Your question has already been answered, but since the confusion seems to have stemmed from Bell's spaceship paradox, let me make some comments on this, as it is a really interesting though experiment.
Janus said:
The trick is in understanding that the distance between the ships stays constant in the rest frame and that this an arbitrary condition placed on the scenario.
That condition isn't really all that arbitrary. The condition is actually that the ships have the same acceleration sequence ... the condition is merely that:
the ships are identical, and start simultaneously in their mutual/intial inertial rest frame.
Yes, from this you can get that the length between the ships is constant according to the initial inertial frame, but it comes about so in a natural way.
What Bell's spaceship paradox shows, is several interesting things:
1) For a rigid object to accelerate, the acceleration must be
different along the length of the object (unlike in Newtonian mechanics). This is fairly counter-intuitive and interesting about relativity. It also leads to the idea of a Rindler horizon.
2) Bell originally made the paradox to argue that length contraction of an object was
physical in at least this sense: from an inertial frame S, measure an inertially moving object ... have it accelerate to a new inertial frame ... if in equilibrium in the object's inertial frames in the beginning and end, the physics according to frame S
must have included a physical contraction (in his spaceship paradox, an elastic string between the ships will shorten the distance ... and only has tension
while accelerating). It is an interesting problem, but wording can be subtle sometimes (it has lead to arguments on this forum before, because even though everyone agrees on the measurements and outcomes, the wording can unfortunately be easily misconstrued as misleading statements. So be careful to work out the physics, so you aren't mislead by unintential semantics of describing this scenario. Its usually the semantics of how to present this that leads to arguments unfortunately.)
Bell's spaceship paradox, and the reason so many people get it wrong, is that often people are too casual with the "length contraction" formula. Lorentz transformation actually just relate
labels for events in one coordinate system to labels for events in another coordinate system. You can use the coordinate transformation formula to relate lengths measured in two inertial frames:
L = L' \sqrt{1 - v^2 / c^2}
For the length of an object under arbitrary motion to follow the equation:
L = L_0 \sqrt{1 - v^2 / c^2}
is something separate. Don't get me wrong, it can be easily derived from the Lorentz transformations
given assumptions about the object and equilibrium, but the point is that the equation (while having the same form) is relating a
different situation. Because instead of just involving transformations of labels, if v is not constant, it is telling us something about how the object
changes as it accelerates. Man, now I see why people argue about the semantics. I know what I'm trying to describe, and knowledgeable people would probably understand as well, but I really don't like how that is worded. Well, if someone else here wants to try to do it better, feel free ... don't worry, I won't argue semantics with you.
Bell used his spaceship example to make this distinction extra clear. It still catches people off gaurd to this day.
