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I am facing trouble in understanding Bell's Spaceship Paradox.
I understand the explanation for the rope breaking in the stationary observer's frame well. I do not understand how to explain it with respect to the spaceships.
One of the explanations I read involved the idea that leading clocks in the moving system. In order to maintain the same speeds in the observer's frame, they will have to accelerate at the same time w.r.t the observer. If the observer looks at the clocks on each spaceship though, the leading spaceship will have to accelerate earlier than the one behind. And that will cause the rope to break.
All this is again w.r.t the observer, right? But how do we explain this in the spaceships' frame? I tried to think of explanations but couldn't. Essentially the stationary observer's frame came popping up and I couldn't rely entirely on the spaceships' frame.

Ambica Govind
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2 Answers2

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Let's say the spaceships use bombs to accelerate. Front ship detonates one bomb every hour, measured by a local clock,rear ship does the same.

Now we can see that in the ships' frame:

  1. Delta v caused by one explosion is same for both ships according to the crew. (except when the ships have gained a very large speed difference)
  2. According to the crew the front clock ticks faster than the rear clock. After some time rear ship has detonated million bombs, while the front ship has detonated million + 100 bombs.

So we can say that the rope breaks because of the rear ship's slow clock.

stuffu
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It is the relativity of simultaneity again.

If the spaceships remained the same distance apart in a moving frame, the gap between them would appear to contract in the Earth's frame owing to length contraction. To maintain a constant gap in the Earth frame they have to spread out in their own frame as they speed up.

From the perspective of the ships, the front ship is always moving faster than the rear ship, so even though they accelerate at the same rate, they are doing so out of synch, as far as their speeds are concerned, so they gradually move further apart. The reason why their acceleration is out of synch is that they start to accelerate at the same time in the Earth's frame, which means that as soon as they start moving, those two instants are no longer simultaneous in their frame: in their new moving frame, the instant the lead ship started to accelerate was earlier than the instant the rear ship started to accelerate.

To put it another way, in the arrangement described by Bell the speeds of the spaceships are always synchronised in the Earth frame, which means they can never be synchronised in the frame of the ships- the leading ship has to reach a given speed v before the lagging ship in the moving frame if the two ships are to have the same speed v simultaneously in the Earth frame.

Professor Sushing
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