11.9.5 Tensor Mixed Law Tensor Type Preservation

Tensor Mixed Law Tensor Type Preservation ensures consistency in tensor transformations by maintaining type integrity across mixed law operations.

Tensor Mixed Law Tensor Type Preservation is the guarantee that a mixed tensor's type, meaning the specific number of upper indices and the specific number of lower indices it carries, remains exactly the same after any coordinate transformation, since the mixed variance transformation law never converts an upper index into a lower index, never converts a lower index into an upper index, and never changes the total count of either kind.

Definition and Statement

What Tensor Type Means

The type of a tensor is conventionally described by a pair of numbers, one counting the upper indices and one counting the lower indices, and type preservation asserts that a tensor entering a coordinate transformation with a given pair of numbers emerges from that transformation with the identical pair of numbers.

type (A_{k}^{j}) = type (A_{l^{'}}^{i^{'}})

Structural Reason for Preservation

Preservation holds because the mixed variance transformation law assigns exactly one direct Jacobian factor to each upper index and exactly one inverse Jacobian factor to each lower index, with no mechanism in the law that would merge an upper index with a lower index or produce a new index of a different kind.

Why Type Cannot Change

No Cross-Type Contraction in the Transformation Law

The transformation law contracts old upper indices only against new upper index labels through the direct factor, and old lower indices only against new lower index labels through the inverse factor, so there is no step in the law by which an upper index could be reassigned to a lower position or vice versa.

A_{l^{'}}^{i^{'}} = \frac{\partial x^{i^{'}}}{\partial x^{j}} \cdot \frac{\partial x^{k}}{\partial x^{l^{'}}} A_{k}^{j}

Consistency Across Repeated Transformations

Since each individual coordinate change preserves the tensor's type, any sequence of successive coordinate changes also preserves the type, because applying a type-preserving operation repeatedly cannot introduce a change in type at any intermediate stage.

Consequences of Type Preservation

Reliability of Tensor Classification

Because type is preserved, it is meaningful to classify a tensor once, at the outset, by its number of upper and lower indices, and this classification remains valid for describing the tensor in every coordinate system, without needing to re-examine or reclassify the tensor after each transformation.

Compatibility With Contraction Rank Reduction

Type preservation under coordinate transformation is distinct from, and fully compatible with, the separate operation of contraction, which does reduce the number of upper and lower indices by one each; contraction is an algebraic operation performed within a fixed coordinate system, whereas type preservation concerns what happens when the coordinate system itself changes.

Role Within Tensor Algebras

Foundation for Classifying Tensor Spaces

Tensor type preservation is what allows tensors to be organized into well-defined spaces indexed by their upper and lower index counts, since a tensor of a given type in one coordinate system can only ever appear as a tensor of that same type in any other coordinate system, never migrating between different tensor spaces.

Relationship to the Definition of a Tensor

Type preservation is an implicit part of what it means for an indexed array to be called a tensor of a specific type in the first place, since the very notion of assigning a fixed number of upper and lower indices to an object presupposes that this assignment remains stable under the mixed variance transformation law.