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I would like to compare the time indicated by two clocks: clock A is located at the top of a mountain, the clock B is in an helicopter flying stationnary at the same altitude than the clock A.

Clock A and B are at rest with each other and located at the same altitude. Clock A has a 0 proper acceleration, and clock B has a non zero proper acceleration (is it correct ?).

If we neglect the Earth rotation/tidal effects, does the general relativity predicts that both clocks will be equivalent thanks to the clock hypothesis ?

Why this simple experiment has never been done, instead of sending 2 planes in opposite directions with complex trajectories ?

Thank you!

François Ritter
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2 Answers2

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Both clocks have the same proper acceleration. The calculation of the proper acceleration is described in What is the weight equation through general relativity? The proper acceleration for an object stationary at a distance $r$ from the centre of the Earth turns out to be:

$$ A = \frac{GM}{r^2}\frac{1}{\sqrt{1-\frac{2GM}{c^2r}}} $$

It makes no difference that one clock is stationary on a mountain at the distance $r$ from the centre of the Earth while the other is stationary in a helicopter at the distance $r$ from the centre of the Earth.

John Rennie
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The reference frames of clock $A$ and clock $B$ are equivalent. They are stationary with respect to the spacetime given by the earth mass/energy at the same radial coordinate. They measure a proper acceleration as the frames are not along a geodesic.

The time of clock $A$ and clock $B$ runs with the same rate.

Michele Grosso
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