Why does the frequency of a light clock change when it is accelerated?

[Mentz114]

That was careless of me ( unless the web page has been changed). It makes no difference to my result because $\dot{x}^2+\dot{y}^2=1$. I don't think the curvature can be constant in time because that would imply a circular path for the falling light beam.

 

[yuiop]

... It makes no difference to my result because $\dot{x}^2+\dot{y}^2=1$.

I think this points to the problem. $\dot{x}^2+\dot{y}^2=1$ is the instantaneous tangential velocity of the light particle and the velocity of light is not c according to an observer accelerating in the x direction, when the x coordinate is not constant. By analogy, the coordinate velocity of a radially moving light particle in the Schwarzschild metric is not equal to c, but c(1-2M/r).

I don't think the curvature can be constant in time because that would imply a circular path for the falling light beam.

The path is circular, but the light does not eventually return to the observer, because the path is cut off by the Rindler horizon, so the path is actually a semi circle.

 

[Mentz114]

I think this points to the problem. $\dot{x}^2+\dot{y}^2=1$ is the instantaneous tangential velocity of the light particle and the velocity of light is not c according to an observer accelerating in the x direction, when the x coordinate is not constant. By analogy, the coordinate velocity of a radially moving light particle in the Schwarzschild metric is not equal to c, but c(1-2M/r).

My analysis is done in flat spacetime so the analogy is inapposite, but you're right about the difference between the accelerated and unaccelerated frame. Remember that we 'Newtonized' the velocities in the x,y plane (following E&R ). The 4-velocity tells a different story. In fact, the curvature of $dx/d\lambda$ and $dy/d\lambda$ agrees with your result ! The curvature is -g, always.

The path is circular, but the light does not eventually return to the observer, because the path is cut off by the Rindler horizon, so the path is actually a semi circle.

I'm not sure if this agrees with E&R bcause they say that the acceleration depends on $t$ but is $g$ at $t=0$

Remember that we can't compare our results directly. ** It seems we can, see below. Your acceleration comes from the $g$ parameter in the metric, mine comes from a boost with $\beta=gt$. I'm working in local frames.

I'm confident that my calculations are correct and with the framework specified, make sense physically.

[edit warning] - I redid the curvature of the velocities parametrized with an affine parameter and it agrees with your result.

 

[yuiop]

It is good to see we are converging on some sort of agreement. As you point out I think we were analysing slightly different cases. Mine was from the point of view of a stationary Rindler observer where g and $\varphi$ are constants. I know from the methods I used in the analysis that no approximations were used and the result is exact. Your analysis is for the point of view of successive inertial reference frames that are momentarily co-moving with the accelerating elevator (I think).