Is there any evidence for the crazy idea that clocks moving relative to each other tell time differently? Indeed, there’s plenty. An experiment in 1971 sent an atomic clock around the world on a jet plane while an identical clock remained in the laboratory. This was a difficult experiment because the traveling clock’s speed was so small compared to c, but measuring the small differences between the time intervals was just barely within the capabilities of atomic clocks. It was also a more complex experiment than we’ve analyzed because the clock accelerated as it moved around a circle. The scientists found that, upon its return, the eastbound clock, traveling faster than the laboratory on a rotating earth, was 60 ns behind the stay-at-home clock, which was exactly as predicted by relativity.

Very detailed studies have been done on unstable particles called muons that are created at the top of the atmosphere, at a height of about 60 km, when high-energy cosmic rays collide with air molecules. It is well known, from laboratory studies, that stationary muons decay with a half-life of $1.5\mu \text{s}$. That is, half the muons decay within $1.5\mu \text{s,}$ half of those remaining decay in the next $1.5\mu \text{s,}$ and so on. The decays can be used as a clock.

The muons travel down through the atmosphere at very nearly the speed of light. The time needed to reach the ground, assuming $v\approx c$, is $\text{Δ}t\approx (60\text{,}000\text{ m})/(3\times {10}^8\text{ m}/\text{s})=200\mu s$. This is 133 half-lives, so the fraction of muons reaching the ground should be $\approx {\left(\frac{1}{2}\right)}^{133}={10}^{-40}$. That is, only 1 out of every ${10}^{40}$ muons should reach the ground. In fact, experiments find that about 1 in 10 muons reach the ground, an experimental result that differs by a factor of ${10}^{39}$ from our prediction!

The discrepancy is due to time dilation. In FIGURE 36.23, the two events “muon is created” and “muon hits ground” take place at two different places in the earth’s reference frame. However, these two events occur at the same position in the muon’s reference frame. (The muon is like the rocket in Example 36.4.) Thus the muon’s internal clock measures the proper time. The time-dilated interval $\text{Δ}t=200\mu \text{s}$ in the earth’s reference frame corresponds to a proper time $\text{Δ}\tau \approx 5\mu \text{s}$ in the muon’s reference frame. That is, in the muon’s reference frame it takes only $5\mu \text{s}$ from creation at the top of the atmosphere until the ground runs into it. This is 3.3 half-lives, so the fraction of muons reaching the ground is ${\left(\frac{1}{2}\right)}^{3.3}=0.1$, or 1 out of 10. We wouldn’t detect muons at the ground at all if not for time dilation.

FIGURE 36.23 We wouldn’t detect muons at the ground if not for time dilation.

The details are beyond the scope of this textbook, but dozens of high-energy particle accelerators around the world that study quarks and other elementary particles have been designed and built on the basis of Einstein’s theory of relativity. The fact that they work exactly as planned is strong testimony to the reality of time dilation.