Evidence for special relativity: muons, accelerators and GPS: VCE Physics Unit 4

What this dot point is asking

VCAA lists three examples you must be able to explain and analyse:

1. muons can reach Earth even though their half-lives suggest they should decay in the upper atmosphere;
2. particle accelerator lengths must be designed with special relativity in mind;
3. GPS satellite time signals must be corrected for the effect of their orbital speed.

For each you should be able to explain it in words and do a calculation with $t = \gamma t_0$ and $L = L_0/\gamma$.

Example 1: muons from the upper atmosphere

Cosmic rays strike nuclei high in the atmosphere and make unstable particles called muons. Muons travel at close to $c$. A muon has a half-life of about $1.5\ \mu s$ measured in its own rest frame (its mean lifetime is about $2.2\ \mu s$, which is a half-life of $2.2 \times \ln 2 = 1.5\ \mu s$). Without relativity, even at nearly $c$ a muon would travel only about $450$ m in one half-life, so almost none should survive a trip of many kilometres. Yet large numbers are detected at the surface.

Explanation in the Earth frame (time dilation). The muon's half-life is its proper time. In our frame its clock runs slow, so the half-life we measure is $\gamma$ times longer, and the muon travels $\gamma$ times further.

Explanation in the muon frame (length contraction). The muon's half-life is unchanged, but the atmosphere rushes towards it at nearly $c$ and is length contracted. The distance to the ground is much shorter, so the muon gets there in only a few half-lives.

Example 2: particle accelerators

In a linear accelerator, particles travel along a long tube of fixed length $L_0$ as measured in the laboratory. As the particles approach $c$, the tube is length contracted in their frame, and the time they spend in it is short. Designers must use the relativistic relations when planning the machine. Classical formulas would give speeds above $c$ at these energies.

Example 3: GPS satellites

GPS positions are found by comparing time signals from several satellites, each carrying an atomic clock. Light travels about $0.3$ m in a nanosecond, so the clocks must agree to a few nanoseconds.

The satellites orbit at about $3.9$ km s$^{-1}$ relative to the centre of the Earth. Their clocks are moving, so seen from the ground they run slow by time dilation. Without correction, this speed effect alone would build up an error of several microseconds each day. The system corrects for it, so the clocks match ground time.

For the very small speeds involved, $\gamma \approx 1 + \tfrac{1}{2}v^2/c^2$, so the fractional slowing is $\tfrac{1}{2}v^2/c^2$. This approximation is a shortcut beyond the study design; you only need to use $t = \gamma t_0$.

The study design asks only about the correction for orbital speed. (General relativity also changes the clock rate because of the satellite's height in Earth's gravity, but that is beyond this key knowledge.)

Try this

Q1. Explain why muons produced in the upper atmosphere can be detected at the surface even though their half-life is only about $1.5\ \mu s$. [3 marks]

• Cue. Their speed is close to $c$. In the Earth frame their clocks run slow, so the half-life measured on Earth is longer (time dilation) and they travel further before decaying. In the muon frame the atmosphere is length contracted so the distance is shorter.

Q2. A muon travels at $0.98c$. Calculate its Lorentz factor, its half-life as measured on Earth and the distance it travels in that time. Use a rest-frame half-life of $1.5\ \mu s$. [4 marks]

• Cue. $\gamma = 5.03$. $t = 5.03 \times 1.5 = 7.5\ \mu s$. Distance $= 0.98 \times 3.00 \times 10^8 \times 7.54 \times 10^{-6} = 2.2$ km.

Q3. Explain why the length of a linear accelerator is not the length the electrons "see". [2 marks]

• Cue. The accelerator is at rest in the lab so its lab length is the proper length. In the frame of the fast electrons it is contracted by a factor of $\gamma$.

Q4. State why the clocks on GPS satellites must be corrected for special relativity. [2 marks]

• Cue. The satellites move at high speed relative to the ground, so by time dilation their clocks run slow relative to ground clocks. Uncorrected, the accumulating timing error would give large position errors.