Spinning tornado wheel - circumference and radius

[ManDay]

Spinning "tornado" wheel - circumference and radius

Another quote from my current lecture "elgant universe":

Page 63, 3rd Paragraph

We ask Slim and Jim, who are currently enyoing a spin on the Tornado, to take a few measurements for us. [...] As Slim begins to measure the circumference, we immediately see from our [stationary] bird's-eye perspective that he is going to get a different answer than we did. As he lays the ruler out along the circumference, we notice that the ruler's length is shortened. [...] This means that Slim will measure a longer circumference than we did. [Jim, however, is measuring the same radius as we did] But now, when Slim and Jim calculate the ratio of the circumference [...] to its radius they will get a number that is larger than our answer of two times pi.

This doesn't make much sense to me. While I can comprehend that the ruler must be shortened as it's layed out tagentially to the circumference, as it's moving in the direction of which it's layed out, I would assume that the circumference too is shortened, implying that Slim will get the same result for the circumference as we would, when the ride is standing still. In other words, let Slim start measuring the circumference when the ride is standing and then spin it up - assuming that his ruler is actually layed out exactly on the circumference, no discrepance should emerge.

We however, perceive a contracted circumference due to Lorentz contraction but the same radius, meaning that in our system of observation, the ratio has changed.

 

[ManDay]

Not a single reply? Can it really be that hard to say whether this is true or false?

 

[mikeph]

To slim, his ruler is true, but to us, his ruler is contracted and no longer true. So we see him using a shortened ruler and are not surprised when he measures a very long circumference. We use a ruler which is in our inertial frame, and is therefore not contracted and appears longer than slim's ruler, so we measure a shorter circumference.

 

[matheinste]

Not a single reply? Can it really be that hard to say whether this is true or false?

In what context is the example given. Is the chapter or section from which it comes dealing with inertial frames or rotating frames.

Matheinste.

 

[ManDay]

To slim, his ruler is true, but to us, his ruler is contracted and no longer true. So we see him using a shortened ruler and are not surprised when he measures a very long circumference. We use a ruler which is in our inertial frame, and is therefore not contracted and appears longer than slim's ruler, so we measure a shorter circumference.

This is how it's presented in the book, but as I pointed out, this conclusion is incomplete and thus wrong. You did not address the issue I raised, just said what's wirtten in the book in your own words. Please read again and try to grasp my point:

Yes, the ruler is shortened as we watch Slim measuring the rotating circumference while he spins with it, but so is the circumference itsself.

Matheinste: The chapter brings about the introduction of general relavitivy, depicting the disortion of space in rotating frames.

 

[matheinste]

Matheinste: The chapter brings about the introduction of general relavitivy, depicting the disortion of space in rotating frames.

Its out of my league then.

Matheinste

 

[ManDay]

It shouldn't be a different "league". All effects observed in rotating frames can allegedly be deduced from the concepts of spec. relativity (at least on the level this is being examined upon). The first quote I gave basically describes the whole idea, eventually concluding that hence space for Jim/Slim must be curved. I, however, object that - regardless of the outcome - the argument that Slim would go measure a longer circumference, is wrong.

 

[mikeph]

This is how it's presented in the book, but as I pointed out, this conclusion is incomplete and thus wrong. You did not address the issue I raised, just said what's wirtten in the book in your own words. Please read again and try to grasp my point:

Yes, the ruler is shortened as we watch Slim measuring the rotating circumference while he spins with it, but so is the circumference itsself.

Matheinste: The chapter brings about the introduction of general relavitivy, depicting the disortion of space in rotating frames.

What makes you think we see a smaller circumference? The circle does not change shape as we look down on it.Also try not to assume I am misunderstanding you, or that the author's interpretation is wrong, just because you don't understand it.

 

[matheinste]

It shouldn't be a different "league". All effects observed in rotating frames can allegedly be deduced from the concepts of spec. relativity (at least on the level this is being examined upon). The first quote I gave basically describes the whole idea, eventually concluding that hence space for Jim/Slim must be curved. I, however, object that - regardless of the outcome - the argument that Slim would go measure a longer circumference, is wrong.

I know my own limitations but I am still learning. I have always found that the best way to learn is to trust the textbooks and if they do not make the subject clear, ask for help here. If I do that and disagree with both, then I assume there is a major lack of understanding on my part. I am always correct in this last assumption.

Matheinste

 

[ManDay]

What makes you think we see a smaller circumference? The circle does not change shape as we look down on it.Also try not to assume I am misunderstanding you, or that the author's interpretation is wrong, just because you don't understand it.

Mikey, regardless of on whose part the misunderstanding lies, your two contributions to this thread have in no kind of way helped resolving anything. I thus ask you to leave, instead of handing out smart advices such as "...just because you don't understand" and in the end not saying anything, until you find yourself apt to join me in the argumentative debate I tried to set off with my introductory post.

