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Suppose on the earth one put a tower of height $h$, there's two clock synchronized with respect to ground observer at $t_0$. Clock $A$ put under the tower, and clock $B$ put on top of the tower.

Since there's a velocity difference, there's time dilation.

However, according to Griffts, $A$ would see $B$ running slow, and $B$ would see $A$ running slow, and they are both right.

But this is not the case, for if $A$ send a single to $B$ or $B$ send a single to $A$(in experiments): $A$ and $B$ will always conclude that $A$ is slower than $B$.

  1. What's went wrong here? a. Is it because the presence of centripetal acceleration changed story, and one must consider the gradient from the view of GR?(gravity filed)

  2. Is there a way to explain it in SR for the "direction" of $A$ goes slower than B?

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    $\begingroup$ I'm curious to know if you have actually computed the difference in speed (as measured from the Earth-centered inertial frame let's say) for 10 meter tower (similar in size to the one used form the earliest gravitational time dilation experiment in the late 1950s). How does that compare to speeds you encounter in your day to day experience? What I'm getting at here is that the gravitational effects (GR) are small enough that it requires a tour de force to measure them, but the kinematic effects (SR) are tiny. $\endgroup$ – dmckee --- ex-moderator kitten Sep 3 '19 at 22:24
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On 2. After a period of acceleration of two clocks such as you describe, with one ahead of the other, the leading clock will show the accelerated observers a later time than the following one. They saw it running faster while accelerating. Observers in the rest frame see both clocks always in agreement.

If after the acceleration stops the moving observers then synchronise their clocks, the observers in the rest frame will now see the Lorenz time difference on the two clocks, with the leading clock showing the rest frame an earlier time than the following clock shows, compatible with the Lorenz foreshortening.

The leading clock is 'above' the other in the accelerating force field.

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