Physlib.SpaceAndTime.GalileanGroup.Basic

Let $R$ be an element of the orthogonal group $O(d)$ acting on the $d$-dimensional Euclidean space $\mathbb{R}^d$. For any scalar $c \in \mathbb{R}$ and any vector $v \in \mathbb{R}^d$, the action of $R$ on the scalar multiple $c \cdot v$ is equal to the scalar multiple of the action of $R$ on $v$, that is, $R(cv) = c(Rv)$.

The identity element of the Galilean group in $d$ dimensions is defined as the transformation $1 = (\mathbb{1}, \mathbf{0}, \mathbf{0}, 0)$, where $\mathbb{1} \in O(d)$ is the identity orthogonal transformation, $\mathbf{0} \in \mathbb{R}^d$ is the zero boost velocity, $\mathbf{0} \in \mathbb{R}^d$ is the zero spatial translation, and $0 \in \mathbb{R}$ is the zero time translation. This transformation maps a spacetime point $(t, x)$ to $(t + 0, \mathbb{1}x + 0t + 0) = (t, x)$.

Let $1 \in \text{Gal}(d)$ be the identity element of the Galilean group in $d$ dimensions. The spatial rotation component of this identity element is the identity transformation $\mathbb{1}$ of the orthogonal group $O(d)$.

The boost velocity component of the identity element $1$ of the Galilean group $\text{Gal}(d)$ is the zero vector $\mathbf{0}$.

The spatial translation component of the identity element $1$ of the Galilean group $\text{Gal}(d)$ is the zero vector $\mathbf{0}$.

The time translation component of the identity element $1$ in the Galilean group $\text{Gal}(d)$ is $0$.

The multiplication operation on the Galilean group $\text{GalileanGroup}(d)$ is defined such that for two transformations $g$ and $h$, their product $g * h$ is given by: - The rotation component is the composition of rotations $R_g R_h$. - The boost velocity component is $R_g v_h + v_g$. - The spatial translation component is $a_g + R_g a_h + b_h v_g$, where $b_h$ is the numerical value of the time translation of $h$. - The time translation component is the sum $b_g + b_h$. This definition ensures that the group product corresponds to the composition of the active transformations on space-time, i.e., $(g * h) \bullet (t, x) = g \bullet (h \bullet (t, x))$.

For any two elements $g$ and $h$ of the Galilean group in $d$ dimensions, the rotation component of their product $g * h$ is equal to the product (composition) of the rotation component of $g$ and the rotation component of $h$. That is, $(g * h).\text{rotation} = g.\text{rotation} \cdot h.\text{rotation}$.

Let $g$ and $h$ be elements of the Galilean group in $d$ spatial dimensions. The velocity (boost) component of their product $g * h$ is given by $R_g v_h + v_g$, where $R_g$ is the spatial rotation (orthogonal transformation) of $g$, and $v_g, v_h$ are the velocity components of $g$ and $h$ respectively.

Let $g$ and $h$ be two Galilean transformations in $d$ spatial dimensions, where each transformation consists of a rotation $R$, a boost velocity $v$, a spatial translation $a$, and a time translation $b$. The spatial translation component of the product $g * h$ is given by: $$a_{g * h} = a_g + R_g a_h + b_h v_g$$ where $a_g$ is the spatial translation of $g$, $R_g$ is the rotation of $g$, $v_g$ is the boost velocity of $g$, $a_h$ is the spatial translation of $h$, and $b_h$ is the numerical value of the time translation of $h$.

For any two Galilean transformations $g$ and $h$ in $d$ spatial dimensions, the time translation component of their product $g * h$ is equal to the sum of their individual time translation components: $$(g * h).\text{timeTranslation} = g.\text{timeTranslation} + h.\text{timeTranslation}$$

The inverse of a Galilean transformation $g = (R, v, a, b)$ in $d$ dimensions, where $R$ is the spatial rotation, $v$ is the boost velocity, $a$ is the spatial translation, and $b$ is the time translation, is defined as: $$g^{-1} = (R^{-1}, -R^{-1}v, -R^{-1}a + b(R^{-1}v), -b)$$ This definition corresponds to the active transformation convention $(t, x) \mapsto (t + b, Rx + vt + a)$, ensuring that $g^{-1}$ is the group-theoretic inverse of $g$.

For any Galilean transformation $g$ in $d$ spatial dimensions, the rotation component of the inverse transformation $g^{-1}$ is equal to the inverse of the rotation component of $g$.

