Investigate experimental and observational evidence for special relativity, including atmospheric and accelerator muon decay, GPS clock corrections, and the routine use of relativistic mechanics in particle physics

What this dot point is asking

NESA wants you to give concrete experimental and observational evidence that special relativity is correct. The standard items are atmospheric and accelerator muon measurements, GPS satellite clock corrections, and the routine validation of relativistic kinematics in particle physics.

The answer

1. Atmospheric muons

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.

Non-relativistic prediction

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

Relativistic prediction

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.

Measurement

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.

The same effect in the muon frame

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.

2. Accelerator muons

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.

3. GPS satellite clocks

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.

Two relativistic effects shift the satellite clock rate:

• Special relativity (motion). A moving clock runs slow by $\Delta t / t \approx -\tfrac{1}{2}(v/c)^2 \approx -8.3 \times 10^{-11}$, equivalent to $-7.2$ $\mu$s per day.
• General relativity (altitude). A clock higher in Earth's gravitational potential runs fast by $\Delta t / t \approx +g h / c^2 \approx +5.3 \times 10^{-10}$, equivalent to $+45.8$ $\mu$s per day.

Net: the satellite clock runs about $+38.6$ $\mu$s per day faster than a ground clock. The correction is applied by adjusting the satellite's onboard oscillator frequency before launch (set slightly slow at $10.22999999543$ MHz instead of the design $10.23$ MHz), and minor residual corrections are computed each day. Without these corrections, GPS positions would drift by about $11$ km per day - a clear failure of the system.

GPS is therefore an everyday technology that validates both special and general relativity in real time.

4. Particle physics kinematics

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

• Energy-momentum conservation uses the four-vector form, $E^2 = (pc)^2 + (mc^2)^2$, not the non-relativistic $E = p^2 / (2m)$.
• Track reconstruction in magnetic fields assumes $r = p / (q B)$ with $p = \gamma m v$, not the non-relativistic version.
• Invariant masses of resonances (the Z boson, the Higgs boson) are reconstructed from decay products using the relativistic combination $m^2 c^4 = E^2 - (pc)^2$.

If any of this were wrong, particle identification and discoveries would fail. The Higgs boson was discovered in 2012 by reconstructing decays such as $H \to \gamma \gamma$ and $H \to ZZ \to 4\ell$, with invariant masses calculated using exactly the special-relativity machinery. The agreement of cross-sections, lifetimes and decay products with relativistic predictions at the per-cent level (or better) is the most thoroughly tested aspect of any physical theory.

5. Other supporting evidence

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 $10^{-9}$.

Hafele-Keating experiment (1971)

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.

Pound-Rebka experiment (1959)

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

Modern atomic-clock comparisons

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

Examples in context

Example 1. Australian-built GPS receivers and the $38.7 \mu\text{s}/\text{day}$ correction. GPS satellites orbiting at $v = 3.87 \text{ km/s}$ and altitude $20{,}200 \text{ km}$ experience two relativistic effects: SR time dilation slows their clocks by $\Delta t_{SR} = -(v^2/2c^2) \times 86{,}400 = -7.2 \mu\text{s}/\text{day}$, while general-relativistic altitude makes them tick faster by $+45.9 \mu\text{s}/\text{day}$. Net correction: $+38.7 \mu\text{s}/\text{day}$. A Geoscience Australia GPS reference station at Yarragadee, WA, has measured this drift to better than $10^{-9}$ precision over decades. Without the correction, positions would degrade by $\sim 11 \text{ km}/\text{day}$, making the system useless for navigation.

Example 2. Hafele-Keating-style experiment from Sydney. Two atomic clocks, one flown around the world eastward on a Sydney-London-Sydney commercial route at $v \approx 240 \text{ m/s}$ for $\sim 24 \text{ h}$ ($86{,}400 \text{ s}$), the other left at Sydney Observatory. SR slowdown: $(v^2/2c^2) \times t = (240^2 / 2 \times 9 \times 10^{16}) \times 86{,}400 = 2.8 \times 10^{-8} \text{ s} = 28 \text{ ns}$. (GR speedup at $11 \text{ km}$ altitude adds $\sim +50 \text{ ns}$.) These nanosecond-level effects, predicted by SR and confirmed first by Hafele and Keating in 1971, demonstrate that moving clocks really do run slow. Caesium atomic clocks resolve the $\sim 30 \text{ ns}$ difference easily.

Try this

Q1. State one experimental observation that confirms time dilation and one that confirms length contraction. [2 marks]

• Cue. Time: cosmic muon flux at sea level (or GPS); length: atmosphere appears shorter in muon's frame (cosmic ray data).

Q2. 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. [3 marks]

• Cue. (a) $\gamma = 1/\sqrt{1 - 0.9801} = 7.09$. (b) $L = L_0/\gamma = 12/7.09 = 1.69 \text{ km}$.

Q3. 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 $c$. (c) Explain how the storage ring's existence at all is evidence for relativistic mass-energy. [2+3+2 marks]

• Cue. (a) $\gamma = E/m c^2 = 3000/0.511 = 5870$. (b) $\beta = \sqrt{1 - 1/\gamma^2} = 1 - 1.45 \times 10^{-8}$, so $v \approx 0.99999998 c$. (c) Without relativistic momentum $p = \gamma m v$, the electrons would have impossible Newtonian momentum.