I'm a mathematician learning physics from scratch, starting from Newtonian mechanics. As far as I understand, Galilean transformations are defined as transformations of space-time that transform from one inertial frame to another. In turn, an inertial frame is a frame of reference in which Newton's first law holds: a body not acted upon by any force will move in linear motion. From those two definitions, it seems like Galilean transformations should be all transformations of space-time that preserve linear motion.
However, Galilean transformations are then described as compositions of:
- Translations of space and time: $(\mathbf{x}, t)\mapsto(\mathbf{x}+\mathbf{x}_0, t+t_0)$.
- Orthogonal transformations of space: $(\mathbf{x}, t)\mapsto(R\mathbf{x}, t)$ where $R:\mathbb{R}^3\rightarrow\mathbb{R}^3$ is an orthogonal transformation. Note that it includes both rotations and reflections of space.
- Rectilinear motion: $(\mathbf{x}, t)\mapsto(\mathbf{x}+t\mathbf{v}_0, t)$.
All of them definitely preserve linear motion, but they are not the only ones. We also have:
- Linear transformation of space: $(\mathbf{x}, t)\mapsto(T\mathbf{x}, t)$ where $T:\mathbb{R}^3\rightarrow\mathbb{R}^3$ is an invertible linear transformation.
- Time stretching: $(\mathbf{x}, t)\mapsto(\mathbf{x}, at)$, $a>0$.
- Time reversal: $(\mathbf{x}, t)\mapsto(\mathbf{x}, -t)$.
I see why the first two are problematic: they do not preserve measurements of distances and time intervals (so the definition of galilean transformations should really mention that too...). But what's wrong with time inversion? It doesn't seem to be any more "problematic" than reflections of space. Both preserve quantitative measurements but invert the orientation.
