Thread: twin paradox for orbit

Let's say we have two observers, Alice and Carol, that are stationary to each other with a distance of 100 meters between them and their clocks are synchronized. Another observer, Bob is circling around Alice at a distance of 100 meters, constantly properly accelerating toward her, and so orbits Alice and passes Carol once per orbit. Bob sets his clock to read the same as Carol's when they coincide, then Bob continues travelling around until he reaches Carol again, where they compare times on their clocks. According to Alice and Carol's frame, Bob is constantly accelerating, so his clock ticks sqrt[1 - g R / c^2] slower than theirs. Since g = v^2 / R, that becomes sqrt[1 - 2 (v/c)^2]. Also, observers in each frame sees clocks in the other ticking sqrt[1 - (v/c)^2] slower because of the time dilation due to the relative speed. Okay, so Alice and Carol see Bob's clock ticking sqrt[(1 - 2 (v/c)^2) (1 - (v/c)^2)] slower than theirs and Bob sees their clocks ticking sqrt[(1 - (v/c)^2) / (1 - 2 (v/c)^2)] faster than his own. The problem here is that when Bob and Carol meet back up and compare times on their clocks, each should say the other is the inverse of their own, so if Carol's says her clock ticked twice as fast as Bob's, then Bob should say his clock ticked at half the rate of Carol's, but that is not how the ratios work out. It does for the time dilation of the fictitious gravity but not when that for the relative speed is included. I have also tried using simultaneity effects to resolve this, but apparently the time that Carol's clock falls behind Bob's as he accelerates away from Carol while orbitting Alice is gained equally upon returning, so simultaneity effects don't seem to figure in this way so far. So what's going on here? How do we make the times between SR and GR work out inversely for the times on the clocks between the frames of the observers?

it has been a long time since I've ever attempted to run through the calculations for such cases, but I do know what the problem is in this case. Bob, is in a constant state of acceleration. The magnitute of his velocity is constant but the direction of his velocity vector is constantly changing. He isn't in an inertial reference frame and the assumption of SR simply don't apply when comparing his clock to the clocks of either Alice or Carol.