Physlib.SpaceAndTime.ReferenceFrame

Given a finite set of trajectories $\{p_i(t)\}_{i \in I}$, where each $p_i: \text{Time} \to \text{Space } d$ is a point in a $d$-dimensional affine space, this function constructs a reference frame. The construction requires that at every time $t$, the set of points $\{p_i(t)\}$ is affinely independent and spans the entire space $\text{Space } d$ (meaning they form an affine basis). One of these trajectories is selected as the origin $\mathbf{o}(t)$ of the frame, and the coordinate basis vectors $\mathbf{b}_j(t)$ are defined as the displacements from this origin to the remaining $d$ points in the set, i.e., $\mathbf{b}_j(t) = p_j(t) - \mathbf{o}(t)$.

The property that a reference frame's basis induces a time-independent inner product on coordinates. A reference frame satisfies this property if for any times $t_1, t_2$ and any indices $i, j$, the inner product of the $i$-th and $j$-th basis vectors at time $t_1$ is equal to their inner product at time $t_2$: $$\langle \mathbf{b}_i(t_1), \mathbf{b}_j(t_1) \rangle = \langle \mathbf{b}_i(t_2), \mathbf{b}_j(t_2) \rangle$$ where $\mathbf{b}_k(t)$ denotes the $k$-th basis vector of the frame at time $t$, and $\langle \cdot, \cdot \rangle$ denotes the inner product on the underlying displacement space.

A reference frame in $d$-dimensional space is said to be orthonormal if, for every time $t$, the basis vectors associated with the frame form an orthonormal basis for the space of displacements. This means that at any given time $t$, the basis $\mathbf{e}_i(t)$ satisfies $\langle \mathbf{e}_i(t), \mathbf{e}_j(t) \rangle = \delta_{ij}$.

If a reference frame in $d$-dimensional space is orthonormal, meaning that at any given time $t$ its basis vectors $\{\mathbf{e}_i(t)\}_{i=1}^d$ satisfy $\langle \mathbf{e}_i(t), \mathbf{e}_j(t) \rangle = \delta_{ij}$, then the frame satisfies the property of metric conservation. That is, for any two times $t_1, t_2$ and any indices $i, j$, the inner product of the basis vectors remains constant: $$\langle \mathbf{e}_i(t_1), \mathbf{e}_j(t_1) \rangle = \langle \mathbf{e}_i(t_2), \mathbf{e}_j(t_2) \rangle$$

If a reference frame in $d$-dimensional space is orthonormal, then its metric is conserved. Specifically, if for every time $t$ the basis vectors $\{\mathbf{e}_i(t)\}$ of the frame satisfy $\langle \mathbf{e}_i(t), \mathbf{e}_j(t) \rangle = \delta_{ij}$, then for any two times $t_1$ and $t_2$, the inner product of the basis vectors remains constant: $\langle \mathbf{e}_i(t_1), \mathbf{e}_j(t_1) \rangle = \langle \mathbf{e}_i(t_2), \mathbf{e}_j(t_2) \rangle$.

For an inertial reference frame in $d$-dimensional space, this definition provides the constant, time-independent velocity vector $\mathbf{v} \in \mathbb{R}^d$ of the frame's coordinate origin. This value is extracted from the property that the origin of an inertial frame moves uniformly through space.

If a reference frame is inertial, then it conserves its coordinate metric. This means that for any two times $t_1$ and $t_2$, and any indices $i, j$, the inner product of the basis vectors $\mathbf{b}_i$ and $\mathbf{b}_j$ of the frame is time-independent: $$\langle \mathbf{b}_i(t_1), \mathbf{b}_j(t_1) \rangle = \langle \mathbf{b}_i(t_2), \mathbf{b}_j(t_2) \rangle$$ where $\langle \cdot, \cdot \rangle$ denotes the inner product on the underlying displacement space.

If a reference frame is inertial, then its metric is conserved. Specifically, for an inertial frame, the inner product of the basis vectors $\mathbf{b}_i(t)$ and $\mathbf{b}_j(t)$ is independent of time $t$, such that for any times $t_1, t_2$ and any indices $i, j$: $$\langle \mathbf{b}_i(t_1), \mathbf{b}_j(t_1) \rangle = \langle \mathbf{b}_i(t_2), \mathbf{b}_j(t_2) \rangle$$ where $\langle \cdot, \cdot \rangle$ denotes the inner product on the underlying displacement space.

For a reference frame $f$ in a $d$-dimensional space, this definition establishes an equivalence (bijection) between the type of vectors associated with that frame ($f.\text{Vector}$) and their representation as coordinate components in $\mathbb{R}^d$, modeled as functions from the index set $\{0, \dots, d-1\}$ to the real numbers $\mathbb{R}$.

