grav

2009-Jun-20, 12:23 AM

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 tried using simultaneity effects to resolve this, but so far haven't been able to make it work. 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?

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 tried using simultaneity effects to resolve this, but so far haven't been able to make it work. 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?