Zoran Skoda Galilean structure

A Galilean structure is a 5-tuple $G = (\mathcal{B}^4, \mathbb{R}^4, \mu, t, (,))$ where $(\mathcal{B}^4, \mathbb{R}^4, \mu)$ is a 4-dimensional affine space, $t : \mathbb{R}^4 \rightarrow \mathbb{R}$ is a nonzero $\mathbb{R}$-linear map and $(,) : ker\, t \times ker\,t \rightarrow \mathbb{R}_{\geq 0}$ is a real inner product on 3-dimensional $\mathbb{R}$-vector space $ker\,t$.

The elements of $\mathcal{B}^4$ will be called events. For any two events $a$ and $b$ we define the time interval between them to be $t(b - a)$. If the time interval between events $a$ and $a'$ is $0$ then we say that $a$ and $a'$ are simultaneous events. Fix an event $a$. The set $\mathcal{B}^3_a$ of all events which are simultaneous with $a$ is simply $a + ker\,t$. Action of $\mathbb{R}^4$ on $\mathcal{B}^4$ thus restricts to an action of $ker\,t$ on $\mathcal{B}^3_a$ which therefore becomes a 3-dimensional affine space.

If $a,a'$ are simultaneous events we can speak of the space distance between them. It is simply the length of the parallel displacement $a' - a$ with respect to the norm induced by inner product on $ker\,t$:

(1)$$d_3(a,a') = \| a' - a \| = \sqrt{(a' - a, a' - a)}.$$

However there is no natural choice of space distance between the events which are not simultaneous. We can not even say for two such events if they took place at the same point. Indeed, imagine that we define such a notion. Then the naturality would mean that that notion is determined solely in terms of Galilean structure. In other words it should be preserved by the maps between Galilean structure which preserve it in the following sense.

A map $A : \mathcal{B}^4 \rightarrow \mathcal{B}^4$ is called a Galilean transformation if it preserves the Galilean structure. More precisely, $A$ is an affine map such that

(2)$$t(b - a) = t(A(b) - A(a))$$

and $(b - a, c - a) = (A(b) - A(a), A(c) - A(a))$ for any three events $a,b,c \in \mathcal{B}^4$.

Definition. Let $v \in ker\,t$. The uniform motion $A_{v,a}$ with velocity $v$ cooccuring with event $a$ is the Galilean transformation given by

(3)$${\cal A}_{v,a}(a + w)= a + w + t(w)v\,.$$

The problem with non-simultaneous events above can now be restated as assertion that for any two non-simultaneous events $a,b \in \mathcal{B}^4$ and any event $b'$ simultaneous with $b$, the uniform motion $A_{v,a}$ where $v = \frac{b' - b}{t(b-a)} \in ker\,t$ leaves $a$ intact and changes $b$ into $b'$. Thus if $a$ and $b$ ‘took place’ at the same point by some magic definition, that notion will be ruined by the Galilean transformation mentioned.

Lemma. Let $p \neq q$ be two points in some affine space $\mathcal{B}$ over a division ring $k$. The set

(4)$$\mathcal{B}^1(p,q) = \{ p + \alpha (q-p) \,|\, \alpha \in k \}$$

has a natural structure of a left 1-dimensional $k$-vector space and therefore also the natural structure of a 1-dimensional $k$-affine space. The inclusion of sets $\iota : \mathcal{B}^1(p,q) \rightarrow \mathcal{B}$ is an affine map.

Lemma. Let $p$ and $q$ be two non-simultaneous events in a Galilean structure $G = (\mathcal{B}^4, \mathbb{R}^4, \mu, t, (,))$. Then for every point $r \in \mathcal{B}^4$ there is exactly one point $r' \in \mathcal{B}^4$ simultaneous with $r$ and belonging to $\mathcal{B}^1(p,q)$.