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Time dilation is always mutual - both clocks are travelling relative to each other and both must observe that the other clock is slower than themselves. Since motion is relative, and it is not possible to determine which one is noving and which one is still, so it is not possible for just one clock to be slower than the other.

Therefore, in the famous experiment, when the atomic clock was flown around the world, was it observed and recorded that (a) the person travelling with the flying clock found that the clock on earth was slower than the flying clock, whereas (b) the person on earth found that the flying clock was slower than the clock on earth?

In particular, I am interested in the first observation, i.e. for the person travelling with the clock the earth clock was really slower than the flying clock, as that is not the result that one usually hears for this experiment.

For this question, let us consider only the special relativistic part of the time difference, and ignore the general relativistic part, although the actual measured time difference was the sum of both.

For example, https://en.wikipedia.org/wiki/Hafele%E2%80%93Keating_experiment gives results only of time gained by the flying clock. It gives no results for time gained by the lab clock, which makes me wonder whether the time dilation was mutual or not.

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    $\begingroup$ >Since motion is relative, and it is not possible to determine which one is noving and which one is still, so it is not possible for just one clock to be slower than the other. But, acceleration is absolute, so it was easy to determine which was which. It is the acceleration that breaks the symmetry $\endgroup$ – zeta-band Aug 20 '18 at 18:58
  • $\begingroup$ researchgate.net/publication/… $\endgroup$ – Albert Aug 20 '18 at 19:30
  • $\begingroup$ Notice that the original papers are linked (in un-paywalled PDFs) in the Wikipedia article. $\endgroup$ – dmckee Aug 21 '18 at 19:06
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Special relativity does not say that all motion is relative. It says that all inertial motion is relative. None of the clocks in the Hafele-Keating experiment were moving inertially. (The clocks that remained in the lab at the US Naval Observatory were accelerating due to the earth's rotation. The clock that flew to the west actually had a lower proper acceleration than the lab.)

I think all of this becomes much more transparent if you talk in terms of the metric. The elapsed time on a clock is given by $s=\int_A^B \sqrt{dt^2-dx^2-dy^2-dz^2} $, where $(t,x,y,z)$ are the Minkowski coordinates, and the integration is along the clock's world-line. This integral does not come out the same if you change the path taken from A to B. It is maximized for an inertial path from A to B.

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  • $\begingroup$ If this is true that none of the clocks were moving inertially, then the experiment was an invalid experiment and does not really prove time dilation in inertial frames moving relative to each other. If it was real time dilation, it would be mutual. It cannot be just one clock slower than the other when both moved relative to each other. $\endgroup$ – Khushro Shahookar Aug 21 '18 at 3:02
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    $\begingroup$ @KhushroShahookar No the experiment was conducted to compare the combined predictions of special and general relativity with the observation. It's a perfectly good experiment, if a bit more complicated that the kind of idealized experiment that makes good pop-sci descriptions. Welcome to the reality of experimental science: the world is a messy and uncooperative place and you have to work hard to extract good data from it. $\endgroup$ – dmckee Aug 21 '18 at 18:59
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    $\begingroup$ In addition to dmckee's remaeks, keep in mind that these experiments were done using passenger jets, so inevitably there was gravitational time dilation involved. Cleaner experiments have been done to separate the gravitational and kinematic effects. Alley et al. did measurements using aircraft over Chesapeake Bay that were designed to be sensitive mainly to the gravitational effect, and there were also mountain-valley experiments. such as Briatore's and Iijima's. A 2010 experiment by Chou moved a clock in a lab to isolate the kinematic effect. $\endgroup$ – Ben Crowell Aug 22 '18 at 23:17
  • $\begingroup$ Perhaps you should simplify the question. You could have two identical spaceships that are lined up parallel to each other, but are positioned far apart and are pointing in opposite directions. They are also tied together via a very long rope. Both then fire their engines for say 5 minutes. Now they are going in circles. At the center of the rope, there is a clock. Question: do those on the space ships see this clock ticking slower, and would anyone positioned at this center clock see the space ship clocks running slower. $\endgroup$ – Sean Aug 25 '18 at 18:17
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Let's ignore the fact that the Earth is rotating, for simplicity.

At the start, the airplane and the airfield are in the same inertial reference frame. The airplane then accelerates and keeps accelerating very slightly as it performs a large circle. It is constantly changing reference frames. In every frame the airplane is in, the airbase is aging more slowly, but the airplane doesn't remain in any frame, and the changes in frame mean that the airbase is a touch older with each new frame.

Eventually, the airplane comes back to the airfield and lands and comes to a stop, again accelerating. Everything's again in the same reference frame, but the airfield has stayed in that frame. The airplane has accelerated through its flight, changing reference frames. Therefore, we're checking the clock that has not accelerated against the clock that has accelerated.

If the plane activated its space drive and headed away from the airfield at a constant speed and direction, then the plane and airfield would each see the other as aging more slowly, but they'd never meet up again to compare clocks. If the airplane turned around, in that turn it would change reference frames, and that change would show the airfield as aging faster.

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  • $\begingroup$ If all this is true, then the experiment was an invalid experiment and does not really prove time dilation in inertial frames moving relative to each other. If it was real time dilation, it would be mutual. It cannot be just one clock slower than the other when both moved relative to each other. $\endgroup$ – Khushro Shahookar Aug 21 '18 at 3:00
  • $\begingroup$ After moving at a constant speed, if the airplane turned around and came back, why would the earth lab age faster and the airplane age slower and not the other way around? What theory explains this and how? Obviously it is not time dilation in inertial frames. $\endgroup$ – Khushro Shahookar Aug 21 '18 at 3:09
  • $\begingroup$ Because the planet is rotating, the experiment is not symmetric. The expectation for east- and west-bound motion differ considerably and this shows up beautifully in the data. $\endgroup$ – dmckee Aug 21 '18 at 19:01

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