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Suppose a space ship is traveling from star A to star B at some significant fraction of the speed of light. In the frame of the ship, the distance A to B is less than the distance in A and B's rest frame. Is it possible for the ship to quickly increase its speed so that in its frame the ship is then closer to A compared to when it was going slower? That is, in the time interval that it is accelerating it moves further away from A due to its speed but the effect of increased length contraction has it end up closer? That sounds like an odd situation to be in.

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This is an interesting puzzle and I don't think I agree with the previous answer.

Suppose the ship is stationary, halfway between A and B, which are $L$ apart. In a very short (negligible) time it accelerates towards B at high speed. A and B are now $L/\gamma$ apart. The ship is still halfway between A and B (playing the 'negligible time' card). So A was $L/2$ away and is now $L/2\gamma$ away. A gets closer.

If you want, you can avoid the 'negligible time' argument and the fact that acceleration is off-limits for special relativity by hypothesising that another ship travelling at constant high speed just happens to pass the stationary one, and the two pilots compare notes and data as they pass.

Odd situation - yes. But not inconsistent.

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Results are somewhat different.

Suppose the observer aligns his x-axis passing through stars and Star A has some negative x-coordinate (-x1) and B has some positive (x2). Clearly observer stands at origin (0). We clearly quantify our definitions:

  1. Distance between A and B: x1+x2
  2. Distance between observer and A: x1

Now we can apply length contraction concept to 1. and not to 2.. Thus the observer in spaceship will see star A and B come closer due to increased length contraction. But A will nonetheless move away from observer in his frame. then in his frame velocity of A was always in negative x-axis before acceleration and acceleration tends to increase its magnitude so distance of A from his origin will increase.

PS: If we where to apply Newtons laws to A and B in observer's frame , both would seem to accelerate back at sane time and undergo acceleration for sane duration. However, according to relativity of simultaniety, B would start to accelerate first than A and hence travel some distance towards A. A similar result is Bell's spaceship paradox https://en.m.wikipedia.org/wiki/Bell%27s_spaceship_paradox

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