Understanding the Path Difference in Michelson Morley Experiment

[physiks]

I'm a little confused about this.

I know how to derive the expression t≈lv²/c³ for the time difference between the arrival of the two beams due to the presence of an ether, with the Earth moving at speed v through the ether, and l the distance between the beam splitter and each mirror. This can be done so by working in the ether frame or the interferometer frame.

Now, ultimately I want to find the expected phase shift if the ether were to exist. To do so I need a path difference from this time difference. My book says it is given by ct, i.e s≈lv²c². I understand this in the ether frame, as the light travels at speed c. But in the interferometer frame, it doesn't have this speed, and so I don't understand what's going on there. I'm very confused. From this path difference, it's easy to get the phase difference.

Thanks

 

[haruspex]

Can't you assume the difference in reference frame speeds is small, so the difference in computed path differences is a second-order small quantity?

 

[physiks]

Can't you assume the difference in reference frame speeds is small, so the difference in computed path differences is a second-order small quantity?

That would make sense!