# Why isn't time reversal a Galilean transformation?

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.

• I don't think this question is well-defined, since different sources define "Galilean transformation" differently. E.g. Wikipedia accepts only rotations and not reflections for $R$. Sources can differ in this because in the end it rarely matters whether reflections and time inversions are included or not. Do you have a specific problem where it would matter? Commented Sep 5, 2021 at 10:31
• The Galilean transform has no problem in time reversion in Newtonian mechanics. The deficiency of Galilean transform is failed in observing the equivalent principle for the Maxwell equation.
– ytlu
Commented Sep 5, 2021 at 11:04
• Is time inversion and reflections even included in other formalisms (like special relativity)? Commented Sep 5, 2021 at 11:11