I don't like the way this worded. "Clock A is at rest in spaces" makes no sense - things can only be at rest relative to something else. I think we can assume that this means that clock A is in an inertial frame - neither accelerating nor being affected by a gravitational field. And the question "how long does it take for B t reach that same point in space" I would also argue is better worded as "how long does it take for B to be 0.5 LY from A?" - this allows for the valid point of view that B is actually at rest and A is moving past at 0.5c.
Imagine that A has a long pole attached to it that is 0.5 LY in length. What A is measuring is the time it takes for B to go from being immediately adjacent to A to B being adjacent to the end of that pole. We are told that A measures the time as 1 year, which is consistent with the relative velocities of A and B being 0.5c.
Now consider B's perspective. As far as he's concerned he is at rest and A is coming towards him at 0.5c, which means that the long pole attached to A's ship appears to be contracted to something less than 0.5 LY in length - I'll leave the math to you. As A and the pole rush past at 0.5c it takes somewhat less time for the the tip of A's pole to be adjacent to B, so he records a shorter elapsed time than does A. Hence note that while both A and B are in inertial frames, they disagree as to the length of the pole and the time it takes to go from A and B being adjacent to each other to B being adjacent to the tip of that pole. So each has a different perspective as to what has happened.
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Last edited by ChipB; Jan 2nd 2016 at 07:42 AM.
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