# Information needed on two wheels and chain apparent paradox in SR

• sar_sar
In summary, the chain will shrink in length in the rest frame of the two gears, so the gears will either be pulled closer together or the chain will break.
sar_sar
I hope someone can point me to some information to assist resolving this apparent SR paradox.

I have two gear wheels with an endless chain passing round them. The axles of the wheels are 100 chain_link_lengths apart, so we have 100 chain links along the top; 100 chain links along the bottom and, say, 20 links around the wheels, for a total of 220 links. I paint numbers 1 to 220 on the links.

If I set the chain moving at such a speed that the links are relatively contracted to 1/3 of their rest length, I will need 300 links along the top. But there are only 220 links in the chain.

How can I explain what is occurring? Note that this is not the usual "paradox" where a stationary and a moving observer see things differently - I stay stationary all the time and I apparently see more links appear.

If I take a photo of the assembly with the chain in motion, how many links do get in my photo? (ie for a given simultaneous time t in my stationary frame, how many links do I see?)

I came across the problem thinking of charges in a circuit. I know that charge conservation will not be violated, but I cannot explain why. When the charges are in motion, the space between them will contract and it appears that I will get an increased charge density. There is a wealth of material explaining what happens in an infinite conductor but none on what happens in a finite, complete circuit.

Any ideas or pointers will be GREATLY appreciated - I have googled extensively without luck.

If you set the loop of chain moving, it actually will shrink in length in the rest frame of the two gears, so the gears will either be pulled closer together or the chain will break.
sar sar said:
I came across the problem thinking of charges in a circuit. I know that charge conservation will not be violated, but I cannot explain why. When the charges are in motion, the space between them will contract and it appears that I will get an increased charge density.
Well, if you have a circle of wire with x number of charges evenly spread around the wire, and then you get all the charges moving at 0.866c, then if the distance between each charge and its nearest neighbors stays the same in its own rest frame, then in the wire's frame the distance between charges will be cut in half, so the moving charges can only occupy half the circle at any given time; on the other hand, if you want to keep it so all x charges are evenly distributed along the wire, then you must double the distance between each charge and its neighbors as seen in the charge's own rest frame (so that in the wire's frame the distance between charges is unchanged).

Last edited:
Thank you - I hadn't thought of solutions based on the chain breaking which opens up lots of ideas - but I am not sure I agree for two reasons.

I don't see how it answers the charge conservation problem - I will have more charges along the moving top chain alone than I put on the stationary chain.

Also, in the frame of the chain (at least until it accelerates around the wheel), the distance between the wheels contracts, so why does the chain break in its own frame?

Sorry...

sar_sar said:
Thank you - I hadn't thought of solutions based on the chain breaking which opens up lots of ideas - but I am not sure I agree for two reasons.

I don't see how it answers the charge conservation problem
Did you see the second paragraph I added to my post in an edit?
sar_sar said:
Also, in the frame of the chain (at least until it accelerates around the wheel), the distance between the wheels contracts, so why does the chain break in its own frame?
The chain doesn't have a single rest frame--the top section of the chain is moving in the opposite direction as the bottom section, and the section along the gear is moving in all different directions at different points (but if the distance between the gears is large compared to the radius of each gears we can ignore these sections). So no matter what inertial frame you pick, the total length of the chain shrinks when it moves.

JesseM

Thank you! (I wrote my earlier comment before seeing your edit.)

I think I now understand it.

Everything presumably happens during the acceleration, so it would be difficult to explain in detail, but the key fact is that when the chain is up to speed, it can only be 1/3 of its rest length.

Option 1 - assume I accelerated each link individually in such a way that it "broke" between link 100 and 1. I then have a chain of length 220/3 going round only occupying 1/3 of the rest frame circumference - 1/3 is chain, 2/3 is empty. This also conserves the charge.

I also think that there is a "time discontinuity" when one goes round in a complete circle (I have read explanations of the Sagnac effect which rely on this). Perhaps this explains why there is a time difference between the end of link1 passing me and the start of link2 arriving.

Option 2 - I could "accelerate" it so I broke it into 100 links in which case each link separates from its neighbour by 2/3 link_length. The time discontinuity I speculated about above is aportioned among all the link_to_link_gaps. Again charge is conserved.

I understand that "SR does away with the concept of rigidity" but I find that intuition is of little help in trying to understand these problems, and I find it difficult deciding on which "obvious" fact to discard and which to keep!

I am still a little confused about what "break" means because if I slow the Option 1 chain to a halt does it "re-join" itself - ie is the "break" real (metal tears apart) or is it just that I observe a gap in time between seeing link 1 go past, and link 100 arriving, and this gap in time reduces to zero as the chain stops?

