Why does length contraction occur and how does it relate to time dilation?

[syfry]

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With length-contraction (in the second diagram)​

[the separation of the worldlines marking the ends of the X-arm (along the relative-velocity axis) is shorter],​

the signal along the relative-velocity axis has a shorter round-trip time so that​

the reception events TY and TX are now coincident, as experimentally observed,​

in accord with the principle of relativity.​

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View attachment 329233 View attachment 329232​

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The interactive demo on your website looks good! Really cool you created that. Even if I don't know how to use it, can still appreciate it.

So if I'm understanding your attached images correctly, the first image is what they expected, but instead they got the second image. After some zooming closer into the image's details, yeah it's apparent that the light would travel a longer distance in the first image, and that the red TX dot has moved down to overlap the green TY dot in the second image. (I'm assuming the light is traveling from bottom left to top right)

Then they modeled a length contraction to explain such a result.

What I don't get is why would any part of the apparatus be length contracted since it's in the same reference frame as the observers? (in the same room)

That's why I was confused and thought maybe they were length contracting the light itself.

 

[Ibix]

What I don't get is why would any part of the apparatus be length contracted since it's in the same reference frame as the observers? (in the same room)

Because in Lorentz and Fitzgerald's original conception anything moving with respect to the ether rest frame is length contracted. In the full relativistic theory, anything moving with respect to you is length contracted. Lorentz-Fitzgerald contraction is not quite the same phenomenon as relativistic length contraction.

 

[Dale]

In the full relativistic theory, anything moving with respect to you is length contracted.

Or more precisely: anything moving with respect to any inertial frame is length contracted relative to that inertial frame.

 

[robphy]

The interactive demo on your website looks good! Really cool you created that. Even if I don't know how to use it, can still appreciate it.

Thanks. It has a lot of features built into it... which you can selectively disable and enable.

So if I'm understanding your attached images correctly, the first image is what they expected, but instead they got the second image. After some zooming closer into the image's details, yeah it's apparent that the light would travel a longer distance in the first image, and that the red TX dot has moved down to overlap the green TY dot in the second image. (I'm assuming the light is traveling from bottom left to top right)

Then they modeled a length contraction to explain such a result.

I would say that the first spacetime diagram summarizes what they expected to find,
although I doubt that anyone diagrammed it like this as a position-vs-time diagram (from what I have seen in the literature).
It should be in agreement with various textbook derivations.

The second spacetime-diagram shows the special-relativistic explanation of null result of the experiment.
It should be in agreement with various textbook derivations.
(The only spacetime-diagram of the Michelson-Morley experiment that I have seen is the sketch in J.L. Synge, Relativity: The Special Theory (1962), pp. 158-162.)
I suspect that neither Lorentz (proposed in 1892) nor Fitzgerald (proposed in 1889) had the spacetime picture, which was articulated by Minkowski (1907), soon after Einstein formulated Special Relativity (1905).
Lorentz and Fitzgerald ( https://en.wikipedia.org/wiki/Length_contraction ) proposed length-contraction
based on ideas of the aether and its presumed electromagnetic properties.

What I don't get is why would any part of the apparatus be length contracted since it's in the same reference frame as the observers? (in the same room)

That's why I was confused and thought maybe they were length contracting the light itself.

Note: a reference frame isn't a "room" (or a diagram of a room).
Without getting into subtle definitions and interpretations, one can think of a "reference frame" akin to "a set of parallel worldlines".
Since the earth and the apparatus are in relative-motion, the earth worldline is not parallel to the apparatus worldline.

Subtle point: Light signals don't get length-contracted.
Length-contraction (involving \gamma) is associated with parallel timelike-worldlines (like the ends of a ruler or the arm of the MM-apparatus).
Light signals, however, are associated with lightlike-worldlines (associated with wavefronts)... they are subject to the Doppler effect (involving the Doppler factor k)... wavelengths (between parallel lightlike-worldlines) can lengthen or shorten, depending on the relative-velocity of the source and receiver...
they are not simply length-contracted.