# Would the fast rotating legs of a human age slower? For how much?

Let's imagine that there is some human that rotates rapidly around his head. That means that the legs have a very large velocity, but the head has the velocity nearly equal to zero. Let's just ignore the spaghettification effect. Using special relativity, we would expect the legs to age slower (using the concept of the time dilation), since the legs have a very large speed. But since the velocity is changing, thus producing an accelerating movement, we should use the general relativity equations here.

I am well versed in the special relativity on the introductory level, but still very far from being professional in the field of general relativity. Thus, I am asking you to explain these two points to me:

1. Would the fast rotating legs age slower than the head, even in the general relativity? (I expect: yes.)
2. How does this effect deviate from the solutions using just special relativity, if there are any deviations at all?

The calculation using special relativity:

The time dilation is given by $$t_{head}=\gamma t_{legs}=\frac{1}{\sqrt{1-\beta^2}}t_{legs}$$ For example, using $$\beta=\frac{v}{c}=\frac{2}{3}$$ ($$\gamma=\sqrt 3$$) and $$t_{legs}=100\rm\,yr$$ $$t_{head}=\gamma t_{legs}=\sqrt 3 \cdot 100\rm\,yr=173.2\rm\,yr$$ So, the legs have just "died" after 100 years, but the head died a long time ago.

• Hi User123. Did you try to do a back-of-an-envelope-calculation? Commented Sep 6, 2021 at 18:18
• @Qmechanic What do you mean by this? Commented Sep 6, 2021 at 18:21
• If the angular velocity is constant (your person is rotating at a constant speed) then there's no need to invoke general relativity or worry about acceleration at all. Time dilation only depends on the speed (absolute value of velocity). See for example physics.stackexchange.com/questions/599344/… Commented Sep 6, 2021 at 18:28
• @EricSmith Yes, that's the insight I was looking for. Try posting an answer. Commented Sep 6, 2021 at 18:29
• My reading of this question suggests that you are under the impression that special relativity is only capable of treating objects moving with constant velocity, and that general relativity is required whenever acceleration is present; this is not true. General relativity is required when one needs to include the effects of (strong) gravitational fields, but special relativity is perfectly capable of treating dynamics in flat spacetime. Commented Sep 6, 2021 at 21:03