Special relativity: relativity principle, time dilation, length contraction, relativistic energy and momentum
What this domain tests
Special relativity covers the two postulates, time dilation, length contraction, relativistic momentum and energy, and mass–energy equivalence. Tasks usually give a concrete scenario - a muon beam, a fast spacecraft, a particle accelerator - and ask for a comparison with the classical result.
Postulates and the Lorentz factor
- The laws of physics are the same in all inertial frames; the speed of light in vacuum is the same for every observer.
- Consequences: simultaneity is relative, moving clocks run slow, moving lengths contract.
- γ = 1/√(1 − v²/c²); at v = 0.6c, γ = 1.25; at 0.99c, γ ≈ 7.1.
Time and length
- Time dilation: Δt = γ·Δt₀, where Δt₀ is the proper time measured by a clock present at both events.
- Length contraction: L = L₀/γ, where L₀ is the proper length measured in the object's rest frame.
- Only the component along the relative velocity is contracted.
Momentum and energy
- p = γ·m·v; E = γ·m·c²; rest energy E₀ = m·c².
- Kinetic energy E_kin = (γ − 1)·m·c², which reduces to ½mv² for v ≪ c.
- E² = (p·c)² + (m·c²)²; for photons m = 0, so E = p·c.
Worked example
A muon with proper lifetime 2.2 µs travels at v = 0.98c. Then γ = 1/√(1 − 0.9604) = 1/√0.0396 ≈ 5.03. In the laboratory its lifetime is γ · 2.2 µs ≈ 11 µs, so the mean decay length is 0.98 · 3.0·10⁸ m/s · 11·10⁻⁶ s ≈ 3.3 km. Classically only about 0.65 km would result, which is why atmospheric muons reach the ground. In the muon's own frame the distance contracts to about 0.65 km instead - both descriptions agree on the observation.
Traps
- Applying γ in the wrong direction, or treating the dilated time as the proper time.
- Using ½mv² for electrons or protons at high speed.
- Mixing SI units with MeV; the electron rest energy is 0.511 MeV.
- Presenting the twin paradox as a contradiction - the travelling twin is not inertial throughout.