Inertial frame of reference

Remarks

Example: if a frame with coordinates $(t, x)$ is inertial and $u \in \R^3$ is fixed, then the frame with coordinates $(t, x')$, where $x' = x - u t$, is inertial. Indeed a free particle $x(t) = x_0 + v t$ has $x'(t) = x_0 + (v - u) t$, again of constant velocity. Non-example: let $x'$ be the coordinates of a frame that rotates about the third axis with angular speed $\omega \ne 0$ relative to an inertial frame. A particle at rest at $x = (1, 0, 0)$ in the inertial frame has $x'(t) = (\cos \omega t, -\sin \omega t, 0)$, a circle, so the rotating frame is not inertial. Newton's first law can be read as the claim that inertial frames exist.