Physlib.SpaceAndTime.TimeAndSpace.EuclideanGroup.Action

The Euclidean group $E(d)$ acts on the product space of time and space, $\text{Time} \times \text{Space } d$. For any element $g \in E(d)$ and a point $(t, x) \in \text{Time} \times \text{Space } d$, where $t \in \text{Time}$ and $x \in \text{Space } d$, the action is defined as: $$g \cdot (t, x) = (t, g \cdot x)$$ This definition implies that the group action leaves the time coordinate $t$ invariant and transforms the spatial coordinate $x$ according to the standard action of the Euclidean group on $\text{Space } d$.

For any element $g \in E(d)$ of the $d$-dimensional Euclidean group, any time coordinate $t \in \text{Time}$, and any spatial coordinate $x \in \text{Space } d$, the action of $g$ on the time-space pair $(t, x) \in \text{Time} \times \text{Space } d$ is given by the coordinate formula $$g \cdot (t, x) = (t, g \cdot x),$$ where $g \cdot x$ denotes the standard action of the Euclidean group on the spatial component.

For any element $g$ of the Euclidean group $E(d)$ and any point $tx$ in the product space $\text{Time} \times \text{Space } d$, the first coordinate of the point resulting from the action $g \cdot tx$ is equal to the first coordinate of $tx$. That is, $(g \cdot tx)_1 = tx_1$.

For any element $g$ of the $d$-dimensional Euclidean group $E(d)$ and any point $tx$ in the product space $\text{Time} \times \text{Space } d$, the second coordinate (the spatial component) of the action of $g$ on $tx$ is equal to the action of $g$ on the spatial component of $tx$. That is, $(g \cdot tx)_2 = g \cdot tx_2$.

For any element $g$ of the $d$-dimensional Euclidean group $E(d)$ and any point $tx$ in the combined time-space manifold $\text{TimeAndSpace } d$, the time component of the point after the group action remains unchanged. That is, $\text{time}(g \cdot tx) = \text{time}(tx)$, where $\text{time}$ is the projection onto the temporal coordinate.

For any element $g$ of the $d$-dimensional Euclidean group $E(d)$ and any point $tx$ in the product space $\text{Time} \times \text{Space } d$, the spatial projection of the action of $g$ on $tx$ is equal to the action of $g$ on the spatial projection of $tx$. That is, $\text{space}(g \cdot tx) = g \cdot \text{space}(tx)$, where $\text{space} : \text{Time} \times \text{Space } d \to \text{Space } d$ denotes the projection onto the spatial component.

Let $E(d)$ be the $d$-dimensional Euclidean group and $\text{Time} \times \text{Space } d$ be the product space of time and space equipped with a product distance. For any element $g \in E(d)$ and any two points $tx, ty \in \text{Time} \times \text{Space } d$, the group action preserves the distance between points: $$\text{dist}(g \cdot tx, g \cdot ty) = \text{dist}(tx, ty)$$ where the action of $g$ on a point $(t, x)$ is defined to fix the time coordinate $t$ and act on the space coordinate $x$ via the standard Euclidean action.

For an element $g$ of the Euclidean group $E(d)$, this definition specifies the linear part of its action on the spacetime $\text{TimeAndSpace } d$, which is the product space $\text{Time} \times \text{Space } d$. The result is a continuous real-linear map $L_g: \text{Time} \times \text{Space } d \to \text{Time} \times \text{Space } d$. For any point $(t, \mathbf{x})$ consisting of a time $t$ and a spatial vector $\mathbf{x}$, the map is defined as $L_g(t, \mathbf{x}) = (t, R\mathbf{x})$, where $R$ is the orthogonal linear transformation (the rotation or reflection) associated with $g$.

For an element $g$ of the Euclidean group $E(d)$, the function `actionTranslation` returns the translation component of the group's action on spacetime $\text{TimeAndSpace } d$. It is defined as the spacetime point $(0, \mathbf{v}) \in \text{TimeAndSpace } d$, where $0$ is the zero element of $\text{Time}$ and $\mathbf{v} \in \text{Space } d$ is the translation vector associated with $g$.

For any element $g$ of the $d$-dimensional Euclidean group $E(d)$ and any point $tx$ in the spacetime $\text{TimeAndSpace } d$, the action of $g$ on $tx$ (denoted $g \cdot tx$) is equal to the sum of the linear part of the action applied to $tx$ and the translation part of the action: $$g \cdot tx = \text{actionLinearMap}(g)(tx) + \text{actionTranslation}(g)$$ where $\text{actionLinearMap}(g)$ is the continuous linear map representing the rotation/reflection component and $\text{actionTranslation}(g)$ is the spacetime translation vector associated with $g$.

For any element $g$ of the $d$-dimensional Euclidean group $E(d)$, the map $p \mapsto g \cdot p$ that defines the action of $g$ on the product space of time and space $\text{TimeAndSpace } d$ has temperate growth.

For any element $g$ of the $d$-dimensional Euclidean group $E(d)$, the action of $g$ on the product space of time and space, $\text{Time} \times \text{Space } d$, is an isometry. Specifically, the map $(t, x) \mapsto (t, g \cdot x)$ preserves the distance between any two points in $\text{Time} \times \text{Space } d$ under the metric induced by the structures of $\text{Time}$ and $\text{Space } d$.

Let $E(d)$ be the $d$-dimensional Euclidean group acting on the product space of time and space, $\text{Time} \times \text{Space } d$. For any element $g \in E(d)$, the map $p \mapsto g \cdot p$ is antilipschitz with constant 1. That is, for any two points $p_1, p_2 \in \text{Time} \times \text{Space } d$, the distance between them satisfies the inequality: $$\text{dist}(p_1, p_2) \le \text{dist}(g \cdot p_1, g \cdot p_2)$$