10.17.2 Tensor Active Transformation Map Action

Tensor Active Transformation Map Action describes how tensors change under coordinate transformations through active mapping, essential in physics and geometry.

Tensor Active Transformation Map Action is the description of how a specific point transformation of the underlying space, such as a rotation, translation, or general diffeomorphism, moves points and thereby induces a corresponding, well-defined action on vectors, covectors, and higher-rank tensors defined at those points, all while a single fixed coordinate system is used to record every quantity involved.

The Underlying Point Map

Moving Points, Not Coordinates

An active transformation begins with a map sending each point of the space to another point of the same space, expressed in the one fixed coordinate system used throughout:

y^{i} = φ^{i} (x^{1}, \dots, x^{n})

where $x^{i}$ and $y^{i}$ are coordinates of two different points, both measured in the same fixed system, rather than the same point described in two coordinate systems as in the passive case.

The Differential of the Map

The induced action on vectors is built from the differential of this point map, namely its Jacobian matrix evaluated at the starting point:

A_{k}^{i} = \frac{\partial φ^{i}}{\partial x^{k}}

which, unlike the Jacobian in the passive coordinate transformation case, relates two different points under one fixed coordinate system rather than one point under two coordinate systems.

Induced Action on Tensors

Pushing Forward a Vector

A vector attached at the starting point is carried to a corresponding vector at the image point through the map's differential, an operation commonly called a pushforward:

V^{' i} = \sum_{k}^{n} A_{k}^{i} V^{k}

so that the map action on points directly determines the map action on tensors attached to those points, rather than the tensor action being separately postulated.

Pulling Back a Covector

A covector is instead pulled back using the inverse of the differential, in the opposite direction from the pushforward of a vector, reflecting the same contragredient relationship between contravariant and covariant objects seen in the passive interpretation, but now applied to two genuinely different points connected by the active map rather than to one point described in two coordinate systems.

Composition of Map Actions

Sequential Point Maps

If a first map sends points to intermediate points and a second map sends those intermediate points onward to final points, the combined map action on tensors is obtained by composing the two individual differentials, following the same chain rule pattern that governs composed coordinate transformations in the passive setting:

A_{k}^{i} = \sum_{m}^{n} {\overset{'}{A}}_{k}^{m} {\overset{''}{A}}_{m}^{i}

Inverse Map Action

If the underlying point map is invertible, the inverse map induces its own action on tensors, exactly undoing the pushforward and pullback of the original map, and applying the map action followed by its inverse map action returns every tensor to its starting value at its starting point.

Diagram of the Map Action

Points and Their Attached Vectors Moving Together

Fixed Points and Local Behavior

Fixed Points of the Map

At a point left unchanged by the map, the point action is trivial, but the induced tensor action need not be: the differential of the map at a fixed point can still be a non-trivial linear transformation, meaning a vector attached at a fixed point of the map can still be actively rotated or stretched even though the point itself does not move.

Local Linear Approximation

Near any point, the map action on tensors is governed entirely by the local value of the differential at that point, so two active maps that agree to first order at a given point induce identical actions on tensors attached there, even if the maps differ elsewhere, in the same way that only the local Jacobian, not the global structure of the map, enters the passive transformation formula at a point.

Relevance to Symmetry and Group Actions

Symmetries as Map Actions

When the point map is required to preserve some structure, such as distances or angles, its induced tensor action correspondingly preserves related tensor invariants, such as vector length or the value of contractions built from a metric tensor, making the active map action the natural language for describing symmetries and their effect on tensor fields defined over the space.