Bell's spaceship paradox --- reversed

birulami

The paradox was discussed already in length, but I would like to add a twist. I take for granted, that the thread breaks, as described by Bell himself.

Suppose the experiment starts out with a bit of slack on the thread connecting the two spaceships. Call the surrounding frame, the one that shall not move, S. Then, as described, rockets accelerate gently until the thread is taut, just before it breaks. Call the frame of the rockets S'. Now we play dumb and forget how the situation evolved. Rather, within S', we see just a taut thread between the rockets, which, from this frame's perspective don't move. Within S', an observer A', equidistant to both rockets starts the (seemingly original) Bell experiment, except, by coincidence, he orders the rockets to accelerate in reverse gear, as seen from S. Of course A' can expect the string to break.

But: invoking our prior knowledge, we know that reverse gear actually means deceleration with S and the string should rather get some slack again. What's wrong here?

My guess is as follows: Judged as seen from S, A' does not and can not order the rockets to accelerate synchronously. Rather they get orders in such a way that A' sees them synchronous, while within S they are not synchronous. Snap.

Harald.

JesseM

But what kind of accelerating frame do you want to use for the rockets? The normal type of coordinate system used for objects undergoing constant coordinate acceleration is Rindler coordinates, which have the nice property that at any given point on the accelerating observer's worldline, Rindler coordinates will define simultaneity and distances to other objects at that moment in the same way as they'd be defined in the observer's instantaneous inertial rest frame at that moment. And in Rindler coordinates, only a family of observers undergoing Born rigid acceleration would remain at rest--Born rigid just meaning that the distance between them stays constant in each observer's instantaneous inertial rest frame from one moment to the next (in this case a taut string between the ships would not break). But in order for this to be true, it works out that different ships must undergo different constant proper accelerations, so from the perspective of the inertial frame S they'd be accelerating at different rates. This page on the Rindler horizon has a diagram of what the paths of ships at rest in Rindler coordinates look like from the perspective of an inertial frame, along with a little explanation:


So, if the ships are instead both undergoing the same coordinate acceleration as seen in the inertial frame S, that means that in either ship's Rindler frame S', the other ship is not maintaining a constant coordinate distance but is instead moving away, which is why the string breaks. If you want to use a different type of accelerating coordinate system than Rindler coordinates you're free to do so, but in a non-Rindler coordinate system it would no longer be true that a constant coordinate distance would mean a constant distance in the ship's instantaneous inertial rest frame from one moment to the next.

Meir Achuz

arXiv:0906.1919

Ich

Exactly.