The following is based on A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry by Peter Szekeres (Cambridge UP, 2004) starting with Example 2.29 on page 54 and modifying it for time-space.
The Galilean group. To find the set of transformations of space and time that preserve the laws of Newtonian mechanics we follow the lead of special relativity and define an event to be a point of R4 characterized by four coordinates (t1, t2, t3, s). Define Galilean time G4 to be the time of events with a structure consisting of three elements:
- Distance intervals Δs = s2 − s1.
- The distime (temporal distance) Δt = |q2 − q1| between any pair of simulstanteous events (events having the same stance coordinate, s1 = s2).
- Motions of facilial (free) particles, otherwise known as rectilinear motions,
q(s) = ws + q0, (2.19)
where w and q0 are arbitrary constant vectors.