Evidence for relativity and mass–energy consequences

Key ideas

• Special relativity makes testable predictions. The classic piece of evidence at this level is the muon decay problem.
  • Muons are unstable particles created high in the atmosphere (around $15$ km up) by cosmic rays, travelling toward Earth at close to $c$ (typically around $0.98c$).
  • A muon's average lifetime, measured at rest (its proper time), is only about $2.2\ \mu\text{s}$.
  • Classically, in $2.2\ \mu\text{s}$ even light travels only $2.2\times10^{-6} \times 3.00\times10^8 \approx 660$ m — nowhere near far enough for most muons to reach the ground before decaying.
  • Observation: large numbers of these muons ARE detected at sea level.
• Relativity resolves this two equivalent ways:
  • From Earth's frame: the muon's clock is moving, so time dilation means Earth observers measure the muon's lifetime as longer than $2.2\ \mu\text{s}$ (by a factor of $\gamma$) — long enough for the muon to cover the $15$ km.
  • From the muon's frame: the muon's own lifetime is still $2.2\ \mu\text{s}$ (its proper time), but the $15$ km distance to the ground is length-contracted (by the same factor of $\gamma$) to a much shorter distance — one the muon can cross in its own short lifetime.
  • Both explanations use the same $\gamma$ and give the same physical outcome: more muons survive to reach the ground than classical physics predicts.
• This agreement between two very different-looking explanations (one using time dilation, one using length contraction) is strong evidence that special relativity, not classical mechanics, correctly describes fast-moving particles.

Mass–energy consequences (qualitative)

• Special relativity also connects mass and energy. As an object's speed increases toward $c$, more and more of the energy put into accelerating it appears as an increase in its relativistic mass/energy, not as speed.
  • This is why no object with mass can reach or exceed $c$: as $v \to c$, $\gamma \to \infty$, so an infinite amount of energy would be required.
  • It means mass and energy are equivalent — mass is a form of stored energy, related by $E = mc^2$ (covered fully in the nuclear physics section of this standard).
• These consequences are why particle accelerators can push particles very close to $c$, but never all the way to it.

Tips

• State both explanations, and say they agree. A common Merit/Excellence question is "explain the muon observation from Earth's frame AND the muon's frame" — full marks need both frames explained, plus the point that they predict the same outcome.
• Do not say "the muon's own clock speeds up" or "the muon shrinks" — nothing changes in the muon's own frame. It is other observers who measure it differently. Keep straight whose measurement is being described.