For any Galilean transformation $g$ in $d$ spatial dimensions, the boost velocity component of its inverse $g^{-1}$ is equal to the negation of the inverse spatial rotation of $g$ acting on the boost velocity of $g$. This is expressed as: $$(g^{-1}).\text{velocity} = -(g.\text{rotation}^{-1} \cdot g.\text{velocity})$$

For any Galilean transformation $g$ in $d$ spatial dimensions, let $R$, $v$, $a$, and $b$ denote its rotation, boost velocity, spatial translation, and time translation components, respectively. The spatial translation component of the inverse transformation $g^{-1}$ is given by: $$(g^{-1}).\text{spaceTranslation} = -R^{-1} a + b(R^{-1} v)$$ where $R^{-1}$ is the inverse of the rotation $R$.

For any Galilean transformation $g$ in $d$ spatial dimensions, the time translation component of its inverse $g^{-1}$ is equal to the negation of the time translation component of $g$. This is expressed as: $$g^{-1}.\text{timeTranslation} = -g.\text{timeTranslation}$$

The Galilean transformations in $d$ spatial dimensions form a group under composition. Each element of the group is a tuple $(R, v, a, b)$ consisting of a spatial rotation $R$ (an orthogonal transformation), a boost velocity $v$, a spatial translation $a$, and a time translation $b$. According to the active transformation convention $(t, x) \mapsto (t + b, Rx + vt + a)$, this structure satisfies the group axioms: composition is associative, there exists an identity element $1 = (\mathbb{1}, \mathbf{0}, \mathbf{0}, 0)$, and every transformation has a unique inverse.

The Galilean group in $d$ spatial dimensions, denoted $\text{Gal}(d)$, is an inhabited type, meaning it possesses a designated default element. This default element is defined to be the identity transformation $1 = (\mathbb{1}, \mathbf{0}, \mathbf{0}, 0)$, where $\mathbb{1}$ is the identity orthogonal transformation, and the velocity boost, spatial translation, and time translation are all zero.

For a Galilean transformation $g$ (comprising an orthogonal rotation $R$, a boost velocity $v$, and a spatial translation $a$), this function computes the transformation of a spatial point $x \in \text{Space } d$ at a given time $t$. The resulting spatial point is given by: $$R \cdot x + t \cdot v + a$$ where $R \cdot x$ is the rotation of the position vector (relative to the origin), $t \cdot v$ is the displacement due to the velocity boost, and $a$ is the constant spatial translation.

For a Galilean transformation $g$ (comprising an orthogonal rotation $R$, a boost velocity $\mathbf{v}$, a spatial translation $\mathbf{a}$, and a time translation $b$), this function defines the active transformation of a spacetime point $(t, \mathbf{x}) \in \text{Time} \times \text{Space } d$. The resulting spacetime point is given by: $$(t, \mathbf{x}) \mapsto (t + b, R\mathbf{x} + t\mathbf{v} + \mathbf{a})$$ where the first component is the time translation $t + b$, and the second component is the spatial transformation $R\mathbf{x} + t\mathbf{v} + \mathbf{a}$ involving rotation, velocity boost, and spatial translation.

The Galilean group $\text{GalileanGroup}(d)$ in $d$ spatial dimensions acts on the spacetime manifold $\text{Time} \times \text{Space } d$. For an element $g = (R, \mathbf{v}, \mathbf{a}, b)$ of the group (comprising an orthogonal rotation $R$, a boost velocity $\mathbf{v}$, a spatial translation $\mathbf{a}$, and a time translation $b$) and a spacetime point $(t, \mathbf{x})$, the action is defined by: $$(t, \mathbf{x}) \mapsto (t + b, R\mathbf{x} + t\mathbf{v} + \mathbf{a})$$ This structure satisfies the axioms of a group action: the identity transformation $1$ acts as the identity map on spacetime, and the action of a product of transformations $g * h$ is equivalent to the composition of their individual actions.

For any Galilean transformation $g \in \text{GalileanGroup}(d)$ and any spacetime point $(t, \mathbf{x}) \in \text{Time} \times \text{Space } d$, the time component of the point after the transformation is given by $(g \cdot (t, \mathbf{x}))_1 = t + b$, where $b$ is the time translation parameter of $g$.