Matheinste, the fact that I opened this thread suggests, that I'd rather believe the textbook's facts than my conclusions, which are in contrary to the former. But as it's popular literature, I think everyone qualifies for drawing his/her own conclusions and try to prove me/the book wrong.

 

[mikeph]

The circle you observe does not "shrink", special relativity analysis can only make conclusions about contractions along tangent lines which represent true inertial frames. Anyway I'm not rising to this, I've tried to help and my advice is free and educated, I'm sorry you don't want it.

 

[Ich]

your two contributions to this thread have in no kind of way helped resolving anything.

Then I htink it's time for you to start again from the beginning: Why do you thing the circumference would shrink? What exactly do you mean by a "shrinking circumference"? Can you try to imagine the situation in spacetime, with this helical simultaneity?
You have to rid yourself of that misconception, as the book's (and MikeyW's) arguments are quite clear.

 

[DrGreg]

Instead of a solid disk, consider a thin rod bent to form a circle of radius r, initially not spinning. The two ends of the rod are not welded together to form a complete circle; there is a tiny 1 mm gap between the two ends. The length of the rod, measured by the bird's eye observer Jim stationary relative to it, is 2πr.

Now start the rod rotating around the centre of the circle. Relative to the bird's eye observer Jim, the rod contracts to a length of 2πr/γ. This means the rod no longer forms a complete circle; the gap has expanded to 2πr(1 − γ⁻¹). However, if an observer, Slim, standing on the rotating rod measures its length by slowing walking around it with a tape measure, the length will be an uncontracted 2πr.

Slim has with him some rods and welding equipment and welds more rods into place to fill the gap and complete the circle. The complete circle has a circumference of 2πr as measured by the bird's eye inertial observer Jim. But this length is contracted from the length of 2πrγ measured by the rotating observer Slim. Both observers agree that the radius of the circle is r as the radial direction is at right angles to the motion.

If you start off with a solid, non-spinning disk, then start spinning it, then either parts of the disk will stretch "to fill the gap", or else the disk will shatter into pieces. There's no such thing as a truly rigid, unbreakable disk.

 

[DaveC426913]

I thnk what the OP is struggling with is this:

Why would the ruler not shorten exactly in proportion to the circumference of the Tornado? They're both going the same speed and in the same direction.

If the circumference is relativistically shortened by half, then the ruler will be shortened by half as well, meaning his measurement of the circumference does not change.

 

[DrGreg]

I thnk what the OP is struggling with is this:

Why would the ruler not shorten exactly in proportion to the circumference of the Tornado? They're both going the same speed and in the same direction.

If the circumference is relativistically shortened by half, then the ruler will be shortened by half as well, meaning his measurement of the circumference does not change.

In my example, the measurement of the bent rod, initially at rest and then made to rotate, does not change, according to Slim. It gets longer only when he welds in the extra rods to complete the circle.

 

[Rasalhague]

The two ends of the rod are not welded together to form a complete circle; there is a tiny 1 mm gap between the two ends. The length of the rod, measured by the bird's eye observer Jim stationary relative to it, is 2πr.

Now start the rod rotating around the centre of the circle. Relative to the bird's eye observer Jim, the rod contracts to a length of 2πr/γ. This means the rod no longer forms a complete circle;

You say it "no longer" forms a complete circle. But the previous paragraph seems to say that it never did form a complete circle.

 

[DrGreg]

You say it "no longer" forms a complete circle. But the previous paragraph seems to say that it never did form a complete circle.

Well it almost did apart from a tiny gap (I said 1 mm but it can be as small as you like). The point is the two ends aren't stuck together, they are free to move apart.

 

[Rasalhague]

Well it almost did apart from a tiny gap (I said 1 mm but it can be as small as you like). The point is the two ends aren't stuck together, they are free to move apart.

Ah, right, so they're effectively touching?

 

[DrGreg]

Ah, right, so they're effectively touching?

Yes. Touching but free to move apart.

 

[ManDay]

Now start the rod rotating around the centre of the circle. Relative to the bird's eye observer Jim, the rod contracts to a length of 2πr/γ. This means the rod no longer forms a complete circle; the gap has expanded to 2πr(1 − γ⁻¹).

Why would the gap expand? "It" too is moving along "its" direction. It's like you were argueing that all objects appear smaller at a distance and thus looking at Emmental cheese at a 10 meters would render its holes smaller and hence create more cheese.

Gap or no gap is not the question. The predictions made my s/r (based upon which, we are trying to deduce the effects of g/r) do not concern objects but they concern space and time itsself. That said, it's irrelevant what we are looking it, whether there is vacuum (a gap) - even air if you like - or rod at the place. Since we regard space and not matter, everything transforms homogeneously.

