# The Naked Scientists Forum

#### Solvay_1927

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« on: 25/08/2006 23:02:59 »
Another relativity paradox, but don’t worry, it’s not the “Twin Paradox” again, it’s different.

Take a length of string (or twine, as I prefer to call it – cos then I can call this the “TWINE Paradox” – geddit?!)

Tie a knot in it every metre, then tie the two ends together so that it forms a loop.  Say the loop is 100m long.

Now fit the loop onto two small pulleys which are 50m apart.  So you can see two lengths of twine between the pulleys – the top one has 50 knots in it, and the bottom one has 50 knots in it.

Now start the pulleys spinning (at the same speed) very fast.

According to relativity, because the twine is moving (relative to me), the length of the twine will become contracted, so that the distance between each knot will now be LESS than 1m.

But the pulleys are in my “stationary” frame of reference, so they remain 50m apart.

So there must now be MORE than 50 knots between the two pulleys along the top edge of the loop (at any given point in time), and also MORE than 50 knots along the bottom edge.

To me, that seems like a paradox.

How do I resolve it?

« Last Edit: 25/08/2006 23:04:43 by Solvay_1927 »

#### another_someone

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« Reply #1 on: 25/08/2006 23:27:31 »
The pulleys are not stationary – they are spinning, and their rims are rotating at the same speed as the twine is moving – so I would imagine the shape of the pulleys will become distorted, as some parts of the pulley are moving faster than others.

Not sure that this is the only effect, since it will not explain everything.

Another problem is that we are not really dealing with special relativity, but with general relativity; since the twine, while moving around the pulley (just as the rim of the pulley itself) must be undergoing massive deceleration and acceleration, as the nots move away from you, then move around the pulley, and then decelerate and accelerate as they do so, and then move towards you.

George

#### thebrain13

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« Reply #2 on: 26/08/2006 00:25:56 »
This problem involves acceleration. The twine and the knots have to accelerate. At the moments the twine is accelerating, the amount of time each knot is going to spend, in that area is going to be increased. Or the length between each knot is going to be stretched when the object is accelerating, and conversely contracted when it isnt. Read about the relativity of simultaneity it pretty much resolves all relativity paradoxes, for example my twin paradox.

#### lightarrow

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« Reply #3 on: 26/08/2006 08:25:14 »
Those explanations don't seem to solve the problem. Let's consider only those regions of the strings where they are straight and without acceleration: according to lenght contraction they should be shorter, but how it's possible if the pulleys are the same distance apart?

My compliments to you, Solvay, that's a really interesting paradox!

I wonder if this related problem is more easily solved:
a circular ring of some matter rotates at high speed. Then, according to lenght contraction, its circumference should shorten. Does it mean that the radius should shorten too?
« Last Edit: 26/08/2006 08:25:44 by lightarrow »

#### another_someone

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« Reply #4 on: 26/08/2006 11:20:02 »
quote:
Originally posted by lightarrow
Those explanations don't seem to solve the problem. Let's consider only those regions of the strings where they are straight and without acceleration: according to lenght contraction they should be shorter, but how it's possible if the pulleys are the same distance apart?

My compliments to you, Solvay, that's a really interesting paradox!

But can we actually only look at those regions that are not accelerating?

The whole problem concerns the connectedness of the system, and without the accelerating parts, there is no connection of the other parts.

If you simply had two metal rods flying at high speed past two stationary pulley, there would be no paradox – the only reason for the paradox is because the string is physically attached to the pulleys, which means that those attachment points are critical to the paradox, and those attachment points must account for the difference in speed between the stationary pulley centres and the fast running twine between the pulleys.

Removing the complexity of the physical attachments – what we really have is simply a system where something starts from a stationary position, accelerates to a high speed, then decelerates to another stationary point.  In between these end points, both of which are at a known distance apart, we have 50 way-points that within the reference of the moving object are equidistant, but we cannot say for sure that they will appear equidistant to us.  So the question you are asking is what will the apparent position of these way-points be?

