# Gravitational time dilation compensated by acceleration

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!

• I am curious why you come to your original assumption: There are two close-by objects which are not moving relative to each other. How could one be accelerated while the the other one is not? Is it that the one on the mountain is suspended by solid matter while the one in the helicopter is (at least indirectly) suspended by gaseous matter? That should not, well, matter. Is it that the helicopter must burn fuel in order to stay aloft? It is a bit unintuitive, but there is no energy transferred to or from the object; it is all wasted by heating the air. – Peter A. Schneider Feb 12 at 2:31

## 2 Answers

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.

• Wikipedia " Gravitation therefore does not cause proper acceleration, since gravity acts upon the inertial observer that any proper acceleration must depart from. A corollary is that all inertial observers always have a proper acceleration of zero."... so gravitation causes a proper acceleration and this wikipedia page is wrong ? – François Ritter Feb 11 at 16:42
• @FrançoisRitter that means an object falling freely under gravity has a zero proper acceleration. The objects in your question are being held stationary not falling freely. – John Rennie Feb 11 at 16:44

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.