However, if an observer, Slim, standing on the rotating rod measures its length by slowing walking around it with a tape measure, the length will be an uncontracted 2πr.

Indeed, Slim will take the exact same measure of the rod's lengh as we did, when it was standing still.

Don't get me wrong: I'm aware of argueing for a lost cause, but so far, I don't see any convincing line of thought - although the results concluded are apparently true.

 

[mikeph]

Why would the gap expand? "It" too is moving along "its" direction.

You can't apply length contraction to air as if it were a solid body, it is 10^20 molecules with no internal binding energy, each traveling in their own direction and with their own individual miniature contractions.

You are not changing space-time by this experiment, space-time is not being warped by a simple ring with angular momentum.

 

[A.T.]

While I can comprehend that the ruler must be shortened as it's layed out tagentially to the circumference, as it's moving in the direction of which it's layed out, I would assume that the circumference too is shortened,

Why would the circumference be shortend in the non-rotating frame? The circumference itself is not a moving object, just an abstract distance. And Distances in the non-rotating frame are defined by rulers at rest in this frame, not some moving rulers.

The tangential rotating rulers are contracted in the non-rotating frame, so you get more of them into the circumference. But they are not measuring distances in the non-rotating frame. They are measuring distances in the frame where they are at rest, the rotating frame. In the rotating frame the circumference is greater than 2*pi*r. The spatial geometry in the rotating frame is non-Euclidean:
http://www.phys.uu.nl/igg/dieks/rotation.pdf (page 10, chapter 6)

 

[yuiop]

Why would the gap expand? "It" too is moving along "its" direction. It's like you were argueing that all objects appear smaller at a distance and thus looking at Emmental cheese at a 10 meters would render its holes smaller and hence create more cheese.

Gap or no gap is not the question. The predictions made my s/r (based upon which, we are trying to deduce the effects of g/r) do not concern objects but they concern space and time itsself. That said, it's irrelevant what we are looking it, whether there is vacuum (a gap) - even air if you like - or rod at the place. Since we regard space and not matter, everything transforms homogeneously...

It is a popular misconception that gaps length contract in Special Relativity and is the cause of confusion in the Bell's rocket paradox. Objects length contract but not space or "Gaps". In the chease example, each hole in the material is surrounded by matter that length contracts and in this case, the holes do length contract, but this is just a case of material to the left and right of the gap being drawn together by material above and below the gap rather than length contraction of the gap itself. In the Bell's rocket paradox the rockets length contract but the gaps between the rockets remain constant to an external inertial observer. Observers on the rockets measure the length of their rockets to remain constant and measure the gap between the rockets, to be increasing.

Manday asked why the circumferance of the Tornado does not length contract along with the ruler. If we assume that the Tornado is some sort of rotating fairground turntable made of solid material, then that is a reasonable question and I think it would be fair to say that is poor illustration of the effect that "The Elegant Universe" is trying to demonstrate. A better illustration would be the Ehrenfest paradox where a train moves on a circular track. The carriages of the train are joined by elastic connections. The carriages length contract but the elastic connections get longer as the train speeds up. The track the train is running on is fixed to the ground and in this case there is no doubt that the length of the track remains constant to the ground observers, but the riders on the train will measure the total length of the train (carriages plus elastic connections) to be longer than the track.

See http://en.wikipedia.org/wiki/Ehrenfest_paradox

 

[ManDay]

That was a very useful reply, kev!

 

[A.T.]

Objects length contract but not space or "Gaps".

What do you mean by: "gaps do not contract" ? The gap between two rockets moving at the same constant speed is contracted compared to the rest frame of the rockets.

I would rather rephrase "Objects length contract" as

"All distances defined by moving objects measured by a resting ruler are contracted, as compared to those measured by a co-moving ruler"

this can be interpreted in two ways:

a) The distances defined by moving objects are contracted in our rest frame.

b) Space is more dense for moving objects so distances defined by them occupy less space, and you get more "movng stuff" into the same amount of space.

There is no quantitative difference between the two interpretations and a) is the more common one. But in the case of the rotating ring b) leads very nicely to the non-Eculiedan space geometry in the rotating frame: Viewed from the non-rotating frame space seems more dense for the rotating rulers so you get more of them into the circumference. More rulers "measure" more length for the circumference, but moving rulers don't count as distance measure. Distances in a certain frame are defined by rulers resting in that frame.

http://en.wikipedia.org/wiki/Ehrenfest_paradox

Personally I don't find the train with flexible couplings useful. I prefer to consider an extendable circular ruler:

http://img688.imageshack.us/img688/4590/circleruler.png

You read it off by adding the blue marks (8) to the black marks which are not hidden in the blue pipe.

