I have long puzzled over a similar non-inertial frame paradox–a version of the twin paradox. Isolate two caesium clocks from the outside environment and send one on a one-G roundtrip to the nearest star. According to the Lorentz calculation, the travelling clock shows far less elapsed time upon reuniting. My question–how did the travelling clock “know” it was the travelling clock so it can show less elapsed time? Both clocks were in a sealed one-G environment.

@dave dont worry about your relative intelligence to these geeks. i just imagine that im stationary and they are all on a train that is moving around a circular track. =]

Instead of thinking of a circle, let’s think on a regulat N-Gon with a seperate car travelling along each side. (I think a square’s conceptually easiest). The observer on each side on the N-Gon is a inertial reference frame relative to the track, because he is travelling in a straight line correponding to the side he’s on.

The both the observer on the car and the observer on the track will observe the other one shrinking in length relative to the other one. This is nothing new, it’s the classic relatively scenario of two spaceships passing each other in space and both saying that the other one’s meter sticks are too short.

Now let’s figure out for the observer in the center what the distance is between each of the space ships on each side. If the car and polygon side are the same length when both are at rest a track-stationary observer will see the cars shorten more and more as it they approach the speed of light.

For simplicity sake let’s say each car has dillated to half the length of the side of the track it’s on. This can be represented by circumscribing a smaller N-gon inside the N-gon where the middle of each side forms the new points. Now if you start with a square you have just shurnk by a lot, but as you go to an Octogon the dilatted train is almost as big as the original tracks. As N -> Infinity, the size of the dilatted train (or circumscribed N-Gon) approaches the size of the original N-gon, which is what you get when you have a circle.

Yes car 181 is moving relative to my inertial reference frame. If you’re talking about the rotating reference frame then it is stationary. But remember a rotational reference frame is not inertial, ie it is accelerating. Hence the reason why someone on a Tilt-A-Whirl feels centripetal force.

The easiest way to think about this problem is to think about the train car’s instantenous inertial frame. That is to do the calculations assuming each car on the train continues to move with the same velocity and no acceleration. The inertial vector for any given car is the tangent of the circle, therefore every car has a different inertial frame, and the further cars are away from each other the greater the difference between their inertial frames become.

I think, although it doesn’t make logical sense, that GregS is right. If I’m on the train, the tracks will appear shortened in the direction they are traveling relative to me, but since the radius is perpendicular to my direction of travel, the radius would not change. It’s hard for me to visualize, though.

I’m still sorting this. I’m I on the right track?