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PeterDonis said:No, because special relativity only applies in flat spacetime, i.e., in the absence of gravitating masses. Here there is a gravitating mass present so spacetime is not flat.
But in the Haele-Keating experiment did they not use the general relativity equations to work out the gravitational time dilation and the special relativity equations to work out the velocity time dilation? And when working out the velocity time dilation for the westbound direction, did they not do it from the plane's perspective?
PeterDonis said:(If spacetime were flat, two objects both in free fall could only meet once. So the fact that the two satellites, both in free fall, meet multiple times is enough to show that spacetime cannot be flat.)
I wasn't thinking that the spacetime was flat, only that scientists had used the special relativity equations for similar situations.
PeterDonis said:Yes, that's true, but it's also true that, while passing each other both times, they are in relative motion, and so each would see the other to have velocity time dilation.
So how could you work out what velocity time dilation they observed? Would it be much different from using the special relativity equations, because presumably if they were bouncing torch beams off mirrored ceilings there wouldn't be much difference from whether they were in flat space passing each other, or making an orbit around a huge sphere.
PeterDonis said:Yes, but this is using the "gravity time dilation" of the clock on the sphere. It is not using any concept of "gravity time dilation" that applies in the satellites' own frames.
Yes I was assuming they would work it out for the clock on the sphere.
PeterDonis said:No, because the "gravity time dilation" they are calculating doesn't apply in the satellites' own frames, as above, but the velocity time dilation is relative to the satellites' own frames. So there's no way to combine the two as you are describing.
I would expect them to be able calculate the gravity time dilation for the clock on the sphere the same as you could. And because the satellite is in relative motion to the sphere clock, also see it as having velocity time dilation.