Q:

Trying to understand the equivalence principle in GRT

Suppose that two observers are on a sphere of radius $r$ with angle $\phi>0$ and distance $d>0$ from each other and  they are accelerated from inside to outside perpendicular to the sphere. Then by the equivalent principle, each of them can not know if he is on a planet or if he is accelerated. Now after some time of acceleration the two observers will be on a sphere with bigger radius $R>r$ and bigger distance $D>d$. The part which I do not understand is this:
If they were on a planet, then the distance and radius would not change. But if they are accelerated, the distance and radius change. How does this fit with the equivalence principle?

A:

The equivalence principle is local, not global. One way of stating the equivalence principle is that spacetime can locally be treated as flat, i.e., that the error incurred by using special relativity approaches zero in the limit as you consider small regions of spacetime. Your thought experiment involves a large region of space, so the e.p. doesn't tell us anything about it. If we take your thought experiment and apply a limit as $\phi\rightarrow0$, then it becomes local, and it's consistent with the e.p.

