# How does Radioactive Decay work under Special Relativity?

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 :)

• Should suffice to just consider the proper time experienced by the substance. Commented Aug 15, 2022 at 6:50