Clock speed changes due to centripetal acceleration

Question

As far as I understand, the speed of your clock relative to other clocks is determined only by relative motion and by gravity differences.

Given a rotating sphere with 1 clock at the center, 1 at a pole, and 1 somewhere on the equator. These clocks do not move relative to each other: at 1 clock you can keep looking at 1 of the other clocks without having to move, and the distance between them remains the same.

The gravity experienced by these clocks is different, so they will run at different rates.

If somehow gravity would be uniform (everything outside of the sphere is made from the same material as the sphere?) without affecting the rotation, would those clocks run at the same speed?

Answer 1

Since you want the gravity to be uniform, we might as well consider the three points to be in empty space. So, we can use special relativity in this situation.
Lets say that the point in the center is O, the one at the pole is P and the one on equator is E.
In STR, two clocks run at the same rate when they are at rest in the same inertial frame. An inertial frame is one where an object at rest stays at rest. The clock at O and P are at rest in the same inertial frame. But the clock at E is not- since it is always accelerating(falling) towards the point at the center. One can define a coordinate system with its origin fixed on E and its x axis along the line joining O and E- and in this coordinate system both points are at rest wrt each other. But this is not an inertial frame. This is because any point other than O in this frame experiences a centrifugal force.
One can however define locally inertial coordinates at any point along the trajectory of E. This frame moves parallel to E with the same linear velocity at any point. But in any such frame, E is accelerating towards O.
In summary, the clock at the center and at the pole will run at same speed but the one at the equator will run slower since it is accelerating. Hope that answered your question.