Atomic clocks are the most accurate of clocks known to man. It has been said that if an atomic clock, such as a caesium clock, were to be flown on an aeroplane, those clocks should move at a different speed relative to those on Earth, and that the slowdown could be attributable to differences both in GR, which predicts that the clocks should go faster at lower gravity, and SR, which predicts that the clocks should move slower due to the speed of the aeroplanes.
In 1971 Hafele & Keating (H&K) conducted tests to measure the effects of relativity on caesium clocks on aeroplanes. The planes flew in east and west directions along the equator, making a round-world trip to their starting point. H&K calculated that, due to the combined effects of SR and GR, the different east/west travel times and different altitudes, the eastward should loose 40 ns and the westward should gain 275 ns. The measured results showed that the eastward lost 59 ns, while the atomic clock transported westward gained 273 ns, compared to the stationary laboratory clocks.
It later transpired that the published results were quite different from the original measurements. H&K made a number of ‘corrections’ to their data to average out the errors between the clocks used, and the variations between the clocks moving in similar directions were large enough to invalidate the overall measurements. If the experiments were as flawed as this article suggests then they cannot be used to prove or disprove time dilation.