Time Dilation
Time dilation is one of the primary phenomena predicted by Albert Einstein's theory of special relativity published in 1905. The theory states that the measured length of time between two events depends upon the frame of reference of the observer. In particular, time appears to go slower for objects moving with respect to the observer. Hence, time appears to have dilated, or lengthened.
An event is anything that takes place at a specific point in space and at a specific moment in time, like the chiming of a clock. The twelve o'clock chime of a clock occurs at the location of the clock and at the moment when the minute and hour hands of the clock are on the twelve.
The time interval between two events, like the twelve and one o'clock chimes of a clock, depend upon the observer's frame of reference. An observer who is stationary with respect to the clock will measure the proper time interval t, which--assuming the clock is well made--will be 1 hour. A second observer, moving at a speed v with respect to the clock will measure the time interval between the two events to be longer. The second observer measures a time interval of t = t / sqrt(1 - v2 / c2 ), where c is the speed of light in vacuum measured to be 300,000 km/s (186,000 mi/s).
The phenomenon of time dilation has been tested and observed in experiments since the beginning of the twentieth century. One striking example of time dilation explains why muons are detected at the surface of Earth. Muons are unstable particles that look like electrons with 207 times more mass, but they have a mean life of only 0.0000022 seconds before decaying into smaller particles. (This length of time is called the lifetime of the unstable particle.) Muons are produced with velocities close to 99.8% the speed of light from cosmic rays collisions with atoms in Earth's upper atmosphere, roughly 10 km (6.3 mi) above sea level. Even at these tremendous speeds, the muons would only travel a distance equal to roughly 660 m (2200 ft) before decaying, and therefore they should not be detectable at the surface of Earth.
However, since the muon is traveling at 99.8% the speed of light with respect to us on the surface of Earth, the lifetime of the muon appears to be longer by a factor of 1 / sqrt(1 - (0.998)2 ) = 1 / 0.063 = 16, which means the muon appears to live for 0.000035 seconds. At these speeds, a muon that lives for this length of time can travel approximately 10.4 km (6.5 mi) before it decays. Hence, time dilation explains why we detect muons at sea level.
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