Thinking of atomic clocks riding on planes and GPS satellites? You've come to the right place!
I guess there must have really been a media blitz regarding this experiment because everyone seems to know about it and the vast majority of people seem to think it is irrefutable evidence of time dilation and therefore relativity. For some reason I've also run into people who believe that how the GPS satellites stay synchronized has something to do with this experiment. It does not, but I'll mention the satellites below.
This has got to be one of the most botched experiments of all time. If you and I were in an airport restaurant and I told you that my watch was an hour slow one day and then an hour fast the next day so therefore it keeps perfect time on average, would you use my watch to make sure you catch your flight?
The results of the HK experiment were so tampered with that it could be considered fraudulent. An engineer, A.G. Kelly, obtained the original 1971 test report from the United States Naval Observatory, on which the 1972 paper was based, and discovered that the original results actually did not support the result computed in the 1972 paper. The portable Ceasium-beam clocks that were carried varied in time so badly that some of them could vary more than the total supposed results during the time of the test. The most stable of the four clocks, no 447, by itself constituting a better experiment than all the clocks together, indicated, as an overall result of the test, zero kinematic time accumulation.
In addition to these completely invalidating facts, the expectations of the experiment were based upon a third reference point called "Proper Time". As I have discussed in length elsewhere, introduction of this extra reference point takes relativity out of the equation because it is analogous to adding a hidden Aether.(universal reference frame)
Hey! What about the GPS satellites?!
Pro-Relativity Resource on atomic clocks and GPS etc: http://math.ucr.edu/home/baez/RelWWW/wrong.html#gps
We have the GPS sats supposedly being adjusted according to the Sagnac Effect and Gravitational calculation proven by Pound-Rebka both of which do not require relativity and have nothing to do with time dilation. Because I haven't been able to yet uncover how much of a role that these two elements play I'll have to do some conjecture here and say that I would bet that the contribution of Sagnac VS Gravitational is completely dependant upon the orientation of the clocks in regard to their travel and the Earth's gravity. If I were a betting man, I'd say that they have the clocks oriented parallel to the surface of the earth so that the Sagnac effect is the primary concern though there would still be a small gravitation effect due to the transverse wave propagation observed in space and therefore a transversal Doppler effect.
Another simple explanation of my own would have to do with the relation of Aether pressure increasing as you get farther from earth. The rigidity of the medium would also be increased therefore increasing the speed of light(radiation). This would make the atomic clocks run faster as they moved away from the planet. This is also conjectural and problematic though because a counterpoint is that if all forms of radiation are "evaporation" or transition of matter into Aether particles then wouldn't the additional pressure be less conducive to radiative evaporation and then slow the cycles of the CS-133 atom? To read more of my mucking around with theories, head to the "Aether" section.