I am not at all satisfied with this... If the chain is accelerated slowly, then what force stretches the chain or pulls the gears together?

sar_sar said:
I also think that there is a "time discontinuity" when one goes round in a complete circle (I have read explanations of the Sagnac effect which rely on this). Perhaps this explains why there is a time difference between the end of link1 passing me and the start of link2 arriving.
I don't understand what you mean by "time discontinuity"--unless the end of link1 breaks apart from the start of link2, both should pass at the same time.
sar_sar said:
I am still a little confused about what "break" means because if I slow the Option 1 chain to a halt does it "re-join" itself - ie is the "break" real (metal tears apart) or is it just that I observe a gap in time between seeing link 1 go past, and link 100 arriving, and this gap in time reduces to zero as the chain stops?
There shouldn't be any time gaps in an inertial frame, so if there's a break, it's real, the chain will still be broken when it comes to rest.

PeteSF said:
I am not at all satisfied with this... If the chain is accelerated slowly, then what force stretches the chain or pulls the gears together?
Lorentz contraction causes each link in the chain to shrink as its velocity increases, unless the links are elastic enough that they can stretch by the right amount (in their own rest frame) to compensate for this, so that in an inertial frame there is no change in size.

This is a hard question... The stress on the chain appears to occur when it whips around the gears.

But, I think that the stresses could be reduced to a manageable level:
1. Let's set up our chain in a vacuum in flat space.
2. Each link is 5mm wide and 20mm long.
3. The shape of the chain is similar to before, but much longer, and guided by a lubricated track in a solid support rather than gears.
4. The chain's path has semi-circular ends of with radius 10^9m. The centres of the semi-circles are 2x10^9m apart.
5. The chain is slowly and uniformly accelerated up to 0.00001c (3km/s)

Now, to briefly analyze:
• Initially, the chain is at rest in the frame of the track. The track length is 10,283,185,307.18m, with 514,159,265,359 links in the chain.
• The chain's acceleration as it swings round the hub is miniscule - about 0.01m/s/s
• Relativistic length contraction suggests that the chain should now be 10,283,185,306.67m long - about 26 links short of the track length.

But, I think I see where this will end up now.
Although the acceleration is miniscule, it is enough to slightly distort the links... What's the relativistic limit on material strength? I have a fuzzy idea that Young's modulus is somehow involved here?

PeteSF said:
I am not at all satisfied with this... If the chain is accelerated slowly, then what force stretches the chain or pulls the gears together?

I am not sure there needs to be a force in that sense other than the force I apply to the chain/wheels to accelerate the chain.

Recall the person passing me in the x direction, carrying a rectangular sheet of plywood x long by y high, and moving the sheet in the y direction. He sees the sheet to be rectangular, with the bottom horizontal. Because my view of simultanaeity is different from his, I see the front corner and the back corner at different times from him, so I think the sheet is rhombus shaped, and the base is not horizontal. No force is needed to change the sheet's shape - it is merely my perception is that the sheet is rhombus shaped - his perception is that the sheet is rectangular.

Problem 5-2 in this pdf is relevant: Special Relativity - David Hogg

Problem 5-2:
Imagine a wheel of radius R consisting of an outer rim of length 2piR and a set of spokes of length R connected to a central hub. If the wheel spins so fast that its rim is traveling at a signicant fraction of c, the rim ought to contract to less than 2pi in length by length contraction, but the spokes ought not change their lengths at all (since they move perpendicular to their lengths). How do you think this problem is resolved given the discussion in this Section? If you find a solution to this problem which does not make use of the concepts introduced in this Section, come see me right away!

The section referred to discussed causality in relation to when one part of an object "knows" about what happens to another part with relation to deforming object, and also mentions Young's Modulus in passing in problem 5-1.

There is considerably more stress when the chain goes around the gears, but that's just a red herring.

Here's the general rule: when an object accelerates relativistically, it likes to have its back end accelerating more than its front end. (By a specific amount, depending on its speed and acceleration)

When the actual acceleration on the back and front is the same (as it would need to be in order to be uniformly accelerated around its loop), it results in a stretching force

As to why objects like to have the back accelerating more than the front, there are various ways you could see why. The geometric argument is the easiest, but probably the least satisfying to those who still have a lingering mistrust of SR. You could do a detailed look at the electromagnetic forces that bind the object together -- as observed in some frame in which the particles are moving, the magnetic forces and propoagation delays that arise from the motion make the system prefer the length-contracted state, rather than in the un-contracted state it prefers when resting.

