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## Homework 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##).[/B]

## Homework Equations

Basic kinematics equations: i.e. ##x(t) = x_0 + vt + \frac{a t^2}{2}## and so on...

## The Attempt at a Solution

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Note: This is a problem that someone already posted: https://www.physicsforums.com/threa...erated-extended-non-ideal-light-clock.893252/, however, I don't quite understand their solution.

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}{c^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}## 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!