# How can a mechanical clock tick slower due to time dilation?

I've rarely seen explicit demonstrations along these lines. It would certainly be very difficult; for most clocks one would at the very least have to use relativistic quantum mechanics.

In general we don't want to re-prove time dilation for every kind of clock. Instead the reasoning runs the other way: if a light clock and another clock are ticking side by side, they'd better also be ticking side by side in another frame, so the other clock must also experience time dilation. Since clocks can rely on classical mechanics, or quantum mechanical effects, or anything else, that means that all our theories should be compatible with special relativity. So we build the theory to be relativistic from the very start. Once you've done that it's not really worth bothering to double-check relativity works in a specific case, as we already know it's going to work for all cases.

Still, this is a neat question, so let's do the check anyway! For simplicity I'll consider a 'magnetic' clock. The idea is that a vertical magnetic field $B_z$ makes a particle move in circles, and the clock ticks every time one circle is completed. In the clock's frame, supposing the particle moves slowly enough to neglect relativity, we have $$q v B = \frac{dp}{dt} = ma, \quad a = \omega v$$ since we're dealing with circular motion, and combining gives $$\omega = \frac{qB}{m}.$$ Now consider a frame where the clock is moving along the $z$ direction with time dilation factor $\gamma$. Then the momentum is now $\mathbf{p} = \gamma m \mathbf{v}$, so the only modified equation is $$\frac{dp}{dt} = \gamma ma$$ and we instead find $$\omega = \frac{qB}{\gamma m}$$ which is exactly as we'd expect by time dilation. Physically, the reason the clock ticks slower is that the particle is harder to turn around, by virtue of its relativistic motion in the $z$ direction. With a bit of handwaving, this explanation works for the quartz clock too. The gears in the clock have nothing to do with it -- what matters are the resonant oscillations in the piece of quartz, which control everything else. So it's plausible these oscillations get slower as each particle participating in them gets harder to accelerate.