Ask a Physicist: Time Warp Brain Teaser

Bill, from the US, wants to know: Would a satellite with a perfectly circular orbit around the center of a circle experience time dilation relative to an observer at the center of the circle? What if the observer were spinning to always be looking at the satellite making them both seem at relative rest? If the satellite does experience time dilation, is it due to the non-inertial acceleration due to centripetal force?  This is a really insightful question--it applies the concepts of relativity ...