Special Relativity: Lorentz Transformations, Time Dilation, Length Contraction, Relativistic Momentum and Energy
Relativity tasks in the Abitur are usually one-dimensional: a muon, a spaceship or a particle beam, where the key decision is which frame measures the proper time or proper length before γ is applied.
Postulates and the Lorentz factor
The laws of physics are the same in all inertial frames, and c is invariant. γ = 1/√(1 − v²/c²) = 1/√(1 − β²). For v = 0.6c, γ = 1.25; for v = 0.99c, γ ≈ 7.09. Relativity of simultaneity follows: events simultaneous in one frame need not be in another.
Time dilation and length contraction
Δt = γ·Δt₀, where Δt₀ is the proper time measured by a clock present at both events (same place in that frame). L = L₀/γ, where L₀ is the proper length measured in the object's rest frame. Moving clocks run slow; moving objects are shortened along the direction of motion only.
Momentum and energy
p = γ·m·v; E = γ·m·c²; rest energy E₀ = m·c²; E_kin = (γ − 1)·m·c²; E² = (p·c)² + (m·c²)². For E_kin ≪ E₀ the classical ½mv² is recovered as a limiting case.
Worked example
A muon with proper lifetime 2.2 μs travels at 0.995c. γ = 1/√(1 − 0.995²) = 1/√(0.009975) = 10.0. The lab-frame lifetime is 22 μs, so the mean distance is 0.995 · 3.0×10⁸ · 22×10⁻⁶ ≈ 6.6 km instead of 0.66 km - which is why muons produced in the upper atmosphere reach sea level. Check: the classical answer is too small by exactly the factor γ, and the same factor appears if you instead contract the atmosphere's thickness in the muon frame.
Traps
- Applying γ to the wrong frame - the muon's own lifetime is the proper time.
- Applying γ twice, for example to time and distance in the same step.
- Adding velocities as u + v instead of (u + v)/(1 + uv/c²).
- Using E = ½mv² for electrons above roughly 10 % of c.
- Treating the twin paradox as needing general relativity; it is the change of frame (acceleration) that breaks the symmetry.