Special relativity and uniform circular motion

Question

Suppose we have two observers A and B. A is in some inertial frame. B is undergoing uniform circular motion with some speed $v$ as observed by A. They each have a clock.

My concern is with what A sees when looking at B's clock and what B sees when looking at A's clock.

I expect A will see B's clock slowed down... according to A it will take $\gamma$ seconds for B's clock to tick 1 second. This is the standard time dilation formula. Is this right?

But how does one analyze how B sees A's clock using only special relativity?

Answer 1

Recall in Euclidean geometry how the shortest path between two points was a straight line?

A similar (but opposite) thing happens in relativity. The curve between two events that has the most proper time is the straight line (or geodesic).

So if observer B has coordinates $(R\cos( \omega t),R\sin(\omega t),0)$ and observer A has coordinates $(R,0,0)$ then A ages the most. And they can compare ages when they meet. It's just that A will age more between times they meet than B will age.

Observer B will compute the rate A ages (the computation is based on when you get a light signal, and then correcting for the travel time of the signal, and yes to do the calculation B will have to take into account that they are not inertial) and see them aging faster.

You can even offset A so they never meet, e.g. let A be at $(5R,0,0)$ or $(0,0,0)$ and the results stay the same.