 

[JustinLevy]

The first quote I gave basically describes the whole idea, eventually concluding that hence space for Jim/Slim must be curved. I, however, object that - regardless of the outcome - the argument that Slim would go measure a longer circumference, is wrong.

I really dislike the "measure the circumference of a rotating ring in SR" problems. Some authors try to use it to argue that the circumference is < 2 pi R and therefore this is leading into curved spacetime. That is wrong. This problem should be used to introduce non-diagonal metrics. But should be careful not to claim it introduces curved spacetime, etc. (since it is the same spacetime regardless of how you map coordinates on it ... so it is still flat here).

It is an interesting problem, but it has been accidentally misused/misunderstood by students enough that it probably isn't worth it in a popular science presentation.

If anything, this problem drives home the point that: length contraction is primarily a problem of different coordinate systems not agreeing on what is "simultaneous". To measure a length, you need to measure the position of both ends "simultaneously".

What is amazing about the rotating frame is that "simultaneity" has some strange problems. If you synchronize clocks with a pulse of light, you can find (going around a circle of 3 clocks) that clock 1 is synced with clock 2 and clock 2 is synced with clock 3 ... BUT clock 3 is NOT synced with clock 1. Send light around the circle in the opposite direction to sync the clocks and you merely reverse the problem.

So to even discuss length in these problems, we need to be very clear on what we mean by length in each "frame", and be aware of the non-diagonal metric when discussing the implications.

 

[pervect]

Yes, the rotating disk in relativity is a classic source of confusion. I think that the historical paper by Gron "Space geometry in rotating frames" is one of the more useful - its referenced in the sci.physics.faq on the rigid rotating disk in relativity.

I don't know if Gron's paper is available on the www.

I'm not sure if my thoughts will help the confusion anyway, but I'll give them anyway. The easiest approach, to me, to get the correct answer seems to be to use the concept of "distance between worldlines".

Part of the problem is even defining by what we mean by "the circumference of the disk" - the worldline approach gives us a way of defining this that I think is reasonably intuitive.

Consider an individual point on the rotating disk. It traces out a helical word-line in space-time as the disk rotate.

Mark more world-lines on the disk, spaced closely together around the circumference. If we can determine the distance between a pair of closely spaced worldlines, we can define the "circumference" of the disk by measuring the distance between pairs of worldlines until we return to the original worldline where we started. Taking the limit for a large number of worldlines, we can measure the circumference of the disk.

Geometrically, the distance between world-lines is determined by drawing a line orthogonal to both worldlines, and measuring the length of this line. I.e. the distance between worldlines is the shortest line joining the two worldlines. Note that because we take the limit, we are interested in the case where the worldlines are close together, close enough that they can be considered to be in the same inertial frame.

Also, note that the symmetry of the problem guarantees that the distance between worldlines is constant and doesn't vary with time.

Inspecting the problem of the distance between worldlines, we realize that in the frame of the worldlines, the velocity will be zero, and the distance between worldlines will be greater than it is in the lab frame. To see this, consider a moving ruler. The two ends of the ruler also trace out world lines. In the rest frame of the ruler, the distance between the worldlines is higher than it is in a frame where the ruler is moving. This is known as "lorentz contraction". The longest distance between worldlines occurs in a frame where the ruler doesn't move.

So, if we mark out two points on the circumference, representing some small angle, we see that the distance between the worldlines must be longest in the rotating frame, and shorter in the lab frame. Or in other words, the circumference of the disk increases when we spin it up.

If we draw a space-time diagram of the helix - it's too much work to do here, see

http://en.wikipedia.org/wiki/File:Langevin_Frame_Cyl_Desynchronization.png

in which the redline represents a "worldline" on the disk - unfortunately only one such line is drawn, the exposition I"m giving requires you to imagine at least one more similar red line

and we draw the path that represents the "circumference" of the disk, a path that is always orthogonal to the worldlines and is the "shortest distance" between them, we find something interesting. This path is not a closed path!

((on the wiki diagram, the blue line represents a path that connects the red worldlines that is the shortest distance between them)).

Tracing out the path of our "circumference", though we return to the same world-line, we do not return to the same point on the worldline. The "blue line" representing the path that we trace out is a helix, not a closed curve.

The proper time along the worldline at which we leave it is not the same as the proper time along the worldline as we rejoin it when we trace out the "circumference". We return to the same worldline to measure the "circumference", but we don't return to the same point on the worldline. This is an important point, and is the basic reason why I think the concept of "distance between worldlines" is useful to understanding the problem.

This is related to our inability to synchronize all clocks on a rotating platform, and is the underlying root difficulty in considering the geometry of the rotating disk. It's not really decomposable into a series of space-like slices in the normal manner.