Shrinking Circles: Exploring Relativity at 3/5 the Speed of Light

[Quinzio]

Hi everyone,
I'm new and this is my first post here.
I'm 37 yo, I'm italian so forgive my english.

My question is about the law of relativity that says that an object moving is shorter, when I observe it.
The shortening is the famous sqrt(1-square(v/c)).
Ok, but now imagine this:
I have 2 big circles made of metal, like 2 hula-hops. They are big so big I can put them around the Earth. Lets' say they are 100 000 km in circumference.
On each circles there are 100 000 000 signs made with a pen, one sign every meter.
(1km =1000m)
The two circles are very close one to the other they are almost superimposed.
As far as they are still, no problem.
Now I start to spin one of the circles, and it spins at 3/5 speed light, so the signs travels at the same speed.

According to relativity, on the spinning circle the signs are now not at 1 meter, but at 0.8 meters. So the circumference of the spinning circle should be not 100 000 but only 80 000 km.
But wait. How is it possible ? Did I say something wrong ?
Do the diameter of the spinning circle get shorter ?
Is the circle broken somewhere ?
How to explain this situation ?
Here is a simple draw of the two circle.
The drawing is bad quality, just to give a clearer idea.
Thanks.

http://img267.imageshack.us/img267/3161/immagine1p.jpg

 

[A.T.]

Do the diameter of the spinning circle get shorter ?

No

Is the circle broken somewhere ?

If the circle in not able to stretch or extend (in it's own frame), it will break. Otherwise it will "adapt" to occupy more space in it's own frame. See picture in this post:

https://www.physicsforums.com/showthread.php?p=2417528

 

[Quinzio]

Thanks for your answers.
I didn't know that this dilemma has already been addressed (Ehrenfest).
So far the explanation of stretching joints do not satisfy me.
But I must read more carefully on the subject.
In the analogy of Emmenthal cheese, it seems to me strange that holes do not contract and cheese do contract. Why ?

 

[S.Vasojevic]

If the circle in not able to stretch or extend (in it's own frame), it will break. Otherwise it will "adapt" to occupy more space in it's own frame.

Ok, now consider this instead. Small turret, 1m in radius is packed with bunch of lasers, which are arranged that each points out in radial direction, at same angular distance. 50000 km from the center of the turret, in the same plane, we put light detector. In order to distinguish between lasers, one laser is green, next one red, and so on. What will the detector measure? Are beams equally separated when turret rotates 0.1 rev/s or 1 rev/s, or the angular distance is shortened?

 

[Quinzio]

Good experiment, Vasojevic.
In fact, relativity talks about space contraction, not object conctraction.
Space contracts undependently of what occupys the space, whether it's a man, a rod, a spacecraft , or nothing.
Space does not have eyes too see what there is in a particular portion of space, whether an object or nothing.

PS Please forgive me: space stretches to make objects contract.

 

[A.T.]

So far the explanation of stretching joints do not satisfy me.

Me neither. What about the https://www.physicsforums.com/showthread.php?p=2417528"?

In fact, relativity talks about space contraction, not object conctraction.

No, it talks about length contraction. I also clarify https://www.physicsforums.com/showthread.php?p=2417528" what that means.

 

[A.T.]

What will the detector measure? Are beams equally separated when turret rotates 0.1 rev/s or 1 rev/s, or the angular distance is shortened?

I assume the detector is hit by two neighboring lasers simultaneously and measures the distance between them?

This distance is constant. The impact points are not moving objects, just abstract points. They can even move faster than c over the detector.

 

[Quinzio]

No, it talks about length contraction. I also clarify https://www.physicsforums.com/showthread.php?p=2417528" what that means.

Every stationary frame must be look identical. A smith casts 10 rulers, then he gives them to ten observers. Observers are tossed in the space at near the speed light, in different directions.
The observers measure their hands with the rulers, moving the hand in different directions. When the rulers are perpendicular to their speed vector, the hand and the rulers contracts in the same way, so the hand measure as usual. But in the frame of the smith that tossed the observers, rulers are shorter when they are parallel to the speed vector.
But an observer is unaware of this contraction.
He alone would never be aware of any contraction, because it's the space that changes (the space viewed from a man on the earth), not the ruler or his body. His space is really "normal".

 

[A.T.]

He alone would never be aware of any contraction, because it's the space that changes, not the ruler or his body.

I guess you mean the same as me in the https://www.physicsforums.com/showthread.php?p=2417528" ?

 

[S.Vasojevic]

This distance is constant. The impact points are not moving objects, just abstract points. They can even move faster than c over the detector.

Yes you are right. I was trying to draw analogy with infinitly rigid object, considering lasers as ruler marks. Anyway, still having problems. Measured from the rest frame object(s) on the edge contract. Circumference should be smaller. Why do I even care for ruler inside rotating frame?

 

[A.T.]

Measured from the rest frame object(s) on the edge contract. Circumference should be smaller.

No, the circumference is not an moving object. See:
https://www.physicsforums.com/showthread.php?p=2417201

Why do I even care for ruler inside rotating frame?