For a given reference frame $f$ in a $d$-dimensional space, the type $f.\text{Vector}$, which represents coordinate vectors relative to that frame, is equipped with the structure of an additive commutative group. This means that vectors in $f.\text{Vector}$ can be added and subtracted, and there exists a zero vector, with these operations following the standard commutative and associative laws. This group structure is derived from the component-wise operations on $\mathbb{R}^d$ via the equivalence $f.\text{Vector} \simeq \mathbb{R}^d$.

For a given reference frame $f$ in a $d$-dimensional space, the type $f.\text{Vector}$ (representing coordinate vectors relative to that frame) is equipped with the structure of a module over the real numbers $\mathbb{R}$. Because $\mathbb{R}$ is a field, this defines $f.\text{Vector}$ as a real vector space. This structure allows for the scalar multiplication of coordinate vectors by real numbers, with the operations being defined component-wise via the equivalence $f.\text{Vector} \simeq \mathbb{R}^d$.

For a reference frame $f$ in a $d$-dimensional space, this definition establishes a linear equivalence (an isomorphism of real vector spaces) between the type of vectors relative to that frame, $f.\text{Vector}$, and the space of coordinate components $\mathbb{R}^d$, modeled as functions from the index set $\{0, \dots, d-1\}$ to the real numbers $\mathbb{R}$.

For a given reference frame $f$ and a time $t$, this definition establishes a linear equivalence (an isomorphism of real vector spaces) between the space of coordinate vectors relative to the frame, $f.\text{Vector}$, and the space of geometric displacements, modeled as the Euclidean space $\mathbb{R}^d$. This equivalence is defined by the basis of the reference frame at time $t$, allowing coordinate-based vectors to be interpreted as physical displacements in space.

For a reference frame $f$ in a $d$-dimensional space, the set of frame vectors $f.\text{Vector}$ is endowed with a topological space structure. This topology is defined as the product topology on the coordinate components, induced by the equivalence between $f.\text{Vector}$ and $\mathbb{R}^d$ (modeled as functions $\text{Fin } d \to \mathbb{R}$).

For a given reference frame $f$ in a $d$-dimensional space, this definition establishes a continuous linear equivalence (a topological vector space isomorphism) between the space of coordinate vectors relative to that frame, $f.\text{Vector}$, and the space of coordinate components $\mathbb{R}^d$, modeled as the type of functions $\text{Fin } d \to \mathbb{R}$.

For a given reference frame $f$ in a $d$-dimensional space, the space of coordinate vectors $f.\text{Vector}$ is a finite-dimensional vector space over the real numbers $\mathbb{R}$.

For a given reference frame $f$ and a time $t$, this definition establishes a continuous linear equivalence (a topological vector space isomorphism) between the space of coordinate vectors $f.\text{Vector}$ and the space of geometric displacements in $d$-dimensional space, represented as the Euclidean space $\mathbb{R}^d$. This mapping is determined by the frame's basis at time $t$.

For a reference frame $f$ that satisfies the property of metric conservation (i.e., $f.\text{IsMetricConserved}$ holds), the space of coordinate vectors $f.\text{Vector}$ is equipped with the structure of a normed additive commutative group. The norm $\|\mathbf{v}\|$ of a vector $\mathbf{v} \in f.\text{Vector}$ is defined as the physical length of its corresponding geometric displacement, pulled back from the Euclidean space of displacements at time $t=0$.

For a reference frame $f$ that satisfies the property of metric conservation (i.e., $f.\text{IsMetricConserved}$ holds), the space of coordinate vectors $f.\text{Vector}$ is equipped with the structure of a real inner product space. The inner product $\langle \mathbf{v}, \mathbf{w} \rangle$ of two vectors $\mathbf{v}, \mathbf{w} \in f.\text{Vector}$ is defined by pulling back the standard inner product from the space of geometric displacements at time $t=0$: $$\langle \mathbf{v}, \mathbf{w} \rangle := \langle \phi_0(\mathbf{v}), \phi_0(\mathbf{w}) \rangle_{\mathbb{R}^d}$$ where $\phi_0 : f.\text{Vector} \xrightarrow{\simeq} \mathbb{R}^d$ is the linear equivalence (`dispEquiv 0`) that maps coordinate vectors to physical displacements at the initial time $t=0$.

In an orthonormal reference frame, for any coordinate vector $v$, the square of its norm is equal to the sum of the squares of its components: $$\|v\|^2 = \sum_{i} v_i^2$$ where $v_i$ denotes the $i$-th component of the vector $v$.

In an orthonormal reference frame, for any two coordinate vectors $v$ and $w$, their inner product $\langle v, w \rangle$ is equal to the sum of the products of their corresponding components: $$\langle v, w \rangle = \sum_{i} v_i w_i$$ where $v_i$ and $w_i$ denote the $i$-th components of the vectors $v$ and $w$ respectively.