10.16.2 Tensor Passive Transformation Coordinate Relabeling

Tensor Passive Transformation Coordinate Relabeling is the narrowest and most literal special case of a passive coordinate transformation, in which the new coordinates are simply a renaming or reordering of the same underlying coordinate lines, with no genuine rotation, stretching, or curvilinear distortion of the coordinate grid, so that the Jacobian relating the two systems reduces to a permutation, a set of independent one-dimensional rescalings, or a combination of both.

What Counts as Relabeling

Pure Permutation of Coordinates

The simplest relabeling exchanges the roles of coordinate labels without altering the coordinate lines themselves, for instance swapping the first and second coordinate:

The Jacobian of this relabeling is a permutation matrix, containing only zeros and ones, and its inverse is simply its own transpose, since a permutation matrix is always orthogonal.

Independent Rescaling of Each Coordinate

A second common form of relabeling rescales each coordinate independently by a fixed constant, without mixing directions:

for constants $$c_i$$, one per direction, with no summation implied across different values of $$i$$, producing a diagonal Jacobian matrix whose inverse is simply the diagonal matrix of reciprocals.

Simplified Transformation Formulas

Diagonal Jacobian Case

Because relabeling produces a diagonal or permutation Jacobian, the general transformation sum collapses to a single term per component rather than a full sum over every original direction:

with no summation, since every off-diagonal Jacobian entry is zero, and this simplification is exactly what makes coordinate relabeling a convenient testing case for verifying a general transformation formula before applying it to a genuinely curvilinear change of basis.

Covariant Components Under Relabeling

The corresponding covariant component picks up the reciprocal factor rather than the same factor:

illustrating, in the simplest possible setting with no mixing between directions, the opposite scaling behavior that distinguishes covariant from contravariant components.

Relation to the General Passive Interpretation

Relabeling as a Degenerate Basis Change

Relabeling is still a genuine passive basis change: the fixed tensor is unaltered, and only the labeling of directions and their unit scale changes, so every conclusion drawn about passive transformation object preservation and basis change applies to relabeling as well, only with a Jacobian matrix restricted to the diagonal-plus-permutation form rather than a fully general invertible matrix.

Distinguishing Relabeling From Genuine Rotation

Coordinate relabeling never introduces off-diagonal Jacobian entries, whereas a true rotation of the coordinate axes necessarily does, since rotation mixes directions that relabeling by design keeps separate; checking whether the Jacobian matrix connecting two coordinate systems is diagonal, up to a permutation, is a direct test for whether a given passive transformation is mere relabeling or a genuine change of orientation.

Diagram of Relabeling

Renamed and Rescaled Axes, No Rotation

Practical Uses of Coordinate Relabeling

Sanity Checking Formulas

Because relabeling produces such a simple Jacobian, it is often used as a first sanity check on a newly derived tensor transformation formula, verifying that swapping index labels or rescaling units produces the expected trivial effect before the formula is trusted for a genuinely curvilinear coordinate change.

Unit Conversion as Relabeling

A common real instance of coordinate relabeling is a change of measurement units along each coordinate axis independently, such as converting one axis from meters to centimeters, which rescales that single direction by a constant factor and leaves every other direction untouched, producing exactly the diagonal Jacobian structure characteristic of relabeling.