Length contraction/time dilation mutually exclusive?

[jollyjolly0]

Hello Physics Forums!

I'll try to cut right to the chase. I'm studying both length contraction and time dilation in my physics class and it seems these two occurrences are mutually exclusive. That is, in any given situation, only time will be dilated or length will be contracted. This doesn't really make sense to me though. For one, "how do it know?" whether to contract or dilate length or time respectively? Is it true that its one or the other, but never both?

 

[PeterDonis]

Is it true that its one or the other, but never both?

No. It's the opposite: they are always present together. That is, if an object is length contracted relative to you (because it is moving relative to you), it is also time dilated relative to you.

 

[ghwellsjr]

Also, keep in mind that the length contraction is only along the direction of motion so if you are considering a light clock with the light bouncing back and forth along the direction of travel between two mirrors, those mirrors will be closer together than in their rest frame but if the light clock was rotated 90 degrees, then the mirrors are the same distance apart no matter what the speed is. (But, the mirrors will be length contracted.)

However, you may have heard that in one rest frame it is only the Time Dilation that matters while in another rest frame it is only the Length Contraction that matters but in those cases, although both effects are present, we only need one to explain the interesting phenomenon. This is the case for the famous muon scenario.

 

[haael]

No. It's the opposite: they are always present together. That is, if an object is length contracted relative to you (because it is moving relative to you), it is also time dilated relative to you.

And the magnitude of the length contraction and time dilation are always in such proportion that the speed of light is constant.

 

[ghwellsjr]

And the magnitude of the length contraction and time dilation are always in such proportion that the speed of light is constant.

You make it sound like the two effects cancel out. But one is smaller while the other is larger. And consider what I said in post #3 where a particular orientation of a light clock does not depend on length contraction. How does your comment fit with these two considerations?

 

[jollyjolly0]

Say I observe a spaceship fly between points A and B at close to the speed of light. The spaceship's clock will be running slower than mine, however I still observe the distance it travels as the proper length between A and B. Now say I'm in the spaceship. Now I will see the distance between A and B contract by gamma, but I won't observe my clocks in the spaceship as slowing down or any nonsense like that.

This is just what I've cobbled together from reading the posts here as well as some additional sources. Is this correct?

 

[PeterDonis]

Say I observe a spaceship fly between points A and B at close to the speed of light. The spaceship's clock will be running slower than mine

From your point of view, yes.

however I still observe the distance it travels as the proper length between A and B.

More correctly, you observe the distance it travels as the length between A and B in your rest frame. If there are objects at points A and B, and the objects are at rest relative to you, then the distance between those objects can be called the proper distance from A to B, yes. But "proper distance" is not just a distance between "points in space", because "points in space" is frame-dependent; you have to actually pick out objects that are at rest relative to each other to define a "proper distance" between them.

Now say I'm in the spaceship. Now I will see the distance between A and B contract by gamma

With respect to the rest frame of the spaceship, yes.

but I won't observe my clocks in the spaceship as slowing down

No, but you will observe clocks at rest relative to the objects at A and B (assuming those objects remain as above, i.e., they are still at rest relative to the original frame, not the spaceship frame) as slowing down.

 

[pervect]

Say I observe a spaceship fly between points A and B at close to the speed of light. The spaceship's clock will be running slower than mine, however I still observe the distance it travels as the proper length between A and B. Now say I'm in the spaceship. Now I will see the distance between A and B contract by gamma, but I won't observe my clocks in the spaceship as slowing down or any nonsense like that.

This is just what I've cobbled together from reading the posts here as well as some additional sources. Is this correct?

Well, I'm not sure what you mean by "or any nonsense like that", but what you wrote is right but incomplete.

Specifically, you didn't indicate your understanding (or ask any questions about) the readings of the clocks for A and B - i.e. do they run slow in your spaceship frame?

In order to address the issue of "mutual exclusiveness" (which, I should add, is not in my opinion a good way to look at things), you need to address this question. Perhaps I'm over analyzing, but you see to be implying that you don't think the clocks for A and B are slow?

 

[russ_watters]

Say I observe a spaceship fly between points A and B at close to the speed of light. The spaceship's clock will be running slower than mine, however I still observe the distance it travels as the proper length between A and B. Now say I'm in the spaceship. Now I will see the distance between A and B contract by gamma, but I won't observe my clocks in the spaceship as slowing down or any nonsense like that.

This is just what I've cobbled together from reading the posts here as well as some additional sources. Is this correct?

You are correct, but you are mixing and matching which frame to care about. Time dilation and length contraction are always all about what happens to the other guy. You see the other guy's clock measuring a different elapsed time than yours and you see the other guy's ruler measuring a different distance than yours.

This isn't just convention either, it's a reflection of the fact that if you stare at your ruler and clock by themselves, there is nothing you can ever do to notice them changing their operation.

 

[ghwellsjr]

Say I observe a spaceship fly between points A and B at close to the speed of light. The spaceship's clock will be running slower than mine, however I still observe the distance it travels as the proper length between A and B. Now say I'm in the spaceship. Now I will see the distance between A and B contract by gamma, but I won't observe my clocks in the spaceship as slowing down or any nonsense like that.

This is just what I've cobbled together from reading the posts here as well as some additional sources. Is this correct?

You provided an example of the type I mentioned in post #3:

However, you may have heard that in one rest frame it is only the Time Dilation that matters while in another rest frame it is only the Length Contraction that matters but in those cases, although both effects are present, we only need one to explain the interesting phenomenon.

In the rest frame of points A and B, its clocks and distances are normal but the moving spaceship's clocks are dilated and its length is contracted whereas in the rest frame of the spaceship, the distance between the moving points A and B is contracted while their times are dilated.

Here is a spacetime diagram depicting the rest frame of A (blue) & B (green) separated by a distance of 5000 feet in which the spaceship (red and black) is traveling at 0.6c. The dots mark of one-microsecond increments of time. The speed of light is 1000 feet per microsecond:

Note that the length of the spaceship is contracted from its Proper Length of 1000 feet to 800 feet and its tick marks are dilated to 1.25 microseconds.

Transforming to the rest frame of the spaceship, we get:

Note that the distance between A & B is contracted to 800 feet and their times are dilated to 1.25 microseconds while the spaceship's length is 1000 feet and its clocks are normal.

So in either case, we normally only focus on one effect to explain what's going on: in the A/B rest frame, we focus on the spaceship's clocks being Time Dilated (even though the spaceship is also Length Contracted) and in the ship's rest frame we focus on the distance between A & B being contracted (even though its clocks are time dilated).

But just to assure you that the two effects are not mutually exclusive, we can transform to the frame in which A & B are traveling at 0.3333c in one direction while the spaceship is traveling at the same speed in the opposite direction:

Here we see that both A & B and the spaceship are Length Contracted to the same extent and their clocks Time Dilated equally.

I've uploaded some more spacetime diagrams at additional speeds to show different patterns of Time Dilation and Length Contraction. Remember, there are no preferred frames in Special Relativity, not even an object's own rest frame. Any frame is just as valid as any other frame.

Does this make perfect sense to you? Any questions?