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

[Herman Trivilino]

Please rephrase that? I'm not sure what you're asking.

You're saying the rod contracts by 2 meters. But all rods of that length moving at that speed have contracted by the same amount. So what are you comparing? Two rods in the same rest frame that have contracted by the same amount or one rod in motion relative to the other?

Obviously, the latter. But since they are in motion relative to each other how shall you compare their lengths? Will you do it while you're at rest relative to the uncontracted rod while the contracted rod comes whizzing past. Do you make the comparison when the leading ends of the rod are aligned, when their centers are aligned, or when their tail ends are aligned?

And does it matter?

 

[Ibix]

In that case, physics obviously works and such measurements are more pain than practical, so I'll assume time dilation will automatically fill any discrepancy such as extra spacing between objects. (that aren't accelerating)

So here's an example of what I was saying. What are you actually measuring that length contraction of the rod is relevant? If I measure the time it takes something to get from A to B I usually mean the time between when one point on the object passes A and when the same point passes B. Then length contraction is irrelevant to one frame and only the length contraction of the distance between A and B matters in any other.

 

[syfry]

You're saying the rod contracts by 2 meters. But all rods of that length moving at that speed have contracted by the same amount. So what are you comparing? Two rods in the same rest frame that have contracted by the same amount or one rod in motion relative to the other?

Obviously, the latter. But since they are in motion relative to each other how shall you compare their lengths? Will you do it while you're at rest relative to the uncontracted rod while the contracted rod comes whizzing past. Do you make the comparison when the leading ends of the rod are aligned, when their centers are aligned, or when their tail ends are aligned?

And does it matter?

I'll try to close any holes in my logic.

First, it's only one rod, and one target. The target can be any rectangular object, a stone, a slab of wood, anything solid.

So let's say we had already previously held the rod in our reference frame. And we had visited the target in our reference frame. We had measured each in our own reference frame.

Now, sometime later, for whatever reason, the objects are in outer space and the rod is on the way to its target.

I'm not sure how to select reference frames that wouldn't cause issues, but let's try this: the target is orbiting the Milky Way galaxy, so I'd suppose its rest frame is the galaxy. The rod is approaching from a direction, and you know which reference frame the rod is in. (one that works and doesn't cause extra issues)

Now, knowing what we know about their previously measured size at rest in our own reference frame (they're no longer in it), what amount of extra spacing will appear between the rod and its target at the rod's current relativistic speed?

I forget what even started that line of thought, to be honest.

 

[syfry]

I've encountered a difficult to analyze part in my reading up on relativity, it's actually an old personal difficulty. In the tidbit below, which seems to potentially be the birth of length contraction:

"The most successful explanation was the Lorentz–FitzGerald contraction. These two physicists indepen-dently proposed that lengths are contracted in the direction of the motion by precisely the right factor, namely√1 v2=c2, to make the travel times in the two arms of the Michelson–Morley setup equal, thus yielding the null result. This explanation was essentially correct, although the reason why it was correct wasn’t known until Einstein came along"

It's from the footnotes at bottom of page 8 in this pdf:

https://scholar.harvard.edu/files/david-morin/files/relativity_chap_1.pdf

What I'm trying to wrap my head around is, what are they supposing was length contracted in their setup? The light itself? Can someone shed light on how they interpreted the situation? (in plain words please)

 

[PeterDonis]

observers sitting on different parts of the rod feel different proper accelerations. If the force is applied to the front end, then only the observer on the front end feels proper acceleration as if the whole mass of the rod was concentrated in that point.

Now I'm the one who is confused. Please give the mathematical expression for "proper acceleration as if the whole mass of the rod was concentrated in that point".

 

[Demystifier]

Now I'm the one who is confused. Please give the mathematical expression for "proper acceleration as if the whole mass of the rod was concentrated in that point".

Take a pen and make fixed marks on the rod, the marks are $n=1,2,...,100$. The mark $n=1$ denotes the front end, while $n=100$ denotes the back end. Now apply the force on the front end of the rode. The whole rod will move and any part of the rod with mark $n$ will have some trajectory $x_n(t)$, where $x$ and $t$ are position and time in the inertial laboratory frame.

