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:
- Would the fast rotating legs age slower than the head, even in the general relativity? (I expect: yes.)
- 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.
