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.

The Bailey et al. experiment (CERN, 1977) stored muons in a circular ring at $\gamma \approx 29.3$ ($v \approx 0.9994 c$). The lab-frame lifetime was measured to be about $64$ $\mu$s, $29.3$ times the rest-frame $2.2$ $\mu$s, in agreement with $t = \gamma t_0$ to better than $0.1\%$. The muons' centripetal acceleration in the storage ring was enormous ($\sim 10^{18} g$), confirming that time dilation depends only on instantaneous speed, not on acceleration.

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:

Ives-Stilwell experiment (1938). 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 $10^{-9}$.

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 $10^{-9}$.

Atomic clocks flown on commercial aircraft eastward and westward around the world differed from a stationary ground clock by amounts predicted by SR (motion) and GR (altitude) combined.

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.

Both frames (lab and muon rest frame) give the same observed transit fraction, by different mechanisms (time dilation versus length contraction).

The GR correction (about $+46$ $\mu$s/day) is larger than the SR correction (about $-7$ $\mu$s/day), and they partially cancel. Both must be applied.