## Question about something in the Elegant Universe book I am reading...(KelseyMH?)

Ok, here it goes: In the description concerning general relativity, the following scenario is depicted: Jim and Slim are riding a "Tornado" ride (goes by different names, but essentially is a centrifuge that pins you against the outer wall and then drops the floor out from under you. Sometimes calle d PUKE machine).

Anyways, looking at the ride from a bird's eye view, one measures the circumference and the radius and we find that the ratio is two times PI or about 6.28.

Now that is from OUR perspective, but from Slim's and Jim's measuring the circumference (so says the book) will give a different answer since they are traveling at high speed (assuming the speed is fast enough to invoke the Lorentz contraction). Thus, he says in the book that his measurement will come up short and therefore not be the same ratio noted. I don't doubt this, but I have a problem with not taking into account the lengthing of the circumference because of IT'S motion forward.

Am I missing something? Why wouldn't the same effect be proportional to the lengthening of the ruler used to measure the circumference, seeing they are traveling at the same speed?

Am I taking his example TOO FAR (in other words, he is assuming the ride is there but not physical)? Maybe I over analyze? :-)

Anyways, looking at the ride from a bird's eye view, one measures the circumference and the radius and we find that the ratio is two times PI or about 6.28.

Now that is from OUR perspective, but from Slim's and Jim's measuring the circumference (so says the book) will give a different answer since they are traveling at high speed (assuming the speed is fast enough to invoke the Lorentz contraction). Thus, he says in the book that his measurement will come up short and therefore not be the same ratio noted. I don't doubt this, but I have a problem with not taking into account the lengthing of the circumference because of IT'S motion forward.

Am I missing something? Why wouldn't the same effect be proportional to the lengthening of the ruler used to measure the circumference, seeing they are traveling at the same speed?

Am I taking his example TOO FAR (in other words, he is assuming the ride is there but not physical)? Maybe I over analyze? :-)

active| newest | oldestI think you're right - if they are measuring the circumference with a ruler

onthe circumference, then they would both be subject to the same effects.Maybe they are comparing the circumference to an external, stationary scale? Kind of like a circular ruler, just outside the spinning area?

For a "stationary" observer far from the rotating ring, we would measure the circumference = pi * diameter.

For an observer co-rotating with the ring, who measured the circumference by

walking around it, they wouldalsoget pi*diameter. That's because (as Kiteman and you noted) that observer is at rest in the frame of the rotating ring.The complex relativistic effects kick in if, for example, the folks on the ring were actually spinning around past a nice marked ruler on the ground below the ring. In that case, if they tried to measure the marked ruler as it spun "backwards" past them, then they would indeed observe a shorter length. Hence, they would infer that the circumference of that painted ruler was < pi*diameter.

There

aredifferences between rotational motion and linear motion in general relativity. In particular, uniform rotation has an accelative component, which we know (from GR) is equivalent to a (kind of weird :-) gravitational field.That means the rotating ride in flat spacetime can be described equivalently as being at rest in a special curved spacetime (please don't ask be to write down the solution!). Therefore, it is plausible that observers on the ring itself might well measure a non-Euclidean circumference, but I don't know how to derive the result quantitatively.