I have a question about the reason Feynman gives for why a perfect clock - a clock that remains in sync while in motion with a stationary clock, cannot exist. It is clear why the "light clock" (described in Section 15-4: Transformation of Time) while in motion must appear to run slower to the stationary observer. Feynman gives the following argument for why any clock (irrespective of how it works) must run slower.
Feynman's argument runs as follows:
Not only does this particular kind of clock run more slowly, but if the theory of relativity is correct, any other clock, operating on any principle whatsoever, would also appear to run slower. [...] Why is this so?
To answer the above question, suppose we had two other clocks made exactly alike [...]. Then we adjust these clocks so they both run in precise synchronism with our first clocks. [...] One of these clocks is taken into the space ship, along with the first kind. Perhaps this clock will not run slower, but will continue to keep the same time as its stationary counterpart, and thus disagree with the other moving clock. Ah no, if that should happen, the man in the ship could use this mismatch between his two clocks to determine the speed of his ship, which we have been supposing is impossible. We need not know anything about the machinery of the new clock that might cause the effect—we simply know that whatever the reason, it will appear to run slow, just like the first one.
Let me call the two light clocks $L_1$ (stationary), $L_2$ (moving), and the two supposedly perfect clocks of unknown mechanism as $P_1$ (stationary), $P_2$ (moving). The observer in the space ship will never see a mismatch between his two local clocks ($L_2$ and $P_2$). It is the the stationary observer who sees $L_2$ slowing down with respect to his local clocks ($L_1$ and $P_1$).
Nothing in the argument seems to prevent $P_2$ appearing normal to the stationary observer (i.e., the stationary observer can see $P_2$ in sync with $L_1$ and $P_1$). To the stationary observer $L_2$ and $P_2$ appear to disagree, but the observer in the space ship sees them in sync. Hence there is no violation of the principle of relativity.