# Affine spaces and time-varying vector fields

Consider the following affine space $\mathbb{G}$
1. a four-dimensional vector space $G_{v}^{4}$ over field $\mathbb{R}$ which acts (sharply transitive) on a set $G_{p}^{4}$
2. a surjective linear functional from $G_{v}^{4}$ to its field, which kernel is isomorphic with three-dimensional Euclidean vector space
$$t_{L}\colon G_{v}^{4}\to \mathbb{R}\colon\text{Ker}(t_{L})\cong E_{v}^{3}$$
(I will treat Euclidean space $\mathbb{E}^{n}$ in the strict sense: an affine space where inner product space $E_{v}^{n}$ acts on point space $E_{p}^{n}$)

This affine space is known as Galilean space-time as defined for example here and here.

This definition induces two metrics in Galilean point space $G_{p}^{4}$. The first is called the time difference:
$$d_{t}\colon G_{p}^{4}\times G_{p}^{4}\to \mathbb{R}\colon (p,q)\mapsto d_{t}(p,q)=t_{L}(q-p)$$
The following equivalence relation is derived from this metric: the points in $G_{p}^{4}$ (called events) are "simultaneous" when their time difference is zero. This partitions $G_{p}^{4}$ in equivalence classes
$$\text{Cl}_{\text{sim}}(p)=\{q\in G_{p}^{4}\colon d_{t}(p,q)=0\}$$
The difference vectors in a class are given by $\text{Ker}(t_{L})$ so a second metric is induces in each equivalence class, which is called the distance between simultaneous events
$$d_{e}\colon \text{Cl}_{\text{sim}}(p)\times \text{Cl}_{\text{sim}}(p)\to \mathbb{R}\colon (p_{1},p_{2})\mapsto d_{e}(p_{1},p_{2})=d(p_{1},p_{2})=||p_{2}-p_{1}||$$
where "d" the Euclidean distance in Euclidean point space, which is itself induced by the norm in Euclidean vector space $E_{v}^{3}$. So Galilean point space $G_{p}^{4}$ is actually a union of classes which are all isomorphic with Euclidean point space $E_{p}^{3}$.

In this rather basic space-time I want to start working with scalar and vector fields, which are defined as the functions
$$f\colon\mathbb{R}^{3}\to\mathbb{R} \colon\vec{x}\mapsto f(\vec{x})$$
$$\vec{F}\colon\mathbb{R}^{3}\to\mathbb{R}^{3}\colon\vec{x}\mapsto\vec{F}(\vec{x})$$
where $\mathbb{R}^{3}$ Euclidean coordinate space isomorphic with $E_{v}^{3}$.

Now my question: how do I connect in a strictly mathematical sense the concept of Galilean space-time with scalar and vector fields (which will be time-varying)?

I tried concidering Galilean coordinate space $\mathbb{R}^{4}$ which is isomorphic with Galilean vector space $G_{v}^{4}$ so that
$$t_{L}\colon \mathbb{R}^{4}\to \mathbb{R}\colon\text{Ker}(t_{L})\cong \mathbb{R}^{3}$$
so I can define a scalar or vector field on the kernel of $t_{L}$ but I don't know how to formalize the time-varying aspect of the scalar or vector field. Any ideas?

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