Now let’s look at the situation from Nancy’s perspective. She acquires a velocity of −gr/c from the frame in which Tom emitted the light, and so Nancy is always traveling away from the light source with velocity gr/c . Again according to the Doppler Effect, she receives flashes with a frequency of Sqrt(1−β/1+β)
Since β>0 we know that Nancy receives flashes slower than once per second: 1−β/1+β<1 . Hence she concludes that Tom’s clock is running slower than hers. Note that the rates are exactly symmetric here: if Tom thinks that Nancy’s clock is running at f ticks per second, Nancy thinks that Tom’s clock is running at 1/f ticks per second.
Now let’s put Tom and Nancy in a lab on the surface of the Earth, at exactly r apart, with Tom below Nancy. Once again, they exchange flashes.
By the equivalence principle, the results must be exactly the same as the results in the rocket ship: Tom must receive flashes at Sqrt(1+β/1−β) ticks per second, and Nancy must receive flashes at √(1−β/1+β) ticks per second.
Just for fun, let’s do the numerical calculation. Assume r=10 , g=9.8 ; we are using meters for our unit of distance and seconds for time, so c=3×10^8 . So β=98/9×10^16 = 1.09×10−15 . if we plug this in, and take the Taylor expansion of √(1+x)/(1−x)=1+x+O(x2) we see that Nancy’s clock runs faster than Tom’s by 1+1.09×10^−15 ; about a quadrillionth of a second per second faster. This is tiny, of course — Nancy will gain about 34 nanoseconds per year. But it’s detectable using precision clocks, and this was done in the Hafele–Keating experiment https://en.wikipedia.org/wiki/Hafele%E2%80%93Keating_experiment. And, of course, the results of the experiment were in full agreement with GR.
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