Why is simultaneity a requirement for the distance function of Galilean space?

Question

At the end of Chapter 2 of his Course in Mathematical Physics, Szekeres discusses the notion of a symmetry group. I present my definition, adapted from his, here:

We say that a transformation $g: X \to X$ (i.e. $g \in$ Transf($X$)) leaves the $Y_i$-valued function $F_i$ invariant if, for all $x \in X$, we have $F_i(x) = F_i(gx)$.We have used the shorthand $gx \equiv g(x)$ since $g \in$ Transf($X$) so that it’s (left) action on $G$ is trivially defined by $g \mapsto g$.

We say that a transformation leaves a space $X$ equipped with a set of functions $\mathcal{F}$ invariant if it leaves each $F_i \in \mathcal{F}$ invariant.

The set of all transformations $g$ which leave $\mathcal{F}$ on $X$ invariant is called the symmetry group of that space. I will use the notation Sym$(X,\mathcal{F})$ for this group.

In general, the "functions on the space" may actually be functions of, e.g., Cartesian products of the underlying set (or subsets of the powerset).

Later, Szekeres defines Galilean space:

Define an event to be a point of $\mathbb{R}^4$ characterized by four coordinates $(x, y, z, t)$. Define Galilean space $\mathbb{G}^4$ to be the space of events with a structure consisting of three functions defined thereon:

1. Time intervals $\Delta t := t_2 − t_1$.
2. The spatial distance $\Delta s = |\mathbf{r}_2 − \mathbf{r}_1|$ between any pair of simultaneous events (events having the same time coordinate, $t_1 = t_2$).
3. Motions of inertial (free) particles, otherwise known as rectilinear motions, $\mathbf{r}(t) = \mathbf{u}t + \mathbf{r}_0$, (2.19) where $\mathbf{u}$ and $\mathbf{r}_0$ are arbitrary constant vectors.

Finally, he adds the following discussion:

Note that only the distance between simultaneous events is relevant. A simple example should make this clear. Consider a train travelling with uniform velocity v between two stations A and B. In the frame of an observer who stays at A the distance between the (non-simultaneous) events E1 = ‘train leaving A’ and E2 = ‘train arriving at B’ is clearly $d = vt$, where t is the time of the journey. However, in the rest frame of the train it hasn’t moved at all and the distance between these two events is zero! Assuming no accelerations at the start and end of the journey, both frames are equally valid Galilean frames of reference.

My two questions are as follows:

(1) I have not studied special relativity or mathematical classical mechanics, but I suppose I am to understand that the Galilean group (symmetries of the space described above) is isomorphic to (or defined as?) the set of valid inertial reference frames?

(2) How am I to understand the reference frame example in the context of the above? I believe Szekeres is pointing out that if we allowed distances to be defined between non-simultaneous events, then we would find that different reference frames would "compute" different distances between events. This would, in turn, contradict the axiom of classical mechanics/Galilean relativity that different inertial frames of reference determine the same distances between points. Is this understanding correct? I have bolded the step which I cannot justify; is this an axiom as stated, does it derive from something I've written above, or is it derivable from something I'm missing?

Edit: I'm adding a "bonus" question (bonus since I don't want to layer on an unrelated question, but it's somewhat related -- feel free to indicate if I should just ask this as a separate question). Szekeres goes on to define the Lorentz group by saying that the space of events has on it a function which encodes the "preservation of the light cone":

The Galilean transformations do not preserve the light cone at the origin $$\Delta x^2 + \Delta y^2 + \Delta z^2 = c^2\Delta t^2.$$

Szekeres then goes on to derive a requirement on the form of the symmetry transformations, deriving the structure of the Poincare group thereform. What I am confused about is whether Minkowski space as so defined is to be understood as the same space as Galilean space above (i.e. the 3 functions defined above are still part of the set of functions $\mathcal{F}$ on the space, and we also add that the function $$\Delta s^2 := \Delta x^2 + \Delta y^2 + \Delta z^2 - c^2\Delta t^2$$ must be invariant under the given symmetry transformation. Or are we removing one of the three functions above?

Answer 1

1. The Galilean group is the set of transformations from one inertial reference frame to another.
2. It's not so much an axiom as part of the definition of "distance." What we're looking for is a measurable quantity which is the same in all frames of reference. The spatial distance between simultaneous events fits this description; the spatial distance between non-simultaneous events does not. It's not illegal to talk about the latter, if you'd like, but since it depends entirely on which reference frame you're calculating it in, it does not have any physical meaning in its own right.
3. We're removing the entire concept of simultaneity. Two events which are simultaneous in one frame are not simultaneous in another.

Answer 2

(1) The term isomorphic is not appropriate. A group isomorphism relates two groups that have the same structure. However, the set of inertial frames do not have a natural operation so do not form a group.

The correct mathematical description is that the Galilean transformations have a natural simple transitive action on the inertial frames simply transitive. A typical example is an affine space and its associated vector space. In particular, there are one to one correspondence between the two, but no distinguished one. Intuitively, inertial frames are Galilean transformations up to the identity.

(2) Yes it is correct. Basically, the spatial distance between two events is frame dependent. This is a reformulation of your bolded sentence. The author gives an explicit example of this discrepancy. If you want a more formal example, say you start with a frame $x,t$ and another frame $x’,t’$ related by a Galilean transformation: $$ \begin{align} x’&=x-vt\\ t’&=t \end{align} $$ The spatial increments between two events $1$ and $2$ are related by: $$ \begin{align} d&=x_1-x_2 \\ d’&=x_1’-x_2’\\ &= d+v(t_2-t_1) \end{align} $$ so are different iff $t_1\neq t_2$ (not simultaneous).

(Bonus) Yes you remove the 3 functions defining Galilean relativity. You only conserve the spacetime metric. To avoid confusion, I would recommend referring to as Galilean/Minkowski spacetime not just space.

Hope this helps.