For a Galilean transformation $g$ and a spacetime point $(t, \mathbf{x}) \in \text{Time} \times \text{Space } d$, the spatial component of the transformed spacetime point $g \cdot (t, \mathbf{x})$ is equal to the spatial action of $g$ on $\mathbf{x}$ at time $t$. This spatial action, denoted as $g.\text{actSpace}(t, \mathbf{x})$, is defined as $R\mathbf{x} + t\mathbf{v} + \mathbf{a}$, where $R$ is the spatial rotation, $\mathbf{v}$ is the boost velocity, and $\mathbf{a}$ is the spatial translation.

For any Galilean transformation $g$ in $d$ spatial dimensions and any spacetime point $(t, \mathbf{x}) \in \text{Time} \times \text{Space } d$, the group action of $g$ on $(t, \mathbf{x})$ is given by: $$g \cdot (t, \mathbf{x}) = (t + b, \text{actSpace}(g, t, \mathbf{x}))$$ where $b$ is the time translation component of $g$, and $\text{actSpace}(g, t, \mathbf{x})$ is the spatial action of the transformation (defined as $R\mathbf{x} + t\mathbf{v} + \mathbf{a}$) at time $t$.

For a Galilean transformation $g$ in $d$ dimensions consisting of a rotation $R$, a boost velocity $v$, and a spatial translation $a$, the $i$-th coordinate of the spatial action of $g$ on a point $x$ at time $t$ is given by: $$(g \cdot x)_i = (R(x - 0))_i + t \cdot v_i + a_i$$ where $i \in \{0, \dots, d-1\}$ is the coordinate index, and $0$ is the origin of the spatial coordinates.

Given a Euclidean transformation $g \in \text{EuclideanGroup}(d)$ with linear part $R$ and spatial translation $a$, this function constructs the corresponding Galilean transformation $(R, v, a, b)$ where the boost velocity $v$ and the time translation $b$ are both zero.

For any Euclidean transformation $g$ in $d$ dimensions, the rotation component of the Galilean transformation constructed from $g$ is equal to the linear part of $g$.

For any Euclidean transformation $g \in \text{EuclideanGroup}(d)$, the boost velocity of the corresponding Galilean transformation $\text{ofEuclidean}(g)$ is $0$.

For any Euclidean transformation $g \in \text{EuclideanGroup}(d)$, the spatial translation component of the Galilean transformation constructed from $g$ is equal to the translation component of $g$.

For any Euclidean transformation $g \in \text{EuclideanGroup}(d)$ in $d$ spatial dimensions, the time translation component of the Galilean transformation constructed from $g$ is $0$.

This definition is the group homomorphism $\iota: \text{EuclideanGroup}(d) \to \text{GalileanGroup}(d)$ that includes the Euclidean group into the Galilean group. For a Euclidean transformation $g = (a, R)$ consisting of a spatial translation $a$ and an orthogonal transformation $R$, the map assigns the Galilean transformation $(R, \mathbf{0}, a, 0)$, where the boost velocity $v$ and time translation $b$ are both set to zero.

Given an orthogonal matrix $R \in \mathrm{O}(d)$, this definition constructs an element of the Galilean group $\mathcal{G}_d$. This transformation represents a pure spatial rotation $R$ with zero boost velocity, zero spatial translation, and zero time translation. In the active convention, its action is given by $(t, x) \mapsto (t, Rx)$.

Let $R \in \mathrm{O}(d)$ be an orthogonal matrix. The rotation component of the Galilean transformation $\mathtt{ofOrthogonal}(R)$ (which represents a pure spatial rotation with zero boost and zero translations) is equal to $R$.

For any orthogonal matrix $R \in \mathrm{O}(d)$, the boost velocity of the Galilean transformation $\mathtt{ofOrthogonal}(R)$ is equal to $0$.

For any orthogonal matrix $R \in \mathrm{O}(d)$, the spatial translation component of the Galilean transformation constructed from $R$ (representing a pure rotation) is equal to zero.

For any orthogonal matrix $R \in \mathrm{O}(d)$, the time translation component of the pure orthogonal Galilean transformation associated with $R$ is $0$.

The group homomorphism from the orthogonal group $\mathrm{O}(d)$ to the Galilean group $\text{GalileanGroup}(d)$ that embeds spatial rotations. It maps an orthogonal matrix $R \in \mathrm{O}(d)$ to the Galilean transformation $(R, \mathbf{0}, \mathbf{0}, 0)$, which represents a pure spatial rotation with zero boost velocity, zero spatial translation, and zero time translation.

