11.18.5 Tensor Variance Type Operation Effect

Tensor Variance Type Operation Effect refers to how tensor variance types influence algebraic operations and transformations in mathematical structures.

Tensor Variance Type Operation Effect is the systematic change produced in a tensor's variance type, its count of upper and lower indices, whenever a standard tensor algebra operation such as tensor product, contraction, raising, or lowering is applied, allowing the resulting type of any composite expression to be predicted directly from the types of its ingredients without recomputing the full transformation law.

Foundational Setting

Operations as Type-Level Rules

Each operation available in tensor algebra acts on the underlying components in a specific way, but every one of these operations also has a predictable, purely arithmetic effect on the type pair $(p, q)$ of the tensors involved. Tracking this effect at the level of type alone is often sufficient to verify that an expression is well-formed before any detailed computation is carried out.

The Operations Considered

The four operations whose type-level effect is most commonly tracked are the tensor product, contraction, index raising, and index lowering, each of which is examined in turn below.

Effect of the Tensor Product

Additive Combination of Types

Forming the tensor product of a tensor of type $(p_{1}, q_{1})$ with a tensor of type $(p_{2}, q_{2})$ produces a tensor of type:

(p_{1} + p_{2}, q_{1} + q_{2})

since no indices are removed or altered, only placed alongside one another.

Effect of Contraction

Symmetric Reduction of Both Counts

Contracting one upper index against one lower index of a tensor of type $(p, q)$ reduces the type to:

(p - 1, q - 1)

since the summed index is removed entirely from the free indices of the result, taking one instance of each count with it.

Repeated Contraction Toward Invariance

Applying contraction repeatedly, as many times as the smaller of $p$ and $q$ allows, drives the type toward $(0, 0)$ whenever $p = q$ , producing an invariant scalar as the end result.

Effect of Raising and Lowering

Raising an Index

Using an inverse metric tensor to raise a lower index converts one covariant index into a contravariant one, changing the type from $(p, q)$ to $(p + 1, q - 1)$ :

v^{i} = \sum_{j} g^{i j} v_{j}

Lowering an Index

Using the metric tensor itself to lower an upper index converts one contravariant index into a covariant one, changing the type from $(p, q)$ to $(p - 1, q + 1)$ :

v_{i} = \sum_{j} g_{i j} v^{j}

Total Rank Preserved

Both raising and lowering keep the total rank $p + q$ constant, since one index is converted from one kind to the other rather than being added or removed, in contrast to contraction, which reduces the total rank by two.

Visual Summary of the Four Effects

Diagram of Type Changes

Combining Multiple Operations

Predicting the Type of a Composite Expression

Because each operation's effect on the type pair is purely arithmetic, the resulting type of an expression built from several operations in sequence, such as taking a tensor product and then contracting twice, can be computed by applying each individual effect in order, without needing to track the full component-level computation at every step.

Consistency Check

This predictable arithmetic also serves as a consistency check: if a proposed tensor equation implies a type mismatch between its two sides after accounting for every operation's effect, the equation cannot be a valid tensor identity.

Summary of Key Traits

Defining Characteristics

Tensor product adds the type pairs of its factors.
Contraction subtracts one from both the contravariant and covariant counts, reducing total rank by two.
Raising and lowering convert one index from one kind to the other while preserving total rank.
The type-level effect of each operation can be composed to predict the type of any composite tensor expression in advance.