11.10.5 Tensor Index Raising Type Change

Tensor Index Raising 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 upper index and one fewer lower index, that necessarily accompanies the application of the index raising operation to any one of a tensor's lower 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 raising operation to one lower index produces a new tensor whose upper index count has increased by exactly one and whose lower 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 raising operation changes the type by shifting exactly one index from the lower count to the upper count, so raising two separate lower indices of the same tensor requires two separate applications of the operation, each producing its own intermediate type change.

Limit Imposed by the Available Lower Indices

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

Reversibility of the Type Change

Symmetric Change Under Lowering

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

Net Type Preservation Across a Raise-Lower Pair

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

Role Within Tensor Algebras

Enabling Flexible Representation of a Single Object

Index raising type change allows a single geometric object, such as a bilinear form, to be represented at will as a purely covariant tensor, a purely contravariant 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 raising 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.