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The other day I had the following thought experiment:

Envision that you have 1 kilogram of radioactive Promethium, which has a half-life of 17.3 years, which you put inside a box and load onto a spaceship. You then accelerate this spaceship to 0.9c, where c is the speed of light in a vacuum, and allow it to stay at this speed for 17.3 years after which the spaceship returns to earth, and you take out your sample of Promethium and measure the amount Promethium left in your original 1 kilogram sample. How much Promethium would you expect to find?

In an attempt to come to some sensible answer to my question, I proceeded with the following. The time elapsed between the first time I measured my sample of Promethium and until the spaceship returned and I measured my sample again was 17.3 years which is precisely the half-life of Promethium, so I would expect to find about 500 grams of Promethium left in my sample. On the other hand, we also know that time dilation effects start becoming important when traveling at a significant fraction of the speed of light. So from the perspective of the sample, the amount of time that has elapsed is $\Delta t^{sample} = 17.3 years * \sqrt{1-0.9^2} \approx 7.5 years$, which means the sample only had about 7.5 years to undergo radioactive decay and so you would expect to have about 740 grams of Promethium left.

So the question becomes, which one is correct? How much Promethium will you find when you measure your sample the second time? A why would also be appreciated; this has been like a bad itch that I would like scratched :)

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    $\begingroup$ Should suffice to just consider the proper time experienced by the substance. $\endgroup$
    – AfterShave
    Aug 15, 2022 at 6:50

3 Answers 3

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In special relativity, the elapsed time between two events that occur in different places is generally frame-dependent, which means that it might be, say, five minutes in one frame and three minutes in another. However, the time experienced by a person or an object is known as 'proper time'. Suppose you climb in a spacehip at noon and zoom off somewhere. The time you experience- your proper time- will be told by your watch. If you stop your spaceship after ten minutes by your watch and climb out, the local time outside will depend on how fast you have been travelling, but to you ten minutes will have passed.

In the example you give, the proper time on board the spaceship will determine how much of the promethium decays. The corresponding time that passes in any other frame (on Earth, for example) is irrelevant.

If ever you get confused about SR you might find it helpful to remember that all its effects are symmetrical and all motion is relative. In the frame of a passing muon, for example, you are actually travelling at nearly c, and a second on your watch might be a minute in the frame of the muon. What counts is the time you experience, which is your proper time.

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  • $\begingroup$ Thank you for the great answer $\endgroup$ Aug 15, 2022 at 11:35
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This is just the twin "paradoxon" instead of the twins you have 2 portions of Promethium, so yes you have more Promethium in the returning ship

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It's just the proper time that matters. Just as in the case of atmospheric muons that reach ground despite you would expect them to have decayd since long in the lab frame on the ground.

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