George

#### lightarrow

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« Reply #5 on: 26/08/2006 14:19:58 »
So, how do you explain my previous question: a circular ring of some matter rotates at high speed. Then, according to lenght contraction, its circumference should shorten. Does it really happen? Does the radius shortens too?

If we can solve this, and it should be easier, we can probably solve Solvay's paradox.

#### thebrain13

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« Reply #6 on: 26/08/2006 16:22:42 »
I believe since the ring is round, and rotating, all parts of the ring are accelerating equally thus, length contraction is counteracted. So the result would be just a normal ring, rotating the same as newton would of suspected. However if there were clocks mounted on the ring they might run a little slow.

#### Solvay_1927

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« Reply #7 on: 26/08/2006 22:42:07 »
quote:
Originally posted by lightarrow
My compliments to you, Solvay, that's a really interesting paradox!

Aw shucks, thanks.  (Secretly, I am quite proud of it.[^])  The following is how I first thought up the twine paradox – maybe it makes the paradox more striking:

Suppose we haven’t tied the two ends of the twine together, so it’s not a loop – i.e. it’s now just a 100m length of twine, with knots in it every 1m apart.

My old mate Albert (Einstein to you!) starts running very fast away from me.  As he leaves me, he picks up the end of this length of twine, which I’m holding loosely between my fingers.

As it’s moving relative to me, I see the twine as being contracted in length, so I measure the knots as being less than 1m apart. (And so, if Albert was running at X metres per second, I’d see a knot passing through my fingers MORE frequently than one every 1/Xth of a second).

Now, after 50m, Albert hits a brick wall (accompanied by a large “Ouch!”, of course).  (So he's been running for 50/X seconds.)

At that point the twine becomes stationary again relative to me, so I should see the length of twine between Albert and me as being 50m, and I should be able to see 50 knots in it.  But I’ve counted MORE than 50 knots slipping through my fingers.  So where have the extra knots gone?

Before anyone answers, can I just say:

1. It doesn’t have to be a length of twine, it could be a rigid steel girder, with notches in it every 1m (or every 0.001m, it doesn’t matter).  So please don’t just suggest that the twine goes slack and that more than 50 metres worth of twine really has passed through my fingers.  That wouldn’t work with a steel girder.

2. Yes, I know there’s acceleration and deceleration involved, which must be what causes the paradox – but I can’t for the life of me work out how this explains the “missing” knots.

I’d be grateful for any help. This paradox has been driving me mad for months.

#### Solvay_1927

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« Reply #8 on: 26/08/2006 22:54:56 »
quote:
Originally posted by lightarrow

So, how do you explain my previous question: a circular ring of some matter rotates at high speed. Then, according to lenght contraction, its circumference should shorten. Does it really happen? Does the radius shortens too?

I'm fairly sure (although I can't immediately find any references) that the weird truth (according to relativity) is that:
a. The circumference DOES get shorter, but
b. The radius stays the SAME.
(To picture how that can happen, imagine a flat disk which, as it rotates faster, bends into a "bowl" shape - but in 4 dimensions, not in 3 dimensions.)
In this way angular momentum is conserved.  My understanding is that angular momentum IS conserved in relativity, even though linear momentum isn't. (As far as I can remember, in relativity it's the "momentum-energy four-vector" that's invariant, rather than (linear) momentum in isolation or energy in isolation. But angular momentum IS invariant on its own.  But please let me know if I've got that wrong.)

#### another_someone

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« Reply #9 on: 27/08/2006 06:24:49 »
quote:
Originally posted by lightarrow
So, how do you explain my previous question: a circular ring of some matter rotates at high speed. Then, according to lenght contraction, its circumference should shorten. Does it really happen? Does the radius shortens too?

If we can solve this, and it should be easier, we can probably solve Solvay's paradox.

My understanding is based upon the assumption that length contraction only exists in the direction towards/away from the observer, and not perpendicular to him.  With this in mind, my intuitive guess would be that the ring would cease to be circular.

