Special Relativity V: Muons

Hello, and welcome back to MPC! Last week, we wrapped up our discussion on length contraction. We have now discussed the two major consequences of special relativity, namely time dilation and length contraction. Today, we will be reviewing what we have discussed by looking at a real-world example of special relativity: muons in the atmosphere.

What are muons? Muons are just particles (at least for our purposes). In other words, you can imagine a muon as being similar to a proton or an electron: muons are just very small specks of matter. Muons are actually constantly being “created” in the Earth’s atmosphere and, in a sense, rain down to the Earth’s surface. Muons, believe it or not, travel towards the surface of the Earth at very, very high speeds. Throughout this post, we will say that muons travel towards the Earth’s surface at 99.99% of the speed of light, or 0.9999c, and that muons are “created” 10 km above the surface of the Earth.

Figure 1: Muons (represent by μ, the Greek letter “mu”) “Raining Down” to the Earth’s Surface

**Note: In the figure above, the muon with a red outline represents a muon that was just “created.”**

Because the muons are moving so fast, they must hit the surface of the Earth rather quickly, right? Well, muons are actually quite interesting. Muons have a very short “lifespan” — about 2 microseconds to be precise (this is actually the “half-life” of a muon, but we will call it “lifespan”). That is 2 * 10^-6 seconds, an extremely low number. To help illustrate the lifespan of muons, let’s imagine a muon as a special clock. This clock will have one hand that rotates around the clock once in 2 microseconds. Let’s say that after one rotation of the arm, the clock “disappears.”

Figure 2: An Illustration of a Clock Analogy for Muons

**Note: The symbol μs represents microseconds.**

With this picture in mind, let’s think about a single, “newly-created” muon traveling towards the Earth’s surface. We already know the speed of this muon and how long it travels (or “lives”) for, so we can easily calculate how far it will travel:

So, this muon will only travel 600 m, or 0.6 km, meaning that, if it were to be “created” 10 km above the Earth’s surface, it would never reach the surface of the Earth.

We have a problem: experimentation confirms that muons do indeed hit the surface of the Earth. There is something wrong with our calculations! We have forgotten about something important!

Recall that the effects of time dilation and length contraction are only “significant” when an object is traveling at extremely high speeds (if this does not make sense, try analyzing the formula for the relativistic factor, or view our discussion from the Consequences of Time Dilation post). Therefore, we do not need to worry about relativity when analyzing, say, the motion of a standard car. However, unlike a car, a muon travels at 99.99% of the speed of light, which is a very high speed!

Let’s try applying our special relativity formulas. First, let’s think about the muon from the perspective of someone standing on the surface of the Earth (let’s say the scientist who is proving that muons hit the surface of the Earth).

Figure 3: The Muon’s Descent from a Stationary Observer’s Perspective

Immediately, we run into the first problem with our initial approach to this scenario: we neglected time dilation. Remember: when one object is moving relative to another, time passes by more slowly for the moving object (we discussed how 1 second to the person inside of a moving rocket ship may actually be 3 seconds to a stationary observer). In our scenario, the muon is traveling relative to the observer. The muon may think that its lifespan is only 2 microseconds (the muon thinks that its clock will only take 2 microseconds to undergo a full clock cycle and disappear), but the muon’s lifespan according to the observer will be greater than 2 microseconds (because, according to the observer, the muon’s clock is running in “slow motion”).

Figure 4: The Effect of Time Dilation on the Muon Clock

**Note: The illustration in Figure 4 is not drawn to scale and is simply meant to illustrate a concept**

Exactly how much longer is the muon’s lifecycle (according to the observer) when it is traveling at 0.9999c relative to the observer?:

According to the stationary observer, the muon’s lifespan has increased seventy times! Using this new lifespan, let’s calculate how far the muon should travel according to the observer:

Because 42 km is greater than 10 km, the observer will indeed see the muon hit the Earth’s surface. This conforms to experimentation, making it a very strong piece of proof for the validity of Einstein’s theory of special relativity!

However, we have another problem. What happens from the muon’s perspective?

Figure 5: The Muon’s Descent from the Muon’s Perspective

Recall that, even though it is moving at 0.9999c, the muon does not think that its own clock is running slowly (the muon does not think that its time is dilated). Therefore, according to the muon, its own lifespan is still only 2 microseconds, and we run into the same problem that we had initially (see Equation 1)! This contradicts our math for the stationary observer: could the muon never hit the surface of the Earth from its own perspective, but still hit the surface of the Earth from a stationary observer’s perspective?

And we have now run into our second problem: we are neglecting length contraction. Remember: when an object is moving quickly towards another object (like the rocket ship approaching the star in last week’s post), the distance between the two objects will appear “contracted” or “shortened” to someone inside of the moving object. In our muon scenario, the muon is quickly approaching the surface of the Earth. Who cares? Well, this means that the distance between the muon and the surface of the Earth will not actually be 10 km. Instead, it will be shortened:

Figure 6: The Effect of Length Contraction on the Distance between the Muon and the Surface of the Earth

**Note: The illustration in Figure 6 is not drawn to scale and is simply meant to illustrate a concept**

To calculate the distance between the muon and the surface of the Earth (from the muon’s perspective):

**Note: In Equation 5, L0 represents the length between the muon and the surface of the Earth according to a stationary observer; this formatting is different from last week’s formatting, and the reason why this formatting was chosen will be discussed in a future post.**

Now, the muon only has to travel 0.14 km to reach the surface of the Earth. But wait! According to Equation 1, the muon is able to travel a distance of 0.6 km when its lifespan is 2 microseconds0.14 km is less than 0.6 km, meaning that the muon will “definitely” be able to travel 0.14 km within 2 microseconds and hit the surface of the Earth!

I hope that you now have a better understanding of time dilation, length contraction, and the relationship between the two. The muon experiment is one of the most popular examples of special relativity in our everyday world. Now that we have a stronger understanding of time dilation length contraction, we are ready to delve into some new and exciting ideas in special relativity, which we will discuss next week. See you then!

For more information on muons and relativity, be sure to check out these resources!:

http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/muon.html

http://www.atmosp.physics.utoronto.ca/people/strong/phy140/lecture32_01.pdf

http://www2.fisica.unlp.edu.ar/~veiga/experiments.html