Quick questions on Evidence for special relativity: muons, GPS and particle physics, HSC Physics Module 7

Cosmic rays striking the upper atmosphere produce muons at altitudes around $10$ to $15$ km. Muons are unstable, with proper lifetime $t_0 = 2.2$ $\mu$s and typical speeds of $0.99c$ or more.

GPS satellites orbit at $\sim 20\,200$ km altitude with orbital speed $\sim 3.87$ km/s. A GPS receiver determines position by measuring the time-of-flight from at least four satellites, so the onboard clocks must agree with ground time to within a few nanoseconds to give metre-level positions.

Every collision experiment at a modern accelerator (LHC at CERN, Belle II at KEK, RHIC at Brookhaven) is analysed with relativistic kinematics:

In one proper lifetime, a muon at $0.99c$ travels about $650$ m, so almost no muons should reach the ground.

At $0.99c$, $\gamma \approx 7.09$. The Earth-frame lifetime is $\gamma t_0 \approx 16$ $\mu$s, and the muon travels about $4.6$ km in one dilated lifetime. A measurable fraction (about $10\%$ on average) survives to sea level.

The Rossi-Hall experiment (1941) compared muon flux at the top of Mount Washington (elevation $1900$ m) and at sea level. The ratio matched the relativistic prediction and ruled out the non-relativistic one by orders of magnitude. Modern detectors confirm this to high precision.

From the muon's point of view, its own lifetime is just $2.2$ $\mu$s. What changes is the distance to the ground: the atmosphere is length-contracted to $10$ km $/ \gamma = 1.4$ km, which a $0.99c$ muon can comfortably cross in one proper lifetime. The two frames agree on the observed outcome (10% transit fraction) by different routes.

A direct test of relativistic Doppler shift using hydrogen ion beams; agreed with relativity to a few per cent at the time and now to better than $10^{-9}$.

Measured gravitational redshift of $14.4$-keV gamma photons over $22.5$ m using the Mossbauer effect; supports general relativity, complementing SR evidence.

Optical lattice clocks at NIST can detect altitude differences of a few centimetres through gravitational time dilation.

State one experimental observation that confirms time dilation and one that confirms length contraction. [2 marks]

Muons are created at $h = 12 \text{ km}$ and travel at $v = 0.99c$. (a) Find $\gamma$. (b) Calculate how long the atmosphere appears in the muon's rest frame.

A particle accelerator at the Australian Synchrotron stores electrons at $E = 3.0 \text{ GeV}$. (a) Calculate $\gamma$ given electron rest energy $0.511 \text{ MeV}$. (b) Find the electron's speed as a fraction of $c$.