VISUAL PHYSICS ONLINE

 

SPECIAL RELATIVITY

MUON DECAY

Experimental evidence for time dilation and length contraction

 

 

 

SUMMARY

 

Inertial frames of reference      You must always identify the frames of reference

 

    station frame (S)

    moving frame (M)  

 

relativity factor                

 

Time dilation effect       

       proper time interval        

       dilated time interval     

 

Length contraction

     proper length              

     contracted length      

 

 

 

 

Does Einstein's theory of special relativity accurately describe the motion of objects traveling close to the speed of light?

 

The theory of special relativity has made many astonishing predictions. Einstein did not receive his Noble Prize for his theory of special relativity but for a minor contribution to our understanding of science – the photoelectric effect. Scientist at the time, were uncertain of Einstein’s predictions as they were so inconceivable and so against long-held believes. For more than a century, experiments have been carried out to test Einstein’s theories. So far, all experimental evidence has been confirmed the predictions of special relativity and general relativity.

 

 

Muons are unstable particles with a rest mass of 207 times that of an electron and a charge of ±1.6x10-19 C.  Muons decay into electrons or positrons with an average lifetime of 2.2 ms as measured in their inertial frame of reference.

 

When high energy particles called cosmic rays (such as protons) enter the atmosphere from outer space, they interact with air molecules in the upper atmosphere at a height of about h = 10 km above the Earth’s surface creating a cosmic ray shower of particles including muons that reach the Earth’s surface. The muons are created in these cosmic ray showers travel at  w.r.t to the Earth. In a stationary frame of reference, on average a muon exists for about  before it decays into other particles.

 

            

             relativity factor    

            

 

 

Newtonian (classical) point of view

We can apply the principles of classical physics to calculate the distance a muon on average will travel before it decays (figure 1)

 

 

Fig. 1. On average a muon will travel 650 m before decaying. So, in terms of classical physics, very few muons should be able to reach the surface of the Earth.  

 

Hence, from a Newtonian point of view, there is a low probability of a muon reaching the Earth’s surface from the upper atmosphere where they are produced. However, experiments show that many muons do reach the Earth’s surface in cosmic ray showers.

 

 

Special relativity: muon frame of reference

In the frame of reference of the muon, the Earth is approaching at speed  and the distance to the Earth as measured in the muon’s frame is the contracted length (figure 2)

 

 

Fig. 2.   Using the principle of length contraction, the muons travel on average a much greater distance compared with the classical prediction .

       

The contracted distance is much less and so many muons will be able to reach the Earth’s surface as 2000 m >> 650 m.

 

 

Special relativity: Earth frame of reference

The observer on Earth needs two clocks to measure the time interval for the muon to decay. The time interval  corresponds to the dilated time interval whereas the proper time interval  is the average lifetime in the muon’s frame (figure 3).

 

 

Fig. 3.  Frame of reference is the Earth. The Earth observer measures the dilated time interval, hence will conclude the muon will travel much further than that given by the classical prediction.

 

The average distance travelled is now long enough for many muons to reach the Earth’s surface.

 

 

 

Another look at muon decay

The decay of muon can be accurately described by the radioactive decay law

         

 

where and  are the number of detected muons at times  and time t respectively. is the half-life (time for the number of muons detected to halve) and  is the decay constant. The measured half-life and decay constant for muons at rest in the laboratory are

         

 

An experiment was performed by placing a Geiger counter to detect muons on the top of a mountain 2000 m high. The muons are assumed to be moving at a speed equal to 0.98c. In a time interval T the Geiger counted 1000 muons. The Geiger counter was moved to the bottom of the mountain, 2000 m below the peak. In the same time interval T, the Geiger counter registered 540 muons. These are the muons that survived the trip without decaying.

 

Classically, we can calculate the number of muons surviving the trip.

 

        distance travelled by muons  

        speed of muons  

        time interval for trip  

 

        Number of surviving muons

      

         

 

So, only 45 muons should survive the trip. Something is wrong !!!

 

Our classical theory predicts 45 muons, but measurements record 540 muons. The problem must be approached using special relativity. The muons are moving at a speed of 0.98c w.r.t the Earth. So, we must consider the time dilation effect.

       

 

 

 

The Earth based clock records a time interval of 6.80x10-6 s for the muon to travel from the top to the bottom of the mountain.

          

 

       

The Earth based observes see the muon’s “moving clock” record the proper time

       

 

 

     

 

 

 

In the muon’s rest frame, the decay of the muon is given the radioactive decay law is

       

         

 

The number of muons surviving the trip is 538, which agrees with observations. An experiment like this was performed by B. Rossi and D. Hall in 1941 at Mount Washington, New Hampshire, U.S.A. Their results agreed with the predictions of special relativity and not the predictions of classical physics.

 

 

 

Fig. 4.   The number of muons detected at the top of the mountain is 1000 whereas at the bottom of the mountain only 540 survived without decaying. The experimental result agrees with the prediction based on the time dilation equation.