# Relativity 2

## Example 1: The Decay of an Elementary Particle

**The Question:** There are only three kinds of elementary particles that appear to be absolutely stable. We're made of electrons, protons, and neutrons, right? Neutrons by themselves don't hang around very long – a 100 neutrons will become 50 neutrons in about 15 minutes! Think of it. One moment a neutron is minding its own business and quite randomly, it suddenly becomes a proton, and electron, and the strange elusive particle called a neutrino. 

Although our theories do suggest that protons might decay, all attempts to confirm that have not seen it yet. The lifetime of a proton is more than $10^{34}$ years, which is impressive since the age of the universe is about $10^{10}$ years. Likewise, theories can accommodate an unstable electron, but searches for that possibility lead to a lifetime that's more than $10^{28}$ years. Lucky for us, as our bodily atoms decaying around us would be a disappointment.

One unstable particle that we'll learn about is called the "pion" ("$\pi$") and it decays into another unstable particle called a "muon" ("$\mu$") which in turn, decays into an electron. The lifetime of a pion is about $2.5 \times 10^{-8}$ seconds...25 nanoseconds. They are readily produced in cosmic rays and artificially in accelerators. So let's do that. 

***The scenario:***

* A pion is produced in an accelerator and moves away from its place of birth at a speed of half that of light. The pion is itself its own rest frame (the "proper frame") 

* The lab in which it was produced is where we are observing. So the lab is Home and the pion's frame is Away and is moving at $u=0.5c$. 

* The pion decays into a muon after one of it's lifetimes of $2.5 \times 10^{-8}$ seconds. So we have a distinct interval: the pion is born and then the pion decays. It's a little clock with one "tick" and never a "tock."

How long does the pion appear to live as observed in the lab, Home, frame?

![galilean_train](pion_decay.png)



------

**The Answer:** 

This is a very standard example of time dilation. In our language now:

* $t_A = 2.5 \times 10^{-8}$ seconds
* $u=0.5c$
* $t_H$ is what we want to determine.

From the time dilation model we know that:

$t_H = \gamma t_A$

so we need to know what $\gamma$ is which we can get from the graphs in the text.

$\beta = 0.5$ gives $\gamma = 1.154$ 

so the time that the pion lives in the laboratory is

$$
\begin{align*}
t_H &= \gamma t_A \\
 &= 1.154 \times 2.5 \times 10^{-8} \\
 t_H &= 2.9 \times 10^{-8} \text{ seconds}
\end{align*}
$$

The pion appears to live *longer* to us than it does to the pion itself.