slots

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

Linearity Within Each Contravariant Slot

Linearity in the Inserted Covector

Filling a contravariant slot with a covector produces a result that is linear in that covector, meaning that inserting a sum of two covectors gives the sum of the individual results, and inserting a scaled covector 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 upper index slots, the linearity behavior of each slot is independent of the others, meaning that a covector inserted into one contravariant slot can be varied while the covectors 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 Contravariant Transformation Law

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

Basis for the Inverse Metric's Two Contravariant Slots

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

Role Within Tensor Algebras

Foundation for Interpreting Tensors as Multilinear Maps

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