Train Stopping Time: Different Observers, Different Times?

[adelmakram]

The ends can accelerate a different rates.

Fine.

So how if the 2 ends are programmed to start moving at the same acceleration rate? Similarly to Bell`s spaceship paradox, there should be a tension on the string because its length now is larger than the contracted length. So How did this contraction come from?

So we are facing 2 situations: either the rear end start to accelerate before the near end which lead to a revealed length contraction or both accelerate at the same rate which create a non-revealed length contraction.

 

[A.T.]

So how if the 2 ends are programmed to start moving at the same acceleration rate? Similarly to Bell`s spaceship paradox, there should be a tension on the string because its length now is larger than the contracted length.

Yes there will be tension in Bells scenario.

So How did this contraction come from?

What?

So we are facing 2 situations: either the rear end start to accelerate before the near end which lead to a revealed length contraction or both accelerate at the same rate which create a non-revealed length contraction.

They can also start accelerating simultaneously, but at different rates, to shorten the string.

 

[Dale]

So how if the 2 ends are programmed to start moving at the same acceleration rate? Similarly to Bell`s spaceship paradox, there should be a tension on the string because its length now is larger than the contracted length. So How did this contraction come from?

The expansion scalar is positive in all frames, despite the fact that the length is unchanged in the starting frame. In a relativistic version of Hooke's law the change in tension in the string/train is proportional to the expansion scalar, not the coordinate strain rate.

 

[adelmakram]

Remember, if you get a contradiction then you made an error. So where do you think the error is?

My first suggestion is that Coulombs law doesn't apply, particularly not during and immediately following the acceleration. If you want to apply Coulombs law then you have to wait long enough for the changes to propagate through, but it seems like this is precisely the period you want to examine.

Second, you have claimed that the results in the platform frame contradict the results on the train frame without ever deriving the result in the train frame. Remember, the train frame is non inertial, so you will need to transform maxwells equations into that frame. You cannot have a one sided contradiction.

It would help if you made the acceleration concrete by specifying an exact equation of motion as well as the metric in the trains frame.

I did not apply Coulomb law immediately after the stop. I compared the electric fields on a charge before and after stop relative to 2 observers. I found that Lienard Wichert potential before the stop relative to the platform observer is larger than Coulomb potential after the stop. For the train observer, the Coulomb potential before A starts to move is smaller than after A comes close to B end. This is a potential paradox.

 

[Dale]

It is not a potential paradox. Maxwells equations are invariant under the Lorentz transform. Therefore, it is not possible to set up a scenario which satisfies Maxwell's equations in one frame and not in another.

Your approach of actually working less than a quarter of the problem and then assuming you know the rest is wrong. That is all you have shown. To work this problem correctly requires the following steps:

1) write the expression of the motion of both charges in the platform frame
2) calculate the Lienard Wiechert potential from one charge
3) evaluate the potential at the other charge
4) write the transformation equations to the train frame
6) calculate the metric in the train frame
7) transform the motion of the charges to the train frame
8) transform the fields to the train frame
9) confirm that the transformed fields satisfy Maxwell's equations in the train frame
10) evaluate the fields at the other charge in the train frame
11) compare the results in each frame

You cannot claim even a possible paradox with less. In particular, you have never once written down the critical steps 1 and 4, and because of that if anyone else were to work it you would simply claim it is irrelevant, as you have done previously. This is a problem that you need to work through, and not just part way.