Change in frequency of light clock due to to acceleration

Question

The problem statement:

A light-clock (a photon travelling between two mirrors) has proper length l and moves longitudinally through an inertial frame with proper acceleration $\alpha$ (ignore any variation of a along the rod). By looking at the time it takes the photon to make one to-and-fro bounce in the instantaneous rest-frame, show that the frequency and proper frequency are related, in lowest approximation, by $\nu = \nu_0 \gamma^{-1}(1 + \frac{\alpha l}{2c^2})$. (so the deviation from idealness is proportional to $\alpha$ and $l$).

I began by trying to figure out the time it takes the photon to go from the left mirror (@ $x = 0$) and the right mirror (@ $x = l$) in the instantaneous rest frame. In this frame, you use the basic kinematic equation giving us $cT = l + \frac{\alpha T^2}{2}$. Solving for $T$ gives us $\frac{c \pm c\sqrt{1 - \frac{2\alpha l}{c^2}}}{\alpha}$. We can do the same for the opposite direction, just solving the equation with $-ct$ instead and we get $\frac{-c \pm c\sqrt{1 - \frac{2\alpha l}{c^2}}}{\alpha}$. Adding these two together we get a total period of $\pm \frac{2c}{\alpha} \sqrt{1-\frac{2 \alpha l}{c^2}}$. We can invert this to find the frequency and expand $\frac{1}{\sqrt{1-x}}$ as $1 + \frac{x}{2}$ (assuming $c^2 >> \alpha l$) giving us an approximate frequency of $\frac{\alpha}{2c}(1+ \frac{\alpha l}{c^2})$. My issue now is that I'm not quite sure what to do up to after here. I have a few ideas: namely getting the velocity in the rest frame and then setting up a similar kinematical equation and solving for the period. However, I don't think this problem is supposed to be that complicated.... Any tips would be greatly appreciated!

Answer 1

If an observer that is closed inside the light clock measures a different time for the to-and-fro bounce, that is because his clock is somehow affected.

In the instantaneous rest frame the clock is affected by its motion.

I suggest using a transverse light clock as a clock that measures the proper time inside the big light clock. That clock is easy to analyze in the instantaneous rest frame, I hope.

Oh yes, the acceleration would destroy a normal transverse light clock, light would fall out of the clock. So the clock has to be designed so that that does not happen.

... Or maybe the clock that measures the proper time could be the longitudinal clock as we let its length to approach zero.