Special Relativity
Special Relativity is about 6% of the GRE Physics Subject Test. It rewards a small set of formulas applied carefully: the two postulates, Lorentz transformations, time dilation, length contraction, relativistic energy and momentum, and the invariant interval.
Core rules
- The speed of light is the same in every inertial frame, and the laws of physics are invariant under Lorentz transformations.
- γ = 1/√(1 − v²/c²). A moving clock runs slow by Δt = γΔτ, and a moving object is shortened along its motion by L = L₀/γ.
- Lorentz transform: x′ = γ(x − vt) and t′ = γ(t − vx/c²). Simultaneity is frame-dependent whenever the events are separated along the boost direction.
- Energy and momentum: E = γmc², p = γmv, and E² = (pc)² + (mc²)². Kinetic energy is K = (γ − 1)mc².
- The interval s² = (cΔt)² − Δx² is the same in all frames and classifies separations as timelike, lightlike, or spacelike.
- Velocity addition: u′ = (u − v)/(1 − uv/c²), which never produces a speed above c.
- Relativistic Doppler: for a receding source, λ