This definition is a group homomorphism from the rotation group $SO(d)$ (represented as a subgroup of the Euclidean group) to the Galilean group $\text{GalileanGroup}(d)$. For a rotation $R \in SO(d)$, the resulting Galilean transformation is a pure spatial rotation with zero boost velocity, zero spatial translation, and zero time translation. Its action on spacetime coordinates $(t, x)$ is given by $(t, x) \mapsto (t, Rx)$.

Given a spatial translation vector $a \in \mathbb{R}^d$, this function constructs a Galilean transformation representing a pure spatial translation. In the representation $(R, v, a, b)$—where $R$ is an orthogonal transformation, $v$ is a boost velocity, $a$ is a spatial translation, and $b$ is a time translation—this transformation is defined as $(1, 0, a, 0)$. Under the active convention, this transformation maps the space-time coordinates $(t, x)$ to $(t, x + a)$.

For any spatial translation vector $a \in \mathbb{R}^d$, the rotation component of the Galilean transformation representing the pure spatial translation by $a$ is the identity rotation $1$.

For any spatial translation vector $a \in \mathbb{R}^d$, the boost velocity component of the Galilean transformation representing the pure spatial translation by $a$ is $0$.

For any spatial translation vector $a \in \mathbb{R}^d$, the spatial translation component of the Galilean transformation representing a pure spatial translation by $a$ is equal to $a$.

For any spatial translation vector $a \in \mathbb{R}^d$, the time translation component of the Galilean transformation representing a pure spatial translation by $a$ is $0$.

The group homomorphism $\iota: \mathbb{R}^d \to \text{GalileanGroup}(d)$ that embeds the additive group of spatial translation vectors into the Galilean group. For a given translation vector $a \in \mathbb{R}^d$, the image $\iota(a)$ is the Galilean transformation defined by the tuple $(\mathbb{1}, \mathbf{0}, a, 0)$, where $\mathbb{1}$ is the identity transformation, $\mathbf{0}$ is the zero boost velocity, and $0$ is the zero time translation. This transformation acts on spacetime coordinates as $(t, x) \mapsto (t, x + a)$.

This definition is the group homomorphism $\iota: T(d) \to \text{GalileanGroup}(d)$ that embeds the spatial translation subgroup $T(d)$ of the Euclidean group $E(d)$ into the Galilean group. It is defined as the composition of the canonical inclusion $T(d) \hookrightarrow E(d)$ and the embedding of the Euclidean group into the Galilean group $E(d) \hookrightarrow \text{GalileanGroup}(d)$. For an element in $T(d)$ representing a spatial translation by $a \in \mathbb{R}^d$, the map assigns the Galilean transformation $(\mathbb{1}, \mathbf{0}, a, 0)$, where $\mathbb{1}$ is the identity rotation, $\mathbf{0}$ is the zero boost velocity, and $0$ is the zero time translation.

For a given time interval $b \in \text{Time}$, this definition constructs a Galilean transformation in $d$ spatial dimensions where the rotation $R$ is the identity, the boost velocity $v$ is zero, and the spatial translation $a$ is zero. This element represents a pure time translation, acting on a point $(t, x)$ in spacetime as $(t, x) \mapsto (t + b, x)$.

For any time interval $b \in \text{Time}$, the spatial rotation component $R$ of the Galilean transformation representing a pure time translation by $b$ is the identity transformation.

For any time interval $b \in \text{Time}$, the boost velocity component $v$ of the Galilean transformation representing a pure time translation by $b$ is equal to zero.

For any time interval $b \in \text{Time}$, the spatial translation component of the Galilean transformation representing a pure time translation by $b$ is equal to $0$.

For any time $b \in \text{Time}$, the time translation component of the Galilean transformation $\text{ofTimeTranslation}(b)$ is equal to $b$.

The group homomorphism $\iota : \text{Time} \to \text{GalileanGroup}(d)$ that embeds the additive group of time translations into the Galilean group. For a given time interval $b \in \text{Time}$, the map returns the Galilean transformation $(\mathbb{1}, \mathbf{0}, \mathbf{0}, b)$, which corresponds to the pure time translation $(t, x) \mapsto (t + b, x)$. Here, $\text{Multiplicative Time}$ is used to treat the additive group of time intervals as a multiplicative group to satisfy the group homomorphism signature.