Experimental Evidence for Relativity (HSC SSCE Physics): Revision Notes

Experimental Evidence for Relativity

The theory of special relativity makes predictions that seem counterintuitive, but a range of experiments and astronomical observations provide strong supporting evidence. This note explores three main sources of experimental evidence: cosmological observations, muon decay experiments, and atomic clock experiments.

Cosmological observations

Astronomical observations provide some of the most compelling evidence for Einstein's second postulate - that the speed of light is constant and independent of the source's motion.

Supernova evidence

When massive stars explode (called a supernova), fragments can fly outward at speeds exceeding 3% of the speed of light. If the speed of light depended on the speed of its source, light from fragments moving away from Earth should travel more slowly than light from fragments moving toward us. This would create observable effects in the light patterns astronomers detect.

No faster-than-light particles

Throughout decades of particle physics experiments and cosmic ray observations, no particle has ever been reliably observed traveling faster than light. This universal speed limit is a fundamental prediction of special relativity.

Muon decay experiments

Muons are subatomic particles that provide excellent evidence for relativistic effects because they travel at speeds close to the speed of light and have measurable decay rates.

What are muons?

Muons are particles with a mass approximately 200 times greater than an electron. They occur naturally when cosmic rays bombard Earth's upper atmosphere. Muons have a mean lifetime of $2.2 \times 10^{-6}$ s (or $2.2$ μs) when measured in the laboratory. The mean lifetime relates to half-life by: $t_{\text{mean}} = 1.44t_{1/2}$.

The Frisch and Smith experiment (1963)

In 1963, physicists DH Frisch and JH Smith conducted a landmark experiment measuring muons at two different altitudes. They collected muons on a mountaintop at 1907 m above sea level and compared the count with measurements at sea level. The muons traveled at $0.995c$ relative to Earth.

Prediction without relativity

Without considering relativistic effects, we can calculate how far muons should travel before decaying:

From an observer traveling with the muon (in the muon's rest frame), the time to reach sea level would be:

$t = \frac{H}{v} = \frac{1907 \text{ m}}{0.995 \times 2.998 \times 10^8 \text{ m s}^{-1}} = 6.39 \text{ μs}$

This represents $\frac{6.39 \text{ μs}}{2.2 \text{ μs}} = 2.9$ mean lifetimes.

The number of muons decays exponentially, so the fraction surviving would be:

$\frac{N_{\text{SL}}}{N_{\text{MT}}} = e^{-t/t_{\text{mean}}} = e^{-2.9} = 0.055$

Prediction with relativity (Earth's frame)

However, the detectors measure time in Earth's rest frame, while the muon decays in its own rest frame. Time dilation affects what we observe.

The mean lifetime as measured in Earth's frame is:

$t_{\text{mean}}^{E} = \gamma t_{\text{mean}}^{\mu} = \frac{t_{\text{mean}}^{\mu}}{\sqrt{1-\frac{v^2}{c^2}}}$

At $v = 0.995c$, the gamma factor is:

$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} = 10$

Therefore: $t_{\text{mean}}^{E} = 10 \times 2.2 \text{ μs} = 22 \text{ μs}$

The trip now takes only:

$\frac{t}{t_{\text{mean}}^{E}} = \frac{6.39 \text{ μs}}{22 \text{ μs}} = 0.29 \text{ mean lifetimes}$

The fraction reaching sea level becomes:

$\frac{N_{\text{SL}}}{N_{\text{MT}}} = e^{-0.29} = 0.748$

This predicts about 74.8% should reach sea level - a dramatically different result!

Experimental results

Frisch and Smith measured an average of 563 muons per hour at the mountaintop detector. At the sea level detector:

• Classical prediction: $563 \times 0.055 \approx 30$ muons per hour
• Relativistic prediction: $563 \times 0.748 \approx 420$ muons per hour
• Actual measurement: 412 muons per hour

The muon's perspective (length contraction)

From the perspective of an observer traveling with the muon, Earth appears to be moving at $0.995c$. In this frame, the distance between detectors appears shortened by a factor of $\gamma$ due to length contraction:

$l = \frac{l_0}{\gamma} = l_0\sqrt{1-\frac{v^2}{c^2}}$

Agreement and disagreement between frames

Both observers (one on Earth, one traveling with the muon) will agree about:

• The relative speed: $0.995c$
• The number of physical muon decays
• The number of elapsed mean lifetimes between detectors

They will disagree about:

• The mean lifetime of the muon
• The distance between detectors

Atomic clock experiments

Atomic clocks are extraordinarily precise timepieces that can measure time accurately to within one second per hundred million years. This precision allows them to detect time dilation even at speeds much smaller than the speed of light.

The Hafele-Keating experiment (1971)

In October 1971, physicists Joseph Hafele and Richard Keating sent atomic clocks on commercial passenger jets flying around the world. Some clocks flew eastward, others westward. After the flights, these clocks were compared with a clock that remained stationary on the ground.

The flying clocks showed time differences consistent with the predictions of special relativity (and general relativity, which also affects the results due to gravity).

Calculating time dilation at low speeds

For a moving clock, time dilation gives:

$t = \gamma t_0 = \frac{t_0}{\sqrt{1-\frac{v^2}{c^2}}}$

However, when the speed of an aircraft is much smaller than $c$ (specifically when $v \< 0.2c$), we can use an approximation:

$\gamma \approx 1 + \frac{1}{2}\left(\frac{v}{c}\right)^2$

This leads to:

$t \approx t_0\left(1 + \frac{1}{2}\left(\frac{v}{c}\right)^2\right)$

The time difference between the moving and stationary clocks is:

$t - t_0 \approx \frac{t_0}{2}\left(\frac{v}{c}\right)^2$

Worked example: time dilation on commercial aircraft