# Galilean group time interval

I have a question that arised by reading A Course in Modern Mathematical Physics by Peter Szekeres.

He defines the galilean space $\mathbb{G}^{4}$ to be the space of events with a structure consisting of three elements:

1. Time intervals $\Delta t=t_2-t_1$
2. The spatial distance $\Delta s=|\mathbf{r_2}-\mathbf{r_1}|$
3. Motions of inertial free particles, i.e rectilinear motions, $\mathbf r (t)=\mathbf u (t) + \mathbf r_0$.

This is all fine but then he proceeds by saying:

Note that $\Delta t$ is a function on all $\mathbb G^4 \times \mathbb G^4$ while $\Delta s$ is a function on the subset of $\mathbb G^4 \times \mathbb G^4$ consisting of simultaneous pairs of events $\{((\mathbf r, t),(\mathbf r',t'))|\Delta t=t'-t=0\}$ [...]

I don't see how the time interval is a function on all $\mathbb G^4 \times \mathbb G^4$

• $\Delta t$ takes two events, e.g elements of Galilean space-time and produces a real number by forming the difference between the time-coordinates of those events. Commented Apr 28, 2014 at 9:15
• Do you understand what the Cartesian product means? Commented Apr 28, 2014 at 9:56
• Perhaps it is meant to say, that in general the same event is assigned different times for different observers due to limited speed of $c$, yet in Galilean world light travels with infinite speed, therefore $t'-t=0$? (Which would suggest there are actually two different $\Delta t$'s in your citation.) Commented Apr 28, 2014 at 10:01
• I got confused because the only things used were the time coordinates so I thought that it woud mean that the time interval was a function of a subset of $\mathbb G^4\times \mathbb G^4$ and not ALL$\mathbb {G}^4 \times \mathbb G^4$. But it can involve all the coordinates of any pair of events even though it doesn't take the spatial coordinates into account when the function is applied. I understand now the only thing that counts for considering the time interval to be a function on a SUBSET of $\mathbb G^4\times \mathbb G^4$ is that the coordinates have restrictions on the values they can take Commented Apr 28, 2014 at 10:30