Brian Greenes' Tornado Analogy

[teodorakis]

On his book of the elegant universe when he was explaining the warping of spacetime he use an analogy of a toy called tornado. Basically a spinning machine like a merry go round.

Anyway he said, let's give one ruler to the slim who is at the wall of the tornado,to measure the circumference, and other ruler to the jim who is at the center of the tornado, to measure the radius, and we observe the situation from above. Then he says slim will measure the circumference shorter than normal due to the fact that he puts the ruler along the direction of the movement and basic lorentz contraction effect kicks in. He says he had to put the ruler more, to measure the circumference and he will measure a longer circumference than we see from above. On the other side jim will measure the radius exactly the same value as we see from above. Because he puts the ruler vertical to the moving direction. So from these two measurement greene conclude that such measurement can not happen in a flat space surface, the space must be warped.
So my questions are:
1. First of all can we derive, at least roughly, that acceleration warps space from special relativity? We just use lorentz contraction and conclude the warping of space due to accelaration in the analogy.

2. How come Slim's ruler contracts but the circumference doesn't?

3. Jim is also moving relative to the outside observer, so isn't there any contraction effect occurs for him as well?

Thank you...

 

[A.T.]

There are many threads here on what rotating rulers measure. The most recent one:
https://www.physicsforums.com/threads/circular-barn-pole.846985/

 

[teodorakis]

I searched the ehrenfest paradox and still don't understand why the periphery remains unchanged while the rules contract.

 

[A.T.]

I searched the ehrenfest paradox and still don't understand why the periphery remains unchanged while the rules contract.

When a physical object (ruler) moves along a path (circumference), the physical object is contracted not the path.

 

[teodorakis]

Isn't circumference is also part of a physical object?

 

[A.T.]

Isn't circumference is also part of a physical object?

No, it's a geometrical concept. When two objects move at different speeds along the same circular path, they are contracted, not the path. Which of the two speeds should determine the path's contraction anyway?

 

[teodorakis]

The object is moving with the same speed in the circular path, there's no relative speed. And it is said that rulers contract but not the circunference of the rotating disk. there's no relative speed between the ruler and the disks circunference.

 

[Ibix]

But the rulers aren't tied together. They can contract and open gaps between them. The disc has disintegrated if that happens to it. So what happens is that the edge of the disc is under strain, over and above the strain expected by a naive calculation of the centrifugal forces.

 

[teodorakis]

But the rulers aren't tied together. They can contract and open gaps between them. The disc has disintegrated if that happens to it. So what happens is that the edge of the disc is under strain, over and above the strain expected by a naive calculation of the centrifugal forces.

So the circumference of the disk contract or not?

 

[Ibix]

Everything is obliged to try to contract when it is moving. It isn't obliged to succeed. Think about a rubber band 10cm long. If I hold the rubber band and pass you at 0.6c it will be length contracted to 8cm. But there's nothing preventing me from stretching the rubber band to 12.5cm (in my frame - which is 10cm in your frame) just so it doesn't look contracted to you. Of course, if you examine it closely you'll notice that it looks stretched, but the length measurement alone could fool you.

In this case, the (implausibly strong) intermolecular forces within the disc prevent the edge of it from contracting. If you look closely you will see that it looks stretched.

 

[teodorakis]

Everything is obliged to try to contract when it is moving. It isn't obliged to succeed. Think about a rubber band 10cm long. If I hold the rubber band and pass you at 0.6c it will be length contracted to 8cm. But there's nothing preventing me from stretching the rubber band to 12.5cm (in my frame - which is 10cm in your frame) just so it doesn't look contracted to you. Of course, if you examine it closely you'll notice that it looks stretched, but the length measurement alone could fool you.

In this case, the (implausibly strong) intermolecular forces within the disc prevent the edge of it from contracting. If you look closely you will see that it looks stretched.

Oh thanks that explains a lot. One more thing in the book greene connects the distortion of Space to the accelaration of objects and from equivalance principle to the gravity by this example, Is this approach correct?

 

[teodorakis]

Oh thanks that explains a lot. One more thing in the book greene connects the distortion of Space to the accelaration of objects and from equivalance principle to the gravity by this example, Is this approach correct?

One more thing so we finally say that due to the contraction in the circumference and radius remaining unchanged the space distorts in a non-eucladian way but the rigid disk "resists" this contraction so we don't notice a change in the spatial dimensions of the disk.

 

[Laurie K]

The spokes of the wheel contract in their "optical appearance" but the wheel itself continues to roll along its axle undistorted.

Øyvind Grøn in "Space geometry in rotating reference frames: A historical appraisal" provides a good compendium of many aspects of relativistic rolling wheels, the Ehrenfest paradox as well as an interesting way to avoid the Born rigidity problem among other things.

http://areeweb.polito.it/ricerca/relgrav/solciclos/gron_d.pdf

His Fig 9. plot titled "Points on a rolling ring. A: Observed by simultaneity in its rotating rest frame K ; B: observed by simultaneity in the frame K0 : C: observed in K0 at retarded points of time." describes, in part C, a similar situation to the OP with one exception, the observer lies stationary on the road frame and is in the same plane as both points on the wheel (rim and axle). If you have seen a verification of his Fig 9 part C plot for 0.86 c you'd say it's a good SR based example of what, when and where you should use length contractions.

By making all observers lie in the same plane and setting the wheel thickness zero Grøn effectively reduced his “optical appearance” (what would we see when) problem down to x, y and t dimensions (z = 0) without inherent Born rigidity issues etc. A is the wheel frame, B is the axle or carriage frame and C is the road or ground frame.

The result is shown in Fig. 9. Part C of the figure shows the “optical appearance” of a rolling ring, i.e. the positions of emission events where the emitted light from all the points arrives at a fixed point of time at the point of contact of the ring with the ground. In other words it is the position of the points when they emitted light that arrives at a camera on the ground just as the ring passes the camera.

So effectively all you need to do is work out how to calculate the length contracted x, y position of a point at the end of a radial spoke, relative to the wheels axle location, with respect to a relatively stationary ground observer in the same plane. As the wheels axle has a fixed relationship with the road frame, i.e. length contraction only effects wheel spoke tips (not the central axle location), accurate solutions can be cross checked by comparing the times and x-axis positions between consecutive emission points (d/t should equal the velocity of the relativistic rolling wheel and the angular velocity of a point on a spoke tip).

It isn't identified in Fig 9 part C that straight lines drawn from each emission point (at retarded points in time) directly to the camera point accurately reflects the time/distance taken for that particular emission to travel to the camera.

 

[Ibix]

Oh thanks that explains a lot. One more thing in the book greene connects the distortion of Space to the accelaration of objects and from equivalance principle to the gravity by this example, Is this approach correct?

There's no distortion of space going on here. However, two things happen. First, you have to use curved coordinates if you wish to try to describe the experience of sitting on the rim of the disk as "being at rest". Second, so called "fictional forces" (centrifugal force in this case) appear when you do so. Since the experience of being pushed outwards in a turning car feels rather like a gravitational force, this was part of the path Einstein took towards realising that gravity could be described by curved spacetime.

Note that it's important (in general) to talk about curved spacetime, not curved space.