Galilean transformation

Remarks

Each Galilean transformation is an affine map, and the Galilean transformations form a group under composition. It maps inertial frames to inertial frames: in terms of the new time $t' = t + s$, a free motion $x(t) = x_0 + u t$ becomes $x'(t') = R x_0 + a + (R u + v)(t' - s)$, again of constant velocity. Example: under the boost with velocity $v$, a particle with constant velocity $u$ gets velocity $u + v$; this is the Galilean rule for adding velocities. Non-example: the map $(t, x) \mapsto (t, x + t^2 e)$ with $e \ne 0$ is not Galilean, since it is not affine in $t$; it describes an accelerating frame. A Lorentz transformation is not Galilean either, since it changes the time coordinate by an amount that depends on $x$. Some texts require $\det R = 1$, that is, they leave out reflections.