Exercise 1.05

Particle physicists are so used to setting $c = 1$ that they measure mass in units of energy. In particular, they tend to use electron volts (1 eV = $1.6\times 10^{-12}$ erg = $1.8\times 10^{-33}$ g), or, more commonly, keV, MeV, and GeV ($10^3$ eV, $10^6$ eV, and $10^9$ eV, respectively). The muon has been measured to have a mass of 0.106 GeV and a rest frame lifetime of 2.19 $\times 10^{-6}$ seconds. Imagine that such a muon is moving in the circular storage ring of a particle accelerator, 1 kilometer in diameter, such that the muon's total energy is 1000 GeV. How long would it appear to live from the experimenter's point of view? How many radians would it travel around the ring?

Let $x^{\mu'}$ be the coordinates of the particle in the experimenter's rest frame and $x^{\mu}$ be its coordinates in its own rest frame.

$$x^{0'}=\gamma(v)\,x^{0}=\frac{x^{0}}{\sqrt{ 1-v^{2} }}$$

Let $p^{0'}=1000$ GeV be the energy of the accelerated particle and $p^{0}=0.106$ GeV be its rest mass.

$$p^{0'}=\gamma(v)\,p^{0}=\frac{p^{0}}{\sqrt{ 1-v^{2} }}$$

By rearranging this equation, we obtain:

$$1-v^{2}=\frac{p^{0}}{p^{0'}}=\frac{0.106}{1000}$$

$$v=\sqrt{ 1-1.06\times 10^{-4} }\approx0.9999469986\,c \approx299,777\,\text{km/s}$$

Therefore we have

$$x^{0'}=\frac{x^{0}}{1.06\times 10^{-4} }=\frac{2.19\times 10^{-6} \,\text{seconds}}{1.06\times 10^{-4} }=0.02066 \,\text{s}$$

and the muon will travel

$$\frac{299,777\,\text{km/s}\,\times 0.02066 \,\text{s}}{0.5\,\text{km}}\approx12,387\,\text{rad}$$