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#### TubbaBlubba

**1. A light-clock (a photon travelling between two mirrors) has proper length l and moves longitudinally through an inertial frame with proper acceleration a (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 F = fγ**

^{-1}(1 + al/2c^{2}). (so the deviation from idealness is proportional to a and l).**2. Homework Equations**

x^2 - c^2 t^2 = c^4 / a^2 = X^2

**3. The Attempt at a Solution**

So the proper frequency is proportional to 1 / l, meaning that under uniform motion the factor would simply be 1/γ. I'm really quite boggled at it - I've tried a lot of approaches, but I can't really figure out where the al/c^2 factor comes from (I just know it as a measure of the "magnitude" of relativistic acceleration), or how to get both that parenthesis (a taylor expansion of... what?) AND a gamma in there. I get the basic idea, it's like Bell's Spaceships - if you begin accelerating uniformly, the front end will accelerate more quickly than the back end. However, if you "ignore the variation of a" then the factor of stretching is simply γ, as for an ideal clock, irrespective of the acceleration. The key, therefore, must be the fact that it takes some finite end for the photon to make the "round trip", and that we have to account for both the back end moving toward it and the front end moving away from it, over some small time interval, but I can't figure out a good way to work that out. The taylor expansion within the parentheses looks like it could be cosh(√(al) / c), but again, not sure where that would come from. The physical interpretation of the dimensionless quantity al/c^2 is actually not totally clear to me.

I realize my inadequacy with regard to hyperbolic geometry plays a part in this, and that's kind of why I've been obsessing over this problem for more than a day with little progress. The text is Wolfgang Rindler's 2006 "Realtivity: Special and General", for what it's worth.

I'm thankful for some hints on how to approach this.

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