Sorry about the brain. You really shouldn't let it go dressed that way. Just sayin'.makc wrote:My brain is raped. You're saying that two sleepers are longer apart on paper, but if I put my measuring rod agains them, they are shorter apart. Well, if this is the case, your paper does not describe my experience and is, as you said, "mathematical abstraction" - so why consider it? Btw, what's that bit about "time coordinates are crazy"?
It's really quite simple, but perhaps I managed to complexify it in my explanation. I'll try again:
Relativity is all about events, that is, something that happens at a particular time and a particular place. A reference frame means a system for labeling events with time and space coordinates. The rotating frame we're talking about here is constructed in the following way:
We asssign each event space coordinates that tell where in the train it happened. All events with the same space coordinates will have happened at the same place in the train, measured for example as meters forward of the midpoint of some chosen car. (The coordinates of events in the next car over will have space coordinates that reflect how long the distance between cars look to the passengers. This is unproblematic as long as we're only interested in the situation after the train is already moving with constant speed). This choice of space coordinates is really unavoidable given that we want to construct a rotating frame that follows the train. But we still have to choose time coordinates for each event. Here we choose to label each event with the timestamp that somebody on the ground would assign to it. This choice is dictated by our wish to exhibit the symmetry of the circular track and train. We can multiply all time coordinates by a common factor in an attempt to account for the time dilation for clocks that travel with the train (let's imagine that we do this, though it won't have any implication for what follows), and we can add a common offset to all time coordinates (pointless but harmless). But if we use different factors or different offsets for any two places on the train, we will have broken the symmetry -- an observer on the platform will see some cars reach 12 o'clock (according to rotating-frame coordinates) before or after it happens in other cars, and this will destroy our assumption of symmetry.
So both time and space coordinates are more or less forced upon us by the idea "let's construct a rotating frame and do our analysis there". However, something horrible has happened: the time and space coordinates do not fit together right. To see this, consider one lightbulb flashing in each end of our railway car, such that the passengers in the car consider the flashes to have happened at the same time (i.e., the light from both flashes reaches the midpoint of the car simultaneously). There's a classic SR gedankenexperiment that concludes in precisely this circumstance that an observer on the platform will not consider the two flash events to be simultaneous. But this means that the two events have different time coordinates in the rotating frame, by definition (above) of said time coordinates. The passenger on the train will have seen two flash events and believe them to be simultaneous -- but they have different time coordinates according to the rotating frame. This is what I mean by the time coordinate being "crazy".
When I say that the distance between sleepers increase in the rotating frame, what I mean is neither more nor less than the following: Select an event on sleeper A. Select an event on sleeper B, such that the two events have identical time coordinate. Subtract the space coordinates of those two events. The difference is larger than the distance found by the ground observer.
More "practically", suppose that the ground observer places a lightbulb on each of the two sleepers, and arranges for them to flash at the same time in his frame. By design those two flashes have the same time coordinate in the rotating system. To find their space coordinates, we must intercut to the passenger. He sees (through a window in the car floor?) flash A happen right under the middle of his car. A bit later, according to his subjective time, he sees flash B happen about a meter to the rear. Therefore he can conclude that the sleeper distance in the rotating coordinate system is one meter. However, he won't himself believe this to be the true distance between sleepers. Why, while he was waiting for sleeper B to flash, sleeper A has been hurtling rearwards, and was perhaps only 30 cm from sleeper B "when" the latter flashed. So the measuring-stick distance he would prefer between the sleeper is 30 cm. (Meanwhile, the observer on the ground measures 60 cm between the sleepers).
In summary, the rotating coordinate system is one that does not match the experience of the passenger. And the reason for the whole mixup was that we wanted to have time coordinates that made the symmetry of the experiment manifest. All we need to do to alleviate the passenger's confusion is to let him use the coordinates of the intertial frame in which the car is momentarily at rest. In this frame the track contracts as it should do, the car in front of us reaches its top speed and turns off its engine a short time before ours does, in short everything behaves as it should.
The only thing we lose in the inertial frame is the symmetry of the experiment. For example, we cannot expect that the number of sleepers we count between our car and the next one at some instant in the inertial comoving frame is also the same as the number of sleepers between two cars at the other side of the circle at the same instant. They are not in a symmetric situation; they have a different velocity with respect to the frame where we do our sleeper-counting.
So, did Einstein err when he constructed the rotating frame in the first place? Not at all, because he was doing GR here, not SR. And in GR it is generally expected that coordinates by themselves are meaningless; you need to transform the coordinates into a locally inertial frame before you try doing SR-like physics with them. Our fallacy was to attempt to extract a "spatial geometry" from the metric by ignoring the time coordinate, and nevertheless expect the spatial geometry to give sensible results about things that involve time (namely: the motion of sleepers under the train).