# Special relativity muon example

• Kara386
In summary, we discussed the concept of proper length and proper time in the context of special relativity, using the example of the muon's mean lifetime and distance traveled. We also clarified the difference between proper length and length measured from a moving frame, as well as proper time and time measured from a different frame.

## Homework Statement

Here's a standard example of special relativity in action:

The mean lifetime of the muon as measured in a laboratory is about 2µs (rounded to 1 s.f.). Thus, the typical distance traveled by a muon should be about ##3\times 10^8ms^{-1}\times 2\times 10^6s = 600m##. The atmosphere is about 20 km thick, so the fraction reaching Earth should be about ##e^{\frac{-20km}{0.6km}} = e^{-33}## ≈ 0. However, we detect ∼ 1% at sea level! How can we explain this?

## Homework Equations

##l = \frac{l_0}{\gamma}##
##t = \gamma t_0##

## The Attempt at a Solution

Here ##\gamma## is about 7. This is probably just me being really stupid, but this is a worked example we've been given and in the solution, it says that the length according to the muon in the muon's frame is ##\frac{20km}{7} = 3km##.

I thought that in the equation ##l = \frac{l_0}{\gamma}##, ##l_0## was the 'proper length' - the length measured in the rest frame of the muon. Well, that would make the length the muon sees ##20 \times 7##. wouldn't it? That's obviously wrong, because length in the moving frame contracts. So do I just have the definitions of ##l## and ##l_0## the wrong way round? Or worse, do I have them the right way round but their definitions wrong?

Kara386 said:
I thought that in the equation ##l = \frac{l_0}{\gamma}##, ##l_0## was the 'proper length' - the length measured in the rest frame of the muon.
Yes, ##l_0## is the proper length, but proper length means the length of something as measuring from its rest frame. Here we are talking about the "length" of the atmosphere, so 20 km is the proper length and ##l## is the distance measured from the muon's moving frame.

Doc Al said:
Yes, ##l_0## is the proper length, but proper length means the length of something as measuring from its rest frame. Here we are talking about the "length" of the atmosphere, so 20 km is the proper length and ##l## is the distance measured from the muon's moving frame.
So the proper length of the atmosphere is the same in lots of different frames, but proper time is specific to each frame? So for the muon, proper times are times measured in its frame, but if we chose Earth's frame, while the proper length of the atmosphere is the same, proper times are different, right?

Kara386 said:
So for the muon, proper times are times measured in its frame, but if we chose Earth's frame, while the proper length of the atmosphere is the same, proper times are different, right?
Clocks always measure 'proper time'. Here, the muon itself acts like a clock, so from the Earth's viewpoint the decaying muon is a clock that runs slow.

## 1. What is the muon example in special relativity?

The muon example in special relativity is a thought experiment used to demonstrate the effects of time dilation. It involves cosmic ray muons, which are subatomic particles that are created in the upper atmosphere and travel at near-light speeds towards the Earth's surface.

## 2. How does the muon example illustrate time dilation?

The muons have a very short half-life of only 2.2 microseconds, which means they should decay before reaching the Earth's surface. However, due to their high speeds, time is dilated for the muons relative to an observer on Earth. This means that from the muons' perspective, they have a longer lifespan and are able to reach the surface of the Earth before decaying.

## 3. What does the muon example tell us about the relationship between time and speed?

The muon example shows that as an object's speed approaches the speed of light, time slows down for that object relative to an observer. This is known as time dilation and is a fundamental concept in special relativity.

## 4. Can the muon example be observed in real life?

Yes, the muon example has been observed in real life through experiments such as the Muon Experiment at CERN. This experiment confirmed the predictions of time dilation and provided further evidence for the validity of special relativity.

## 5. How does the muon example relate to Einstein's theory of special relativity?

The muon example is one of the most well-known and frequently used examples in special relativity. It was first proposed by physicist George Sudarshan in 1955 and has been used to illustrate the concepts of time dilation and the relativity of simultaneity, both of which are key principles in Einstein's theory of special relativity.

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