If one were to arbitrarily assign the direction from the foci of the ring to yourself as north, and regard that the ring is rotating from East to North, then the eastern point of the ring is travelling towards you, and will be foreshortened, while the northern point is travelling across your field of vision, and will not be foreshortened, while the western point is travelling away from you, and is thus foreshortened.

Furthermore, since one assumes the ring to be of finite thickness, the outer part of the ring will be travelling faster than the inner part, and thus will be subject to greater foreshortening.  This would not be a problem is one assumes that the ring is anyway becoming an ellipse, and the outer portion being closer to the apex of the ellipse.

In the extreme situation, one would then expect that if the ring was rotating with a tangential speed of the speed of light, the ring would become simply a line from east to west, of zero thickness.

A caveat to this is that this would be the view at infinite distance.  As one approaches closer to the ring, the points at which the ring is moving directly towards and directly away from you will move from the east and west position ever closer to the north position, and so breaking the symmetry of the ellipse or line, making if curve around a crescent shape.

George

#### Solvay_1927

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« Reply #10 on: 27/08/2006 17:44:05 »
Good analysis, George, but I'm not sure it's complete.  Does it address the situation where you're standing inside the ring (between the ring and the centre) or above the ring (say, hovering above the circumference)?

I've managed to find some more info on the relativistic relationship between radius and circumference of a rotating disk/cylinder.  It gives rise to the "Ehrenfest Paradox" (the resolution to which is not straightforward, it seems).  For example, see:

« Last Edit: 27/08/2006 17:45:23 by Solvay_1927 »

#### thebrain13

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« Reply #11 on: 27/08/2006 22:58:12 »
Try to follow this, this is the first time I ever explained this to anybody. Distance=Time literally. And the two units d and t are unified at the speed of light. that means d=c/t and t=c/d in otherwords 300,000 kilometers equals one second, 600,000 km= two seconds and on an on. It is not incorrect to merge the two units, replace whatever. It is just as correct to say light takes eight and a half minutes to reach the earth as it is to say it takes light eight and a half light minutes(which is a unit of distance) to reach the earth. Got it?

So lets consider your steel girder, and lets say it is one second long or 300,000 km long(get it cause its the same thing) and lets say that you accelerate the girder to a speed of .99c. How fast would the girder slip away from your fingertips? The answer is the end that your holding would slip out at .99c. but what is peculiar really is whats happening to the end that is the farthest away from you. You see it is seperated from you by one second or 300000 km, so any change to the front of the girder would take place at a different time. In this case one second, time is altered as a function of distance. The front of the girder since it is somewhat far away is seperated from you in time. So in the moments you see the girder begin to accelerate, you wont see the front moving at all! The size of the girder causes the girder to occupy different realms of time from front to back, a delay of exactly one second. So at one half second the girder would apear slightly bigger than one half its origninal size, at .75 seconds, slightly more thean 3/4ths its orignal size(if the girder accelerated to 1c it would be exactly one half and .75, but then again thats impossible) The girder would reach its full length contracted size at or after 1 second time. Get it?

#### another_someone

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« Reply #12 on: 28/08/2006 01:14:43 »
quote:
Originally posted by Solvay_1927
I've managed to find some more info on the relativistic relationship between radius and circumference of a rotating disk/cylinder.  It gives rise to the "Ehrenfest Paradox" (the resolution to which is not straightforward, it seems).  For example, see:

If I understand the article correctly, in principle, the resolution looks quite obvious when one thinks about it.  The actual argument seems to be say that assuming that the material has sufficient strength, there will be no contraction at all.  If I understand what they are saying correctly, they work on the basis that length contraction only occurs if the material is not under strain (i.e. the material will try to contract, but if there is a counteracting force, then the material will not be able to contract).  Since the bulk material is held in a rigid shape, therefore there is a force that prevents the contraction from happening, and all you see are increased stresses in the material rather than actual changes in the dimensions of the material.

