That Can’t Be Right! – A Look into Relativity

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A whole lot didn’t happen this week, so I am going to explain a really cool physics problem* we went over while studying relativity. I am currently in Modern Physics (PHY 2003), taught by Dr. Baarmand this semester as part of my astronomy and astrophysics major. This class is the third physics class that many majors at Florida Tech must take, and it covers topics such as relativity, an introduction to quantum physics, molecules and solids, atomic structure and an introduction to nuclear and particle physics.

The problem was actually quite simple to solve, but it illustrates the importance of relativity and why we cannot just ignore its effects. The question begins by explaining that the subatomic particles called muons have a very short lifetime of approximately 2.2 μs (micro-seconds). After a muon has been around for that long, it decays into other particles. The first part of the question asks what is the furthest possible distance a muon could travel in its lifetime, which is simply solved by multiplying the lifetime by the speed of light, the fastest anything can possibly go. The result is 0.66 km.

This creates a problem with a scientific observation: muons have been found in cosmic rays that hit the Earth’s surface. **Cosmic rays are high-energy particles from space that are constantly hitting the Earth’s atmosphere. If muons can only travel 0.66 km in their lifetime, how could they possibly go through 10 km of our atmosphere to be measured at Earth’s surface? This is the main part of the physics problem. The 0.66 km was calculated assuming we were in the muon’s frame of reference. “Frame of reference” means the lifetime given to us is what would be measured if we were literally riding with the muon as it traveled through its life. But in reality, we are observers on Earth’s surface, watching as the muons are carried in on cosmic rays. So the 2.2 μs lifetime is called the “proper time” of the muon, or Δt­0.

In order to figure out what lifetime we will observe from Earth’s frame of reference, we need to calculate the time dilation, Δt. The equation for this is Δt = ɣ Δt0, where ɣ (gamma) = 1/sqrt[1-(v/c)2], with v as the speed of the particle and c is the speed of light. Assuming the muon travels near the speed of light, the problem gives the speed of 0.999c. Plugging all of this in, the lifetime of the muon as seen from Earth is 49.2 μs, over twenty-two times longer than what the muon sees from its frame of reference! Using 49.2 μs to calculate the distance traveled now gives 14.8 km rather than 0.66 km, hence why muons can travel from space all the way through our atmosphere. But not only is there relativity of time at work here, but also relativity of length. We see the muon travel 14.8 km, but how far does the muon think it traveled at its proper time, 2.2 μs? The equation for this is L = L­0/ ɣ, where Lo is the proper length of the atmosphere (10 km in this case), L is the length contraction, and ɣ is the same value as mentioned earlier. The result is 0.447 km.

What does this all boil down to? In summary, the muon travels at 2.2 μs and goes a distance of 0.447 km before decaying. Meanwhile, due to relativity, we as observers on Earth’s surface see the muon traveling at 49.2 μs and going 14.8 km before decaying. How is that even possible, right? It’s all based on Einstein’s general theory of relativity. At its most basic, the theory of relativity predicts the dilation of space and time due to how the observer is moving. It also gives us the idea that we live in a universe with three dimensions of space and one dimension of time (called “spacetime”). The curvature of spacetime depends a lot on the mass and energy of the object – so the more massive, the more spacetime bends around it. The theory of relativity is primarily what got me interested in studying physics in the first place. It describes our world in a way that our senses say cannot possibly be true, but with so much that is experimental, we must accept it as reality or fall behind in the newest discoveries of science. I love that feeling of having my mind completely blown when I learn something new that I never would’ve thought possible. It happens quite often in trying to understand how the world actually works, because sadly a lot of people are very ignorant when it comes to scientific matters. Keep learning and fight the ignorance!

Photo credit to University of Oregon












*Information from Young and Freedman’s 13th edition of University Physics.
**Source: R.A. Mewaldt’s “Cosmic Rays” found at

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