1.6.3 Tensor Object Invariance

Tensor Object Invariance ensures tensors retain their structure and meaning under coordinate changes, preserving consistency across reference frames.

Tensor Object Invariance is the property, more fundamental than the invariance of any particular tensor's components, that the tensor itself, considered as an abstract multilinear object, exists independently of any coordinate system or basis used to describe it, so that no choice of basis is needed to define what the tensor is, only to compute a numerical representation of it. Where other treatments of invariance examine how components change or which special tensors have unchanging components, tensor object invariance addresses the ontological question underlying all of them: what kind of thing a tensor is, such that the question of invariance can even be posed.

The Tensor as a Basis-Free Object

Multilinear Maps Without Coordinates

A tensor of type (p, q) on a vector space V can be defined directly as a multilinear map taking p covectors and q vectors as arguments and returning a scalar, a definition that makes no reference whatsoever to a basis. The object is characterized entirely by its linearity in each argument and by the values it produces, not by any array of numbers.

T : \underset{p copies}{V^{*} \times \dots \times V^{*}} \times \underset{q copies}{V \times \dots \times V} \to F

The Tensor Product Space as an Abstract Construction

Equivalently, a tensor of type (p, q) can be defined as an element of the tensor product space V^{⊗p} ⊗ (V^*)^{⊗q}, a vector space built from V and V^* by a universal construction that itself requires no basis, only the vector spaces V and V^* as inputs.

The Universal Property as the Source of Invariance

What the Universal Property Guarantees

The tensor product space is characterized by a universal property: any multilinear map out of the product of the factor spaces factors uniquely through the tensor product. This property is stated purely in terms of vector spaces and linear maps, with no basis appearing anywhere in its formulation, which is why every object and operation defined through it inherits basis independence automatically.

Components as a Consequence, Not a Definition

Choosing a basis of V induces a basis of the tensor product space, and expressing a tensor in that induced basis yields its component array. Components are therefore a derived consequence of an independently existing object, obtained by an act of description, rather than the substance of the object itself.

Equivalence Classes of Component Arrays

Arrays Related by the Transformation Law Describe One Object

Two component arrays, one relative to a basis e_i and one relative to a basis e′_i, describe the same tensor object precisely when they are related by the tensor transformation law connecting those two bases. This relation partitions the collection of all possible component arrays, across all possible bases, into equivalence classes.

[T_{j}^{i}] = {all arrays related to T_{j}^{i} by the transformation law}

The Tensor as the Equivalence Class

Under this view, the invariant tensor object can be identified with the entire equivalence class of component arrays rather than with any single array, making tensor object invariance the statement that it is the equivalence class, not any one representative of it, that constitutes the mathematically meaningful object.

Distinguishing Object Invariance from Related Notions

Object Invariance Versus Component Invariance

An arbitrary tensor has object invariance in the sense described here, since it exists as one fixed abstract entity, even though its components generally differ from basis to basis. This is distinct from the narrower property of having invariant components, which holds only for special tensors such as the Kronecker delta within a given transformation group.

Object Invariance Versus Equation Invariance

Object invariance concerns a single tensor's identity across bases. It is the prerequisite for, but not the same as, the invariance of an equation relating several tensors, which additionally requires that the relation between the objects, not merely each object individually, be preserved under change of basis.

Consequences for How Tensors Are Used

Basis-Independent Statements Are the Primary Content

Because the tensor object is what is invariant, meaningful mathematical and physical statements about tensors are properly made about the objects themselves, definitions, equalities, and operations stated in coordinate-free language, with component expressions serving only as a computational tool for evaluating those statements in a convenient frame.

Safety of Switching Representations

Tensor object invariance is what licenses the common practice of switching between different bases or coordinate systems in the middle of a calculation: since the object being computed with does not depend on the representation, any basis may be adopted at any stage without altering the final invariant result.

Diagrammatic Summary

The diagram places a single invariant tensor object at the center, with each basis-dependent component array beneath it standing as one representative of the same equivalence class, all related to one another by the tensor transformation law and all describing the one object above.