11.19.2 Tensor Variance Convention Basis Direction

Tensor Variance Convention defines how tensor components transform with basis changes, ensuring consistency in mathematical models.

Tensor Variance Convention Basis Direction is the part of the variance convention that fixes, for any given transformation of the underlying basis, which direction that transformation is considered to run, from an original basis toward a new one, so that the labels covariant and contravariant attach consistently to the same components regardless of which of two related bases is treated as the starting point.

Foundational Setting

Why a Direction Must Be Fixed

A change of basis relates two bases through a matrix and its inverse, and either matrix could, in principle, be called the transformation. Without a fixed convention for which direction is designated as "the" transformation, the labels covariant and contravariant would be ambiguous, since what counts as the direct matrix in one direction is the inverse matrix in the other.

The Standard Choice

The standard basis direction convention designates the map from the old basis vectors to the new basis vectors as the primary transformation matrix $A$ , defined by:

{\tilde{e}}_{i} = \sum_{j} A_{i}^{j} e_{j}

with all covariant and contravariant transformation laws then defined relative to this fixed direction.

Consequences of the Fixed Direction

Covariant Components Follow the Named Direction

Given this basis direction convention, covariant components are defined as those transforming with $A$ exactly as named, in the same direction as the basis vectors themselves move:

{\tilde{ω}}_{i} = \sum_{j} A_{i}^{j} ω_{j}

Contravariant Components Follow the Opposite Direction

Contravariant components, by the same fixed convention, transform with the inverse $A^{- 1}$ , moving against the named basis direction:

{\tilde{v}}^{i} = \sum_{j} {(A^{- 1})}_{j}^{i} v^{j}

Reversibility of the Direction Choice

Swapping the Roles of Old and New

If the roles of the old and new bases are exchanged, so that what was called new becomes the starting point, the matrix that was $A$ becomes $A^{- 1}$ and vice versa, but the covariant and contravariant labels attached to specific physical quantities, such as a given gradient or a given displacement vector, do not change, since those labels are tied to the type of object, not to an arbitrary choice of which basis is called original.

Interaction with Coordinate Changes

Extending the Direction Convention to Coordinates

When coordinates rather than abstract basis vectors are involved, the analogous basis direction convention designates the Jacobian of the new coordinates with respect to the old coordinates as the primary transformation, with covariant and contravariant tensor components then defined relative to this same fixed directional choice, matching the abstract basis case exactly.

Why Consistency Across Contexts Matters

Maintaining the same directional convention across both abstract basis changes and coordinate changes ensures that a single tensor equation can be interpreted identically whether it is being read as a statement about linear algebra on a fixed vector space or as a statement about coordinate transformations on a manifold.

Summary of Key Traits

Defining Characteristics

The basis direction convention fixes which of two mutually inverse matrices is designated the primary transformation, typically the map from old basis to new basis.
Covariant components are defined to transform with this primary matrix; contravariant components transform with its inverse.
Swapping which basis is called old and which is called new inverts the matrix labels but leaves the covariant or contravariant classification of any given physical quantity unchanged.
The same directional convention extends consistently to coordinate transformations via the Jacobian matrix.