6.8.4 Tensor Covariant Basis Dependence

Tensor Covariant Basis Dependence is the fact that the numerical components of a covariant tensor are not fixed quantities attached permanently to the tensor, but are values that arise only once a particular basis of vectors has been chosen, so that the same underlying multilinear object yields different arrays of numbers when evaluated against different bases, even though the object it represents remains one and the same. Covariant basis dependence is the precise statement of how and why this happens, and it is what necessitates the transformation law relating the components obtained from one basis choice to the components obtained from another.

How Basis Choice Enters Covariant Components

Components as Evaluations on Basis Vectors

The components of a covariant tensor of a given slot count are obtained by evaluating the tensor's multilinear map on every possible combination of basis vectors drawn from a chosen basis, one basis vector supplied to each slot. Selecting a different basis for the same underlying vector space means supplying different vectors into the slots, and since the tensor is a fixed multilinear map, evaluating it on different inputs generally returns different numbers, which is exactly the origin of covariant basis dependence.

The Underlying Object Is Basis-Independent

Although the components change with the basis, the tensor itself, understood as an abstract multilinear map acting on vectors, does not depend on any basis at all; it is defined purely by its action on arbitrary vector inputs, prior to and independently of any coordinate system being introduced. Covariant basis dependence therefore describes a property of the numerical representation of the tensor, not a property of the tensor as a mathematical object, and this distinction is the reason a single transformation law can relate all the different component sets to one another without any of them being more fundamental than the rest.

The Mechanism Linking Different Bases

Change of Basis and the Induced Change of Components

If a new basis is obtained from an old basis by a linear substitution, the components of a covariant tensor computed in the new basis are obtained from the components computed in the old basis by contracting with the same linear substitution matrix used to relate the bases, applied once for every lower slot the tensor possesses. This is precisely the inverse-Jacobian transformation law associated with covariant indices, and it exists solely because covariant components depend on basis choice in this structured, predictable way.

Basis Dependence Is Fully Predictable, Not Arbitrary

Basis dependence does not mean the components can take on unrelated or unpredictable values from one basis to the next; rather, it means the components change according to a completely determined rule once the relation between the two bases is known. Given the components in one basis and the linear relation connecting that basis to another, the components in the second basis are computed exactly, with no residual freedom or ambiguity, which is what allows the same tensor to be described consistently across arbitrarily many different bases.

Consequences of Covariant Basis Dependence

Component Values Alone Do Not Identify a Tensor

Because covariant components depend on the basis used to obtain them, a bare list of numbers cannot be interpreted as a specific tensor until the basis in which those numbers were computed is also specified. The same list of numbers can represent entirely different tensors depending on the basis assumed, and conversely the same tensor is represented by different lists of numbers in different bases, so any faithful description of a covariant tensor must state both the components and the basis together.

Special Bases and Simplified Components

Certain bases, chosen to align with the structure of a particular problem, such as an orthonormal basis aligned with the principal directions of a symmetric covariant tensor, make the components of that tensor take an especially simple form, often diagonal or otherwise sparse. This simplification is entirely a byproduct of covariant basis dependence: no property of the tensor itself has changed, only the arithmetic convenience of the description, since a generic basis would recover the identical tensor through more complicated, non-diagonal components.

Invariants as Basis-Independent Combinations

Because raw components depend on basis while the tensor does not, meaningful basis-independent quantities can only be extracted by forming particular combinations of the components that happen to cancel the basis dependence exactly, such as full contractions of a covariant tensor against an appropriate number of vectors or against the inverse metric. Such invariant combinations retain the identical numerical value regardless of which basis was used to compute the intermediate components, and their existence is what makes it possible to state physically or geometrically meaningful facts using components at all, despite the components themselves having no basis-independent meaning in isolation.