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I've come across a weird paradox that I can't answer, I will explain it via the following thought experiment:

There is a space-train and an observer 1 light year apart with synchronised clocks. The train travels to the observer at just below the speed of light. In the reference frame of the observer the train is subject to time dilation and length contraction. When it arrives its clock reads 1 second as its time was basically standing still and the observers clock reads one year. From the trains reference frame it sees the 'observer' travelling towards the train at near light speed, so the train should expect to read the observers clock as 1 second and the trains clock as a year as the train sees the observer to have undergone time dilation. So what do the clocks actually read and why? The only way I can imagine the paradox with relativity not occurring is if both clocks read the same otherwise you could tell what speed you were going?

Qmechanic
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Harrison Harris
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  • This is exactly the twin paradox – Jim Aug 18 '15 at 15:44
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    There is no way for the space-train and an observer 1 light year apart to synchronise their clocks, so none of the rest of your argument follows. – John Rennie Aug 18 '15 at 15:45
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    possible duplicate of How is the classical twin paradox resolved? – Jim Aug 18 '15 at 15:46
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    @JohnRennie They can be reasonable synchronized to within acceptably measurement precision. You can't be 100% sure they are synchronized, but you can be sure enough that an engineer would put their stamp of approval on the experiment (if you care about such things) – Jim Aug 18 '15 at 15:49
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    More like a duplicate of this – Jahan Claes Aug 18 '15 at 17:19
  • @Jim: What does "synchronized" mean? – WillO Aug 18 '15 at 21:58
  • @WillO synchronized means an observer views them as simultaneous. I believe what he's referring to is, if you start with two synchronized clocks right next to each other, and move them apart sufficiently slowly, then you can get them arbitrarily far apart and arbitrarily well synchronized. It's possible because time dilation is second-order in velocity. – Jahan Claes Aug 18 '15 at 22:27
  • @JahanClaes: "synchronized means an observer views them as simultaneous"--- which observer? – WillO Aug 18 '15 at 23:51
  • @WillO an observer in the frame in which the clocks started out at rest. You're of course right that synchronized-ness depends on the frame, and that is the key to answering the question. But in general we can construct clocks that are synchronized in any frame. – Jahan Claes Aug 18 '15 at 23:53
  • @JahanClaes: If I understand the question correctly, there is no frame in which both clocks started out at rest. – WillO Aug 18 '15 at 23:55
  • @WillO, that's exactly it, see my answer below. Because of that, one observer can say the clocks are synchronized, but we can't have BOTH observers say the clocks are synchronized. – Jahan Claes Aug 18 '15 at 23:56
  • @JahanClaes: Yes, I'd already seen (and upvoted) your answer. – WillO Aug 19 '15 at 00:00
  • @JahanClaes Actually, time dilation also has a certain dependence on distance apart for two clocks that were initially adjacent in the same rest frame. Regardless how slowly you move them, by the time you reach a specified distance apart, there will always be a small discrepancy in the elapsed time they measure. Luckily, this can be mitigated to ensure that the expected discrepancy is below the maximum allowable experiment uncertainty – Jim Aug 19 '15 at 13:38

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This is not exactly the twin paradox, but it's close.

First, let's make the problem more precise. Let's assume the train is one light year away from a planet, traveling near light-speed, and at the very beginning of the journey, a person on the planet views the clocks as synchronized. Then a person on the train does not view the clocks as synchronized. This is the famous "relativity of simultaneity".

So here's what happens. At the start of the journey, someone on the planet thinks both clocks read 0 years. At the end of the journey, the person on the planet thinks the planet clock reads 1 year, the train clock reads 1 minute.

The person on the TRAIN, on the other hand, starts out thinking that their clock reads 0 years, but the planet clock is just a small fraction of a minute shy of 1 year. Then the person on the train makes the journey, and it takes one minute from their perspective, and even less time passed on the planet from their perspective, and they reach the planet and find that the planet clock reads 1 year, and the train clock reads 1 minute. So no paradox occurred!

You could also set up the problem so that the person on the train views the clocks as synchronized, and get a similar result.

Jahan Claes
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