11.11.5 Tensor Index Lowering Type Change

Tensor Index Lowering Type Change is the shift in a tensor's classification, from a type carrying a given number of upper and lower indices to a type carrying one additional lower index and one fewer upper index, that necessarily accompanies the application of the index lowering operation to any one of a tensor's upper indices.

Definition and Statement

How the Type Numbers Shift

If a tensor begins with a given count of upper indices and a given count of lower indices, applying the lowering operation to one upper index produces a new tensor whose lower index count has increased by exactly one and whose upper index count has decreased by exactly one, with the total number of indices unchanged.

Distinguishing Type Change From Coordinate Transformation

This type change is a purely algebraic reclassification performed within a single coordinate system through contraction with the metric, and is entirely distinct from the coordinate transformation laws, which never alter a tensor's type but only recompute its components within a fixed type when the coordinate system itself changes.

Mechanics of the Type Change

One Index at a Time

Each individual application of the lowering operation changes the type by shifting exactly one index from the upper count to the lower count, so lowering two separate upper indices of the same tensor requires two separate applications of the operation, each producing its own intermediate type change.

Limit Imposed by the Available Upper Indices

The type change through lowering can only be performed as many times as there are upper indices available on the original tensor, since each lowering operation consumes one upper index, and once every upper index has been lowered, the tensor reaches a purely covariant type beyond which no further lowering is possible.

Reversibility of the Type Change

Symmetric Change Under Raising

The type change produced by lowering is exactly reversed by applying the corresponding raising operation to the newly lowered lower index, which shifts the lower index count back down by one and the upper index count back up by one, restoring the original type.

Net Type Preservation Across a Lower-Raise Pair

Performing a lowering operation immediately followed by a raising operation on the same index leaves the type of the tensor completely unchanged, since the two type changes, one increasing and one decreasing the lower index count by one, cancel exactly.

Role Within Tensor Algebras

Enabling Flexible Representation of a Single Object

Index lowering type change allows a single geometric object, such as a linear map, to be represented at will as a purely contravariant tensor, a purely covariant tensor, or a mixed tensor, depending on which type is most convenient for a given calculation, all while referring to the same underlying entity.

Contrast With Mixed Variance Transformation Type Preservation

Index lowering type change stands in direct contrast to the mixed variance transformation law, which strictly preserves tensor type under a change of coordinates; the two facts together clarify that type is fixed under coordinate changes but freely adjustable through metric-based raising and lowering.