aaj said:
Please refer to
http://en.wikipedia.org/wiki/Ehrenfest_paradox for a description of this very interesting paradox.
Although I understood the apparent paradox, I could not get a grasp of the resolution of the paradox, as explained on Wikipedia. Could any of the pros on this forum explain the resolution in more simple terms?
I still could not understand how an observer at rest will be able to reconcile the fact that she is observing a circumference less than 2pi times the radius that she observes.
She (Mary) does not see two measurements. She sees the length of the track as 960 inches and she sees the length of the moving train as 960 inches. Although she measures each of the box cars as length contracted she sees the elastic couplings have stretched to keep the total length of the train constant.
aaj said:
No shape in Euclidean geometery would satisfy what she observes. And how could the space between the axis of rotation of the train and the circular path become bent simply due to the motion of the train?
Well circular motion is a form of acceleration so being on the train is like being in a gravity "field". Imagine being on a very massive planet that is so massive that some of the light coming from the ground is bent back towards the surface. (Assume the planet is almost perfectly spherical with no hills or valleys but with such a large radius it is "locally flat") The human brain always interprets light as traveling in straight lines, so it looks to you like you are standing at the deepest point in a depression, no matter where you stand. That is you seeing the curvature of space.
Now imagine when the train is stationary, that you place a laser on one box car and aim it so the laser fires pulses at a spot marked on another box car. Now after the train has accelerated to its final cruising speed you notice that the laser pulses no longer hit the same spot that you marked on the box car. On board the train light seems to follow curved paths just as on the massive planet. (Not exactly the same, because the curved light paths are assymetric on the train but we won't worry about that right now). Anyway, to you on board the train space certainly looks curved. To Mary, each photon pulse follows a straight line, so everything looks all nice and Euclidean to her, and the corners of triangles add up to 180 degrees (but not for you on board the train).
aaj said:
More puzzling is the view from the perspective of an observer on the rim of the disc. She actually measures a circumference greater then 2pi.
With due respect to sexual equality I am going to demand that 50% of the observers are male :P
Bob is the observer on the train. Initially the train is stationary when he gets on board. He has a truck load of rulers and places them end to end all the way around the train. Each ruler is 10 inches long so it take 96 rulers. When the train is moving the rulers have shrunk leaving gaps between the rulers. He pushes all the rulers together to close the gaps and then finds he need more rulers to span the total length of the train. In fact he has to add another 24 rulers making a total of 120 rulers. By his measurement the length of the train is 1200 inches. How can that be? The rulers have shrunk but the so should the train. Well, the box cars have shrunk, and they will appear to be unchanged to Bob but the elastic couplings have stretched and the couplings will seem much longer to Bob now the train is moving than when it was at rest. This is because the couplings have not been allowed to naturally length contract and they will be under a lot of stress.
So here is the important part. Mary sees the total length of the train as shorter than Bob's measurement by a factor of gamma, but unlike in regular straight line inertial relativity she sees no change in the total length of the train when it is stationary or when it is moving. It is Bob that sees the change.
Another important part is this. Bob will see the radius as larger than the radius measured by Mary. Imagine a bridge is built from one box car to a pivot at the centre of the circular track. Bob find he needs more rulers to span the length of the bridge when it is moving relative to the ground than when it was stationary. This is because of a general relativity effect that causes objects in a gravitational "field" to length contract in the vertical direction. Remember, being in a rotational frame is like being in a "wierd" gravity field. He will measure the radius as being a bit longer, but probably not enough longer to rescue Euclidean geometry. You would have to work out the integral to find out exactly what he measures the radius to be, because the rulers nearest the moving train will shorter than the rulers nearest the central pivot. Maybe, an expert would be able to tell us what Bob measures the radius to be in inches?
The rulers along the perimeter (onboard the train) are not affected by the "gravitational" length contraction because length contraction does not happen in the horizontal direction in a gravitational field.
As you probably guessed, I am no expert :P
An expert would say something like the resultion of the paradox can be summed up as "the geometry 'in the small' is approximately given by the Langevin-Landau-Lifschitz metric".