Relativity and Muon`s Decay Paradox

Einstein`s theory of relativity in 1905 revolutionized physics in many ways by changing our understanding of space-time. The Time hereto considered absolute was no longer the same for two observers in different frames of reference. For example, let’s say you are traveling in a car and passing by your friend on the street. The hands of the clock on your wristwatch and that of your friend will tick at different rates. As time slows down for objects in motion. This will have implications on the length of the object also. One of the postulates of The Special Theory of Relativity says that the speed of light in free space has the same value in all inertial frames of reference. We will understand it through the example of Muon.

 Muon`s Decay Paradox

Muons are subatomic particles that are created in the Earth’s atmosphere through a process known as cosmic ray shower. Cosmic ray muons have an average lifetime of 2.2 microseconds. They travel at a speed of about 0.998c (c=speed of light) and reach sea level in profusion-one of them passes through each square centimeter of the earth`s surface on average slightly more often than once a minute. But in 2.2 microseconds, muons can only travel a distance of 0.998c*2.2 microsecond= 0.66km. Whereas they are created at altitudes of 6 km or more.

Einstein’s theory of relativity saves us from this paradox. Because muons travel at a considerable speed close to the speed of light. According to our frame of reference on earth, the average lifetime of a muon would be 34.8 microseconds and not 2.2 microseconds as time dilation takes place. And in 34.8 microseconds a muon with a speed of 0.998c can cover a distance of 10.4 km. 

What if somebody were to accompany a muon in its descent at a speed of 0.998c so that to him or her the muon is at rest? To an observer, the muon can travel only 0.66km before decaying. The only way to account for this is if the length is shortened by virtue of its motion. The principle of relativity tells us that the shortening must be of the same factor that the muon lifetime is extended from the point of view of a stationary observer on Earth. Hence, both time and length are variable quantities, and the speed of light is constant in all frames of reference.

The application of special relativity has had profound impacts across various scientific fields. It has enabled precise calculations in particle physics, guided the development of high-energy particle accelerators, contributed to the understanding of cosmic phenomena, and played a crucial role in the accurate functioning of GPS systems. Its influence continues to shape our modern understanding of the universe.

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