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 R

^{4}characterized by four coordinates (

*t*

_{1},

*t*

_{2},

*t*

_{3},

*s*). Define

**Galilean time**G

^{4}to be the time of events with a structure consisting of three elements:

- Distance intervals Δ
*s*=*s*_{2}−*s*_{1}. - The duration intervals Δ
*t*= |**q**_{2}−**q**_{1}| between any pair of**simulstanceous events**(events having the same stance coordinate,*s*_{1}=*s*_{2}). - Motions of facilial (free) particles, otherwise known as rectilinear motions,

**q**(*s*) =**w***s*+**q**_{0}, (2.19)

where**w**and**q**_{0}are arbitrary constant vectors.