Is it possible that inertial motion under zero external forcing can change the rate of a mechanical or atomic clock? No! Because with zero external forcing there is nothing which can affect the inner working of a clock and thus the clock rate must be independent of inertial motion. It is inconceivable that clock rate can be affected by inertial motion.
An illusion of changed clock rate is created by the Doppler effect: The light from a receding body will be red-shifted and thus will give the illusion of a changed frequency or change of clock rate. But this is only an illusion like the illusion of the size of an object decreasing with distance.
What about special relativity then? The reading of the same event in space-time in two inertial systems $S$ and $S^\prime$ with coordinates $(x,t)$ and $(x^\prime ,t^\prime )$ moving with constant velocity $v$ with respect to each other, are here supposed to be connected by the Lorentz transformation (in one space dimension)
- $x^\prime =\gamma (x - vt)$,
- $t^\prime =\gamma (t - vx)$,
What to make out of this? Is the clock rate really affected by motion, or is it only an illusion? Lorentz gave the answer by pointing out that the transformed time $t^\prime$ is not physical time, but only illusionary time. It was Einstein who against the judgement of Lorentz claimed that the transformed time $t^\prime$ is physical time, and out came the special theory of relativity, which is no theory according e.g. Louis Essen as shown in the preceding post. An mere illusion is not a physical theory.
Real clock rate cannot change by inertial motion, only imaginary illusionary invented non-physical clock rate can change by inertial motion. This statement is believed to be wrong by 99% of living physicists. No wonder that modern physics is in a state of crisis.
Physicists have been complaining since the mid 1920s when quantum mechanics was born, that quantum mechanics is incompatible with relativity theory, without however doing anything about it.
But if two theories of physics are incompatible or contradictory, then one must go. Your choice?
See also Questioning Relativity 1-18.