12.1 Tensor Operation Scope

Tensor Operation Scope is the set of prerequisite conditions that must hold before a given tensor operation, such as addition, tensor product, contraction, or index raising and lowering, can be validly applied, specifying which combinations of tensors, index types, and underlying vector spaces each operation is actually defined for.

Foundational Setting

Why Scope Must Be Specified

Not every pair of tensors can be combined by every operation. Addition requires matching type; contraction requires a genuine upper-lower pair; raising and lowering require a metric. The operation scope makes explicit, for each operation, exactly which prerequisites must be satisfied, so that an expression can be checked for validity before any computation is attempted.

Scope as a Gatekeeping Concept

Understanding operation scope prevents the construction of expressions that are notationally plausible but mathematically meaningless, such as attempting to add two tensors of different rank or contracting two indices of the same type.

Scope of Addition and Scalar Multiplication

Matching Type Requirement

Addition is defined only between two tensors sharing the identical variance type $$(p,q)$$ and built over the same underlying vector space:

Attempting to add tensors of different type, or of the same type but built over different vector spaces, falls outside this operation's scope and produces no well-defined result.

Scalar Multiplication's Broader Scope

Scalar multiplication, by contrast, has no type-matching requirement between two tensors, since it involves only a single tensor and a number, and so applies to a tensor of any variance type without restriction.

Scope of the Tensor Product

No Type-Matching Requirement

Unlike addition, the tensor product places no restriction requiring its two input tensors to share the same type, since the operation is specifically designed to combine tensors of differing types into a single, larger object:

The Shared Vector Space Requirement

The tensor product does, however, require both factors to be built from the same underlying vector space and its dual, since the operation relies on a shared notion of what a basis change means for both factors simultaneously.

Scope of Contraction

Requiring a Genuine Upper-Lower Pair

Contraction is defined only for a specific pair of indices consisting of one upper index and one lower index on the tensor, or across a product of tensors, being summed:

Attempting to contract two indices of the same type, both upper or both lower, falls outside the operation's defined scope, since no consistent, basis-independent result would emerge from such a sum.

Dimension Compatibility

Contraction additionally requires that the range of the summed index match the dimension of the underlying vector space consistently between the upper and lower slots being paired, ensuring the summation is well-posed.

Scope of Raising and Lowering

Requiring an Available Metric

Raising and lowering an index are defined only when a metric tensor, and correspondingly its inverse, is available on the underlying vector space:

In a vector space with no metric structure, these two operations simply have no defined scope and cannot be applied at all.

Invertibility of the Metric

Even where a metric exists, raising and lowering remain within scope only where that metric is invertible, excluding points or regions where the metric degenerates.

Visual Overview

Diagram of Operation Prerequisites

Consequences for Composite Expressions

Scope of a Chained Operation Sequence

When several operations are chained together, the overall expression falls within scope only if every individual operation in the sequence falls within its own scope at the point it is applied, so scope must be checked stage by stage rather than only for the final assembled expression.

Early Detection of Invalid Expressions

Because operation scope depends only on type, index structure, and available auxiliary objects such as a metric, rather than on specific numerical values, checking scope is a purely structural exercise that can catch an invalid expression before any numerical computation is attempted.

Summary of Key Traits

Defining Characteristics

• Addition requires matching variance type; scalar multiplication has no such restriction.
• The tensor product requires a shared underlying vector space but not matching type between its factors.
• Contraction requires a genuine upper-lower index pair with compatible dimension.
• Raising and lowering require an available, invertible metric tensor, and are undefined in its absence.