# Length contraction in Michelson-Morley experiment

The length contraction in longitudinal arm is required to explain the null result, i.e. no destructive interference. Can't we just explain the null result without involving the length contraction because both arms are equal length and hence light only has to travel the same distance for both arms. Or, perhaps I can put the question differently, does the special theory of relativity require the length contraction to explain the null result? I don't think it does because the observer and apparatus are in the same inertial frame of reference.

I understand that Lorentz came up with the length contraction hypothesis to save the aether theory. In the experiment the following times were calculated where '$$c$$' is speed of light, '$$v$$' is orbiting speed of earth around the sun with respect to absolute frame of reference of aether, and 'd' is length of both arms.

for longitudinal arm along the direction of motion $$t_{total1} = 2dc/(c^2 - v^2)$$

for transverse arm $$t_{total2} = 2d/\sqrt{c^2 - v^2}$$

From the equations, it could be seen that $$t_{total1}$$ would be greater than $$t_{total_2}$$.

Earth is almost an inertial frame of reference therefore if "$$v$$" is removed, we are left with $$t_{total1}=t_{total2}$$. In my view, the aether was the only reason "$$v$$" was introduced in the equation. It was assumed that the distance along the direction of longitudinal mirror would take longer time because during half part of its trip the light move against the aether or aether wind, but during the second half of its round trip aether wind moves with the light.

In short, using our current understanding, Lorentz and others had unknowingly changed the scenario into one which assumed as if the experiment was being looked at by some stationary space observer who was looking at the apparatus moving at speed of '$$v$$' relative to him. For such an observer, length contraction in the direction of motion is definitely required. The stationary space observer would also notice time dilation for the frame of reference of apparatus.

Do I make any sense?

• Where did you read or hear that "length contraction in the longitudinal arm is required to explain the null result"? Of course, that's what those who believed in ether thought over a century ago, but I don't know why anyone would think that now. Commented Jan 16, 2022 at 12:33

In one sense you are right. The interference pattern in the interferometer does not changes as the apparatus is rotated. A modern observer who is stationary with respect to the apparatus does not need to invoke length contraction to explain this. A modern observer expects the speed of light to be the same in every direction.

However, an observer moving relative to the interferometer also sees that the interference pattern does not change. And from their point view the speed of light in the frame of the interferometer does depend on direction. So they have to invoke length contraction to explain the stationary interference pattern.

Michelson and Morley, however, expected the speed of light in the frame of the interferometer to depend on direction, not because they were moving relative to the interferometer, but because they and it (and the whole Earth) were moving relative to the aether. So they expected to see an interference pattern that changed as the apparatus was rotated. A null result on one day could be explained if the Earth, by coincidence, was stationary with respect to the aether on that one day, but M & M carried out repeated experiments over a period of several months to rule this out.

Lorentz introduced length contraction to explain the result of the Michelson-Morley experiment. As he originally conceived it, this was an actual physical change in the length of any physical object that moved through the aether, due to a hypothetical change in the strength of intermolecular forces. Length contraction in special relativity happens to be expressed by the same formula as the Lorentz contraction, but is observed by an observer who is moving relative to the apparatus. not stationary, and is introduced for completely different reasons.

The transverse interferometer arm cannot logically experience length contraction. This arm is perpendicular to the direction of motion. The distance between mirrors cannot contract. When this arm is set in motion, the light path increases. This moving path is longer than the motionless path, but the longitudinal arm's path is mathematically longer than the perpendicular, transverse path.

Length contraction makes these paths equal. But the motionless interferometer's light paths are still symmetrical and shorter. Time dilation would appear to be a convenient way to solve this problem, making an orbiting Earth equal to a motionless one.

This graphic from Wikipedia illustrates what I mean: https://en.wikipedia.org/wiki/Length_contraction#/media/File:Relativistic_wheels.gif

I think Lorentz (or someone else) was confused when they came up with this ad-hoc idea. Length contraction cannot change the velocity of light, nor can it cause the moving transverse light path to shrink since this path is not derived from the geometry of the interferometer. Notice that in the graphic, the shape of the wheels is scaled only in direction of motion, along the X axis. The wheel is not scaled along the Y.

This interferometer graphic also illustrates my point: https://i.sstatic.net/r4eoj.png

This link does a good job explaining this: https://www.researchgate.net/publication/335475185_From_Michelson-Morley_experiment_to_length_contraction