The problem statement of Bell's Spaceship paradox is this:
Two spaceships float in space and are at rest relative to each other. They are connected by a string. The string is strong, but it cannot withstand an arbitrary amount of stretching. At a given instant, the spaceships simultaneously (with respect to their initial inertial frame) start accelerating (along the direction of the line between them) with the same acceleration. (Assume they bought identical engines from the same store, and they put them on the same setting.) Will the string eventually break?
And the solution is here: http://www.physics.harvard.edu/uploads/files/undergrad/probweek/sol11.pdf
The very first statement made in the solution to this problem is "To an observer in the original rest frame, the spaceships stay the same distance, d, apart.". But why do they stay the same distance apart to an observer in the original rest frame? Shouldn't the distance between the spaceships undergo length contraction, as they are connected by a rope? I asked a similar question here, and the answer that I got was that
Length contraction only applies to situations where you have a system with two "ends" that are moving at the same velocity, and you know the distance L between these ends in the frame S where they are at rest, and want to know the distance L' between them at any given instant in some other frame S' where they are moving at velocity v along the axis joining the two ends.
Well the two ends of the rope are moving at the same velocity, I know the distance between them at frame $S$ when they are at rest, and I do want to know the distance between them at any given instant when they are moving at velocity $v$ along the axis joining the two ends. How then can I make the statement that for an observer in the rest frame that the spaceships stay the same distance, $d$, apart?
Answer
"To an observer in the original rest frame, the spaceships stay the same distance, d, apart.". But why do they stay the same distance apart to an observer in the original rest frame?
The spaceships move with constant mutual distance in the original rest frame, since their corresponding parts have the same velocity function of time. The description of the situation in the original question directly implies this.
Shouldn't the distance between the spaceships undergo length contraction, as they are connected by a rope?
No, this would be contrary to the specified situation.
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