Unraveling the Michelson-Moorley Experiment

[Paul77]

The Michelson Moorley experiment to detect the presence of aether is the main experiment
that gave rise to the theory of relativity. I've been trying to understand this experiment.

The idea of the experiment was to split a beam of light and then align one the split beams with the aether and show that this would speed up one beam compared with the other but the interference pattern of the re-combined beams did not change, hence the aether was proved not to exist.

I think this then left the unexpected result that when one of the split beams was aligned
with say the path of the Earth round the sun this should alter the distance traveled by one beam compared with the other and therefore the alignment of the wavelengths and so effect the interference pattern but it did'nt.

The experimental layout I have been looking at is on this webpage:-
http://www.phys.unsw.edu.au/einsteinlight/jw/module3_M&M.htm

In trying to understand the experiment I came up with the following. If the horizontal arm is inline with say the path of the Earth round the sun then the mirror on this arm will be traveling at around 1 x 10^4 m/s and when a photon leaves the beam splitter it will be traveling at 3 x 10^8 m/s. Hence if the mirror it bounces back off is 1m away from the splitter, in the time it takes the photon to travel 1m, 3 x 10^-8 s, the mirror will have moved 1 x 10^4 x 3 x 10^-8 = 3 x 10^-4 m or about 1000 wavelengths of visible light. So the horizontal beam will be traveling a different length compared with the vertical beam. However when the horizontal beam starts to travel back the splitter will travel the same distance towards it and hence cancel the effect out so I would expect the two beams to be back in synch!

There are other things to consider such as the alignment of the beams with the rotation of the Earth on its axis. The experiment always produces the same result - no change in the fringe pattern but its confusing that the setup in the simplified way I looked at it does not appear to setup the two beams to be on different length paths.

Have I mis-interpreted this experiment or does anyone have a qualitative explaintation of why this setup does alter the path lengths and hence why it was expected to produce a change in the fringe pattern?

 

[Dale]

If the horizontal arm is inline with say the path of the Earth round the sun then the mirror on this arm will be traveling at around 1 x 10^4 m/s and when a photon leaves the beam splitter it will be traveling at 3 x 10^8 m/s. Hence if the mirror it bounces back off is 1m away from the splitter, in the time it takes the photon to travel 1m, 3 x 10^-8 s, the mirror will have moved 1 x 10^4 x 3 x 10^-8 = 3 x 10^-4 m or about 1000 wavelengths of visible light. So the horizontal beam will be traveling a different length compared with the vertical beam. However when the horizontal beam starts to travel back the splitter will travel the same distance towards it and hence cancel the effect out so I would expect the two beams to be back in synch!

Don't forget that the vertical beam travels a distance that is greater than the length of the vertical arm.

 

[Ibix]

So the horizontal beam will be traveling a different length compared with the vertical beam. However when the horizontal beam starts to travel back the splitter will travel the same distance towards it and hence cancel the effect out so I would expect the two beams to be back in synch!

I'm not sure that's quite correct. I make the total path length on the "parallel to motion" 2l\frac{c^2}{c^2-v^2} with v being the velocity of the apparatus with respect to the aether. Also, as DaleSpam noted, the "perpendicular to motion" beam is not traveling out-and-back. It needs to follow a triangular path to "keep up" with the moving apparatus.

This was literally a back-of-the-envelope calculation. It's possible I've messed something up - show your working if you disagree, and we'll see.

 

[Paul77]

Thanks I forgot about the effect of the motion of the Earth on the vertical path and hence there is a path length difference for the vertical compared with the horizontal. I found this video on youtube which calculates the time difference between the two paths and expected fringe shift - which is not seen hence no aether and the result that relative motion has no effect on the re-combining light paths.