If you are not in the rotating frame you don't care. You measure the circumference with a ruler at rest in your frame and get 2*pi.

But if you are in the rotating frame you use ruler at rest in that frame, and you get a circumference of more than 2*pi, which implies negative spatial curvature.

 

[yuiop]

Thanks for your answers.
I didn't know that this dilemma has already been addressed (Ehrenfest).
So far the explanation of stretching joints do not satisfy me.
But I must read more carefully on the subject.
In the analogy of Emmenthal cheese, it seems to me strange that holes do not contract and cheese do contract. Why ?

I made the claim in the other thread that the holes in the cheese do not contract so I guess I should try and justify that. I based the claim on the Bell's rocket paradox. Imagine you have 2 rockets at rest in your frame separated by a distance x. They both take off at the same time and have identical motors and accelerate at exactly the same rate. When the fuel of the rockets runs out, the rockets are still separated (centre to centre) by the distance x according to observers that remained in the original rest frame but they observe that the rockets have length contracted. So by accelerating the rockets into a new inertial frame, the rockets being made of solid material have length contracted and the space between the rockets has not.

Now we repeat the experiment, but this time the rockets are joined by a strong cable. The rockets are fully automated and programmed to burn their fuel at a fixed identical rate. This time the gap between the rockets does length contract because the joining cable length contracts and physically draws the rockets closer together. When Bell first introduced his rocket paradox a lot of experts did not understand it, because they had been brain washed with "space contraction" and got it wrong, but eventually conceded that Bell was right. Now we introduce a third rocket and all 3 rockets are initially at rest with respect to each other and not accelerating (and not joined together with cables). The third rocket accelerates away and now measures the two other rockets to have length contracted but this time the gap between the two unconnected rockets is length contracted. What is the cause of the length contraction of the physically unconnected gap between the two rockets on this occasion. The difference is that this time it is the observers in the third rocket that feels acceleration and this causes it clocks to slow down and its rulers to length contract which in turn causes the third rocket to measure the first two rockets and gap between them as length contracted. Nothing physical has actually happened to the other two rockets as they have not felt any acceleration. To give another example. Imagine some particle is emitted by some natural process in space and is accelerated to some relativistic velocity relative to the Earth. Does the acceleration of this single particle cause the Earth to physically length contract. Probably not. Millions of particles are wizzing past the Earth every second in random directions and the Earth can not possibly physically length contract in all these different directions at once in response to all these particles. It is the acceleratd particles that length contract and if an observer was accelerated so that they were co-moving with one of the particles, then changes in their clocks and rulers would cause them to perceive that the Earth was length contracted.

 

[yuiop]

According to relativity, on the spinning circle the signs are now not at 1 meter, but at 0.8 meters. So the circumference of the spinning circle should be not 100 000 but only 80 000 km.
But wait. How is it possible ? Did I say something wrong ?
Do the diameter of the spinning circle get shorter ?
Is the circle broken somewhere ?
How to explain this situation ?

If the spinning circle was made of very tough material then it conceivable that as the circumference length contracts, then the circle radius will shrink too. In this situation, the observer on the spinning ring will measure the circumference to be same as it was before the circle was spinning, but now he measures the radius to be less than it was before it was spinning. (The observer on the spinning circle measures the length contracted circumference with a length contracted ruler so he does not notice any change in the circumference.) Either way, the observer on the spinning circle does not measure the circumference as equal to 2*PI*R as you would expect in Euclidean geometry.

 

[S.Vasojevic]

OK, I see. I was referring to circumference, as circumference of some sort of solid ring. So we can extrapolate all this talk to one real example we have here on Earth, and that would be LHC beam. It would mean that beam is not Lorentz contracted, but individual protons are, and further more that space is expanding between them?

 

[A.T.]

I made the claim in the other thread that the holes in the cheese do not contract so I guess I should try and justify that. I based the claim on the Bell's rocket paradox. Imagine you have 2 rockets at rest in your frame separated by a distance x. They both take off at the same time and have identical motors and accelerate at exactly the same rate. When the fuel of the rockets runs out, the rockets are still separated (centre to centre) by the distance x according to observers that remained in the original rest frame but they observe that the rockets have length contracted. So by accelerating the rockets into a new inertial frame, the rockets being made of solid material have length contracted and the space between the rockets has not.

Just because the distance between the rockets stays constant in the initial frame, doesn't mean that the gap is not subject to Lorentz contraction. At the end, the gap measured in the initial frame is smaller then measured in the rockets' frame (gap's proper length ). That is length contraction.

Length contraction is a ratio:

Lo / Lp = contraction_ratio

where

Lo : length measured by the observer
Lp : proper length measured in the frame where the distance defining objects are at rest.

If contraction_ratio < 1, we say that something is Lorentz contracted.

For the rockets : Lp is constant while Lo gets smaller -> contraction_ratio < 1 -> rockets are Lorentz contracted.

For the gap : Lo is constant while Lp gets bigger -> contraction_ratio < 1 -> gap is Lorentz contracted

 

[A.T.]

OK, I see. I was referring to circumference, as circumference of some sort of solid ring.

Okay, contrary to what I said in #2 I think that the radius of a spinning ring might contract, to adapt the circumference of the ring to the non-Euclidean spatial geometry in the rings frame. To do this the ring would have to overcome the centrifugal force, which is proportional to mass. So a very light, yet strong ring might do this.

So we can extrapolate all this talk to one real example we have here on Earth, and that would be LHC beam. It would mean that beam is not Lorentz contracted, but individual protons are, and further more that space is expanding between them?

I don't know how they are accelerated in the LHC, but assume it is similar to the case with the rockets, I discuss in post #15: In the Earth's frame the distances between them are constant, but in protons' frame these distances grow. Space is expanding in the tangential direction in the rotating frame, where all the protons are at rest.

 

[yuiop]

The ratios are the same but the gap behaves differently to the solid rockets.

In the initial frame, the rockets get shorter and the gap stays the same.

In the rocket frame, the rockets stay the same and the gap gets larger.

That is not something you would expect if you take the simplistic attitude that space contracts and solid objects and gaps length contract equally.

It was that misconception, that caused many so called experts to incorrectly assume that the sting joining the two rockets would not snap, in the original Bell's rocket paradox.

 

[Quinzio]

Me neither. What about the https://www.physicsforums.com/showthread.php?p=2417528"?

Yes, that satisfies me better.
IMO, what expands or contract is the space "under your feet", so length changes.
Just like if the floor was made of rubber you can squeeze or stretch. Unity of measure changes, not object.
I know it is not true as well, because moving objects do shrinks.
Cannot think anything else.

No, it talks about length contraction. I also clarify https://www.physicsforums.com/showthread.php?p=2417528" what that means.

 

[Quinzio]

I guess you mean the same as me in the https://www.physicsforums.com/showthread.php?p=2417528" ?

Yes, I do.
But let me make a point.
IMO, from the non rotating frame, or from an Earth frame, moving frame space doesn't get denser, but more rarefied, stretched, so objects do measure as shorter.
But the concept is the same.

In the case of two circular rods, one still (my frame), one rotating, the rotating space is rarefied and the rod measure as shorter.
Space would superimpose on itself, but I can see less problems in this case. Space is just what we call our unit of measure.

The whole problem is puzzling, really.

 

[Quinzio]

I cut some parts of interest

Imagine some particle is emitted by some natural process in space and is accelerated to some relativistic velocity relative to the Earth. Does the acceleration of this single particle cause the Earth to physically length contract. Probably not. Millions of particles are wizzing past the Earth every second in random directions and the Earth can not possibly physically length contract in all these different directions at once in response to all these particles. It is the acceleratd particles that length contract and if an observer was accelerated so that they were co-moving with one of the particles, then changes in their clocks and rulers would cause them to perceive that the Earth was length contracted.

Does the acceleration of this single particle cause the Earth to physically length contract. ?
Not for us that we live on Earth. We can really ignore what a distant particle is doing and apply our ordinary physical rules.
But from the particles view point things do really change and our Earth is really not as before.
Relativity states that there is not a preferred frame. Each frame is as good as anyone else.
Special relativity doesn't talk about acceleration. It doesn't really matter who accelerates and why.
It matters that in a certain istant, two inertial frames are moving one toward the other.
That's what I think.

Millions of particles are wizzing past the Earth every second in random directions and the Earth can not possibly physically length contract in all these different directions at once in response to all these particles.

That's true. Earth do not length contract if you observe the Earth from the Earth, as we all usually do.
But each particle interacts with the Earth as if it was length contracted in its direction.
But how it is possible ?
Realtivity talks about moving frames, not objects.
An object doesn't belong to any particular frame.
A particle is a particle, it doesn't belong to frame "A" or to frame "B".
It happends that in a particular frame, that particle is still and you can apply our ordinary Newton laws.
As it happends that in a particular frame, the Earth is still.
But a particle belongs to all the frames you can imagine out there.
I think that relativity is all about "observing" the rest of the world from a particular point of view. Observing and measuring.

 

[A.T.]

The ratios are the same but the gap behaves differently to the solid rockets.

The gap behaves differently because in that particular setup it changes it's proper length during acceleration. That has nothing to do with length contraction being applied differently to gap vs. rocket. This is just manipulating the proper length to cancel out the length contraction.

You could build extending / telescopic rockets which can change their proper length, and increase it during the acceleration, to cancel out their length contraction too. This is exactly what the https://www.physicsforums.com/showthread.php?p=2417528" does. It increases it's own proper length (= what the ruler measures) to keep a constant length in the non-rotating frame (= circumference).

That is not something you would expect if you take the simplistic attitude that space contracts and solid objects and gaps length contract equally.

But objects and gaps do length contract equally. The ratios are the same is the key point here. However if one of them changes its proper-length while the other doesn't, they already behave differently in their own frame, without any length contraction. Length contraction in other frames still applies equally to both.

I would agree with you "space contracts" is bad. If you want an alternative to "objects contract" you would rather say rather "space elongates" or "for moving objects space has more capacity ". You get more of the stuff to the same space, if the stuff moves.