Time Dilation ExperimentsThe abandonment of the concept of universal time embodied in the time dilation expression is so counterintuitive that one must look at the experiments to confirm this extraordinary prediction of special relativity. These are some of the experiments which confirm time dilation. 
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Twin ParadoxThe story is that one of a pair of twins leaves on a high speed space journey during which he travels at a large fraction of the speed of light while the other remains on the Earth. Because of time dilation, time is running more slowly in the spacecraft as seen by the earthbound twin and the traveling twin will find that the earthbound twin will be older upon return from the journey. The common question: Is this real? Would one twin really be younger? The basic question about whether time dilation is real is settled by the muon experiment. The clear implication is that the traveling twin would indeed be younger, but the scenario is complicated by the fact that the traveling twin must be accelerated up to traveling speed, turned around, and decelerated again upon return to Earth. Accelerations are outside the realm of special relativity and require general relativity. Despite the experimental difficulties, an experiment on a commercial airline confirms the existence of a time difference between ground observers and a reference frame moving with respect to them.

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Atomic Fine StructureOne of the great successes of the quantum theory was the prediction of the energy levels of the hydrogen atom. When attempts were made to explain the fine structure of the hydrogen spectral lines, it was found that the splitting of the lines was in error by a factor of two. It was realized that relativistic time dilation must be used in calculating the frequencies, and calculations showed that this relativistic correction, called Thomas precession, was the factor of two which was needed for agreement with experiment. 
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Kaivola Time Dilation ExperimentA precise measurement of time dilation was made in a doublephoton experiment by Kaviola et al. in 1985. An atomic beam of neon atoms at v= 0.004c was excited by two lasers which were colinear with the beam and incident upon the atoms from opposite directions. The absorption frequencies for neon are shifted by both the Doppler effect and time dilation, with the Doppler shift being much larger. By measuring a double photon transition with the two lasers from opposite directions, the Doppler shift was canceled and the time dilation shift measured precisely . The measurement was made by measuring the beat frequency between the two tunable lasers. This experiment confirmed the expected time dilation within 4 parts in 100,000. 
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