Now take another, more compact body, shaped like a small ball, rather than a rod. Let this ball have the same mass as the rod, and apply the same force on it. It will have some trajectory $x_{\rm ball}(t)$.

Now, since the force is applied to the front end of the rod, in the approximation of Born rigidity I claim that
$$x_1(t)=x_{\rm ball}(t)$$
For any trajectory $x(t)$ I'm sure you know how to calculate the proper acceleration $a(t)$, so the equality above implies
$$a_1(t)=a_{\rm ball}(t)$$
In fact, if the force is constant, the proper accelerations $a_n(t)$ and $a_{\rm ball}(t)$ do not depend on $t$. But $a_n$ depends on $n$, that's what I mean by the statement that different parts of the rod have different proper accelerations.

Is it clear now?

 

[Ibix]

What I'm trying to wrap my head around is, what are they supposing was length contracted in their setup? The light itself? Can someone shed light on how they interpreted the situation? (in plain words please)

Anything that can be described as at rest when you use one frame of reference will be length contracted when using any other. Anything else (a beam of light, or the distance between a pair of objects in motion with respect to one another) will do something similar, but it won't obey the same formula because other factors come into play.

So Lorentz and Fitzgerald were proposing that the interferometer was contracted. I'm not sure what they thought would happen to the light itself - it doesn't matter for this experiment. You can work out what happens to it in a full relativistic model easily enough if you want.

Note that if you idealise the interferometer to just a few mirrors floating in space, not physically connected but at rest with respect to one another, the distance between the components will also length contract as observed using a frame where they are in motion. This must be the case because I could lay a ruler between them, and if the ruler touches both mirrors all frames must describe this - so if a ruler just touches both mirrors and you determine that the ruler is length contracted then you must also determine that the gap is length contracted or the ruler would not fit exactly.

 

[robphy]

I've encountered a difficult to analyze part in my reading up on relativity, it's actually an old personal difficulty. In the tidbit below, which seems to potentially be the birth of length contraction:
It's from the footnotes at bottom of page 8 in this pdf:

https://scholar.harvard.edu/files/david-morin/files/relativity_chap_1.pdf

What I'm trying to wrap my head around is, what are they supposing was length contracted in their setup? The light itself? Can someone shed light on how they interpreted the situation? (in plain words please)

These visualizations might help:

Here is a (2+1)-d spacetime diagram (a position-vs-time graph) of the Michelson-Morley experiment.
All light signals are sloped at 45-degrees.

• https://www.geogebra.org/m/XFXzXGTq
  Relativity-LightClock-MichelsonMorley-2018 (robphy)

Without length-contraction (in the first diagram),​

light signals at the origin event emitted along the two arms and reflected by the mirrors​

are received at distinct events TY, then TX.​

The time-difference depends on the velocity of apparatus, as they expected,​

in violation of the principle of relativity.​

​

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.​

​

​

​

​

 

[PeterDonis]

since the force is applied to the front end of the rod, in the approximation of Born rigidity I claim that
$$x_1(t)=x_{\rm ball}(t)$$

For the case of the rod this can only be true, if it's true at all, in the equilibrium state after the rod has stopped oscillating. But that's not how you're asserting it. Are you claiming the rod does not oscillate? That as soon as a force is applied to the front end, the entire rod immediately assumes Born rigid motion? You know that's impossible, right? To make the motion Born rigid from the outset, you would need to apply a precisely calibrated force to each point on the rod, not just the front.

\In fact, if the force is constant, the proper accelerations $a_n(t)$ and $a_{\rm ball}(t)$ do not depend on $t$.

Again, if this is true at all, it can only be true in the equilibrium state after the rod has stopped oscillating. But you're not asserting it that way. Or are you?

 

[Demystifier]

Again, if this is true at all, it can only be true in the equilibrium state after the rod has stopped oscillating. But you're not asserting it that way. Or are you?

I am. See Sec. 6 in my paper.

 

[syfry]

​

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.​

​

​

​

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.