PeteSF said:
Problem 5-2 in this pdf is relevant: Special Relativity - David Hogg

Problem 5-2:
Imagine a wheel of radius R consisting of an outer rim of length 2piR and a set of spokes of length R connected to a central hub. If the wheel spins so fast that its rim is traveling at a signicant fraction of c, the rim ought to contract to less than 2pi in length by length contraction, but the spokes ought not change their lengths at all (since they move perpendicular to their lengths).

I think this is a much better answer, especially for the case where I have a wheel which has, say, a diameter of about 100 chain_link_lengths, and 314 links around its circumference. When I set the wheel in motion, the distance round the circumference will be less than 2*pi*r and I have non-Euclidean space. I therefore still have only 314 links, but each is, say, 1/3 of its rest length because the total distance round the circumference is only 1/3 of what it was at rest. The diameter is still the same as it was at rest. Hence there is no problem that the 314 shortened links at 1/3 of their rest length need to "stretch" a distance 2 * pi * 50. The chain does not need to break, and if I have charges, I still have charge conservation. Excellent!

However, I will need to think further about how this solution changes if I revert to the two chain wheels of the original question. I can make the lengths of the straight bits as long as I wish, and their contractions will presumably swamp the distance contraction around the 2 half-circumferences of the two wheels??

One other post asks about time discontinuities on rotating disks. As I understand it, you cannot achieve simultaneity in a rotating frame. Assume you have a clock at the centre, and an identical one half way towards the edge, and one at the edge. The one at the centre keeps "stationary" time, the one half way out runs slower, and the one at the edge slower still - they are identical but they will not stay in synch. If I adjust the rate of an outer clock to force it into sync, and move it radially it goes out of sync again - you cannot sync clocks in a rotating frame.

You need GR to analyse a rotating frame - not SR which does not apply. http://www.mathpages.com discusses it in relation to Sagnac.

I will be abroad for the next week so unable to respond to posts...

sar_sar said:
I think this is a much better answer, especially for the case where I have a wheel which has, say, a diameter of about 100 chain_link_lengths, and 314 links around its circumference. When I set the wheel in motion, the distance round the circumference will be less than 2*pi*r and I have non-Euclidean space.
I think they just mean less than 2*pi*r if you use the original r when the wheel isn't rotating, if the radius of the wheel shrinks to r' the circumference will still be 2*pi*r'. You shouldn't need to bring in non-euclidean geometry for a situation that does not involve curved spacetime, and thus can be analyzed using SR only.
sar_sar said:
One other post asks about time discontinuities on rotating disks. As I understand it, you cannot achieve simultaneity in a rotating frame.
Maybe not, but I thought we were talking about how the situation would appear from the point of view of an inertial observer watching the wheels begin to turn, not how it would look in a rotating frame which as you say is outside the scope of SR. The situation should certainly be possible to understand from the perspective of such an inertial frame, and there couldn't be any "time discontinuities" in this frame.

Thanks Hurkyl, that helps.

## 1. What is the apparent paradox in special relativity (SR) related to two wheels and a chain?

The apparent paradox in SR related to two wheels and a chain is known as the "bicycle and barn paradox." It involves a bicycle with two wheels and a chain riding through a barn that is smaller than the length of the bike. According to SR, the length of the bike should appear shorter to an observer inside the barn, but the wheels should appear to be moving slower due to time dilation. This leads to the paradox of the bike being able to fit inside the barn even though it appears to be longer than the barn.

## 2. How does the apparent paradox in SR relate to the concept of length contraction?

The apparent paradox in SR is related to the concept of length contraction because it involves the discrepancy between the perceived length of an object in motion and its actual length. In the bicycle and barn paradox, the length of the bike appears shorter due to length contraction, but it is still able to fit inside a smaller space due to time dilation affecting the speed of the wheels.

## 3. What is the role of time dilation in the apparent paradox in SR?

Time dilation plays a crucial role in the apparent paradox in SR because it affects the perceived speed of the wheels on the bicycle. According to SR, time moves slower for objects in motion, so the wheels of the bike would appear to be moving slower to an observer inside the barn. This leads to the paradox of the bike being able to fit inside the barn even though it appears to be longer than the barn.

## 4. How does the bicycle and barn paradox challenge our understanding of space and time?

The bicycle and barn paradox challenges our understanding of space and time because it presents a scenario where an object can appear to be both longer and shorter than another object at the same time. This goes against our classical understanding of space and time, where an object can only have one definitive length and travel at one definitive speed. It also highlights the strange and counterintuitive effects of special relativity on our perception of space and time.

## 5. Can the apparent paradox in SR be resolved?

Yes, the apparent paradox in SR can be resolved by taking into account the fact that the observer inside the barn is also in motion. In this scenario, both the bike and the barn are moving relative to each other, and therefore, both will experience length contraction and time dilation. When this is taken into consideration, the bike will not appear longer than the barn, and the paradox is resolved.

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