This would appear to me to mean that the forces an stationary observer sees in the bulk material trying to tear the ring or disc apart are different from the forces that someone fixed to the ring or disc would see.  If this is case, and given that the point at which the ring will break apart should be the same no matter whom the observer is, it seems to imply that the strength of the material would have to appear to be different to the two observers.

George
« Last Edit: 28/08/2006 01:16:21 by another_someone »

#### lightarrow

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« Reply #13 on: 28/08/2006 01:40:01 »
quote:
Originally posted by Solvay_1927

In this way angular momentum is conserved.  My understanding is that angular momentum IS conserved in relativity, even though linear momentum isn't. (As far as I can remember, in relativity it's the "momentum-energy four-vector" that's invariant, rather than (linear) momentum in isolation or energy in isolation. But angular momentum IS invariant on its own.  But please let me know if I've got that wrong.)
Sincerely, I don't know about angular momentum conservation in relativity.

Wikipedia (http://en.wikipedia.org/wiki/Angular_momentum) says:

"...In modern (late 20th century) theoretical physics, angular momentum is described using a different formalism. Under this formalism, angular momentum is the 2-form Noether charge associated with rotational invariance (As a result, angular momentum isn't conserved for general curved spacetimes, unless it happens to be asymptotically rotationally invariant). "

About linear momentum, instead, that is conserved for sure. The relativistic momentum, of course:
p = mv/sqrt(1-v^2/c^2).
(By the way, in other posts when I wrote "square" I intended "sqrt").

The momentum-energy four-vector is used because its conservation means momentum and energy conservations at the same time, and because its square (this time is correct!) is (m*c^2)^2 which is invariant, so computations are easier.
« Last Edit: 31/08/2006 08:21:45 by lightarrow »

#### lightarrow

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« Reply #14 on: 28/08/2006 15:52:07 »
quote:
Originally posted by another_someone

If I understand the article correctly, in principle, the resolution looks quite obvious when one thinks about it.  The actual argument seems to be say that assuming that the material has sufficient strength, there will be no contraction at all.  If I understand what they are saying correctly, they work on the basis that length contraction only occurs if the material is not under strain (i.e. the material will try to contract, but if there is a counteracting force, then the material will not be able to contract).  Since the bulk material is held in a rigid shape, therefore there is a force that prevents the contraction from happening, and all you see are increased stresses in the material rather than actual changes in the dimensions of the material.

This would appear to me to mean that the forces an stationary observer sees in the bulk material trying to tear the ring or disc apart are different from the forces that someone fixed to the ring or disc would see.  If this is case, and given that the point at which the ring will break apart should be the same no matter whom the observer is, it seems to imply that the strength of the material would have to appear to be different to the two observers.
If you are assuming that Hook's law and F = m*a law are Lorentz-invariant, it's not so.

#### another_someone

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« Reply #15 on: 29/08/2006 13:12:43 »
quote:
Originally posted by lightarrow
If you are assuming that Hook's law and F = m*a law are Lorentz-invariant, it's not so.

I am not sure what you mean by the laws not being Lorentz-invariant.

Clearly, F=M*A would be modified by the change in mass effected by relativity, but otherwise, are you saying that taking into account that change in mass, there are other relitavistic changes that invlaidte the direct proportionality?

As for Hooke's law, it was only ever an approximation in typical cases, and it is clear from what I have said that there does seem to be what I would interpret to be a change in the modulus of elasticity brought about by relativity.  Are you saying that it is more complicated than merely a relativistic adjustment of the modulus of elasticity?

I do agree that prior to this discussion, it would not have occurred to me that such a relativistic change in elasticity would occur, but this became evident when I read the explanation for the Ehrenfest Paradox.

