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Muon Experiment
Muons enter the atmosphere (about 10km) at a velocity of 0.98c and have at rest lifetime of 1.56 * 10-6 seconds. To travel 10 km at 0.98c, this will take 34 * 10-6 seconds, which is about 21.8 half-lives. From this, it would appear that 2-21.8 of the original population would be left, or about 0.3 out of every million muons that enter that atmosphere.

However, this is not what is experimentally observed, and instead it is several orders of magnitude grater. The number is closer to 50,000 muons out of a million that make it. So what is happening?

From the ground point of view, it appears that the muon's clock is time dilated and running much slower by a factor of 5. When translated to actual time, only 4.36 half-lives have passed for the muon to move 10km. Thus instead we get 2-4.36 which is 0.049, or about 49,000 out of every million muons.

Realize that to the muon's point of view, time has not slowed down, but rather the distance has been shortened by a factor of 5. Thus, to move this distance it only takes 6.8 * 10-6 seconds, which happens to be 4.36 half-lives.

This experiment was first conducted in 1941 by B. Rossi and D.B Hall. Since then, this has also been demonstrated with muons in particle accelerators.

Hafele and Keating Experiment
"During October, 1971, four cesium atomic beam clocks were flown on regularly scheduled commercial jet flights around the world twice, once eastward and once westward, to test Einstein's theory of relativity with macroscopic clocks. From the actual flight paths of each trip, the theory predicted that the flying clocks, compared with reference clocks at the U.S. Naval Observatory, should have lost 40+/-23 nanoseconds during the eastward trip and should have gained 275+/-21 nanoseconds during the westward trip ... Relative to the atomic time scale of the U.S. Naval Observatory, the flying clocks lost 59+/-10 nanoseconds during the eastward trip and gained 273+/-7 nanosecond during the westward trip, where the errors are the corresponding standard deviations. These results provide an unambiguous empirical resolution of the famous clock "paradox" with macroscopic clocks."

J.C. Hafele and R. E. Keating, Science 177, 166 (1972)

```           East          West
Predicted  -40 +/- 23ns  +275 +/- 21ns
Measured   -59 +/- 10ns  +273 +/-  7ns
```

In this, several forms of time dilation occur:

• Gravitational
• Kinematic
The sum of these moves to the predicted values:
```               East          West
Gravitational +144 +/- 14ns  +179 +/- 18ns
Kinematic     -184 +/- 18ns  + 96 +/- 10ns
==========================================
Predicted     - 40 +/- 23ns  +275 +/- 21ns
```

The Gravitational component is associated with the altitude above the earth. This plays a large role in GPS signals (and was not initially corrected for resulting in large cumulative errors - it has been corrected).

Kinematic time shift is the 'classic' one that we are more or less familiar with. As an object moves faster, its clock slows down with respect to the rest frame.

Atomic Fine Structure
With quantum theory came the prediction of the energy levels of a hydrogen atom. Attempts were made to explain the actual structure of the hydrogen spectral lines, an error was discovered that was off by a factor of two. The solution to this error turned out to be that time dilation must be used for the calculation of the frequency.

Kaivola Time Dilation Experiment
In 1985 time dilation was again measured with a double photon experiment conducted by Kaviola. In this experiment, a beam of neon atoms moving at 0.004c was excited by two lasers from opposite directions. The absorption frequency which were observed were shifted by the Doppler effect and time dilation. When the two directions were measured, the Doppler shift was canceled out leaving only time dilation which was measured by examining the beat frequency between the two tuneable lasers. The experiment confirmed the expected time dilation within 0.0004%.

Kaivola, M., et al., Phys. Rev. Lett. 54, 255 (1985).

Primary source: http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/twin.html

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