We explore a simplified version of spacetime with only 1 space dimension.
Mathematics of Spacetime in 1+1 Dimensions
In 1+1 spacetime we consider only the following 3 motions:
- d -> displacement one unit in space (instantly)
- h -> displacement one unit in time
- b -> acceleration of one unit of speed (instantly)
These motions respect the following 3 identites:
- Moving then waiting is the same as waiting then moving: \(dh=hd\)
- Moving then accelerating is the same as accelerating then moving: \(db=bd\)
- Accelerating then waiting gets you further than waiting then accelerating: \(hb=dbh\)
These 3 motions with their 3 “commutation relations” are enough to fully describe the group. The details are developped in the Galilean 1+1 notebook.
This notebook dwelves into the infinitesimal aspect of the group with a study of its lie algebra.
Living on A Finite Line
Here we make the Galilean group (artificially) finite by imposing the relations $B^3=D^3=H^3=e$. It gives a 27-element finite group which has the same commutation relationships as the Galilean group.
In the Finite Galilean notebook we explore topics of representation theory, non-commutative harmonic analysis, character theory, complex and real irreps, all in the finite context.