George

#### lightarrow

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« Reply #16 on: 29/08/2006 13:32:06 »
quote:
Originally posted by another_someone
quote:
Originally posted by lightarrowIf you are assuming that Hook's law and F = m*a law are Lorentz-invariant, it's not so.
I am not sure what you mean by the laws not being Lorentz-invariant.
Clearly, F=M*A would be modified by the change in mass effected by relativity, but otherwise, are you saying that taking into account that change in mass, there are other relitavistic changes that invlaidte the direct proportionality?
Lorentz-invariant means invariant in the transformation from one reference system to another one, in inertial motion with respect the first, according to Lorentz rules of transformations. Of course, the case of an accelerated motion is something different, but, at least, if you consider a small angle of rotation, and so you can approximate the arc with a straight line, you can apply those transformations rules.

F = m*a is not valid, even when you consider m as the relativistic mass:

F = dp/dt, where p is the relativistic momentum. Of course, you should write these formulas with vectors, here is because I have written them with bold letters.
Deriving the relativistic momentum ( p = m*v/sqrt(1-(v/c)^2)), you have:

dp/dt = m*a/sqrt(1-(v/c)^2) - (v°a)*m*v/(c^2)*sqrt(1-(v/c)^2),
where ° means "scalar product".
« Last Edit: 29/08/2006 17:10:51 by lightarrow »

#### lightarrow

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« Reply #17 on: 30/08/2006 06:25:47 »
About Hooke's Law, what i meant is that you shouldn't think of  relativistic contraction as an elastic contraction you can have in one specific frame of reference.

The relativistic contraction has nothing to do with an elastic contraction and strenght of materials.

Of course, the fact that a material in an accelerated frame of reference is under the effect of forces and hence can be modified elastically, it's another story.

#### Soul Surfer

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« Reply #18 on: 30/08/2006 08:32:09 »
The problem is not realistic because it is not possible to perform the experiments because the foces invoved to perform it vastly exceed the strengths of all materials.

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evolution rules in all things
God says so!

#### thebrain13

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« Reply #19 on: 30/08/2006 19:24:19 »
but that cant invalidate the hypothetical. All materials beings strong enough is assumed.

#### Solvay_1927

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« Reply #20 on: 31/08/2006 01:48:32 »
George - that was a useful explanation of the Ehrenfest paradox, thanks. (I did find that article a bit confusing until you rephrased it that way.)
And thanks lightarrow (Alberto) for the link re: angular momentum.

But I still don't think I've got a resolution to the twine paradox (especially not the 2nd version above, where Albert runs 50m with the twine).
Regardless of what happens during acceleration and deceleration, the fact is that a constant velocity can be achieved for long enough for the problem still to occur: For a given time interval, there will be a fixed contraction in length which will lead to more than 50 knots passing through my fingers during that time.

Ian - I'm not sure I agree that the problem is "not realistic".  It may be a bit impractical (to say the least!), but that's not the same as being unrealistic.  In theory you don't have to be moving at an incredibly high speed - so long as your metal rod is long enough and has markings very close together (every tiny fraction of a millimetre, say) - and you do the experiment in deep space, say (i.e. effectively zero gravity).

In any case, even just as a thought experiment, this paradox needs a resolution - it can't just be ignored.  If a resolution can't be found (within the framework of relativity), then relativity must be "incomplete" in some way, mustn't it?

#### thebrain13

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« Reply #21 on: 31/08/2006 03:43:40 »
dude, whats wrong with my answer?

#### lightarrow

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« Reply #22 on: 30/08/2006 06:25:47 »
About Hooke's Law, what i meant is that you shouldn't think of  relativistic contraction as an elastic contraction you can have in one specific frame of reference.

The relativistic contraction has nothing to do with an elastic contraction and strenght of materials.

Of course, the fact that a material in an accelerated frame of reference is under the effect of forces and hence can be modified elastically, it's another story.

#### Soul Surfer

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« Reply #23 on: 30/08/2006 08:32:09 »
The problem is not realistic because it is not possible to perform the experiments because the foces invoved to perform it vastly exceed the strengths of all materials.

Learn, create, test and tell
evolution rules in all things
God says so!

#### thebrain13

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« Reply #24 on: 30/08/2006 19:24:19 »
but that cant invalidate the hypothetical. All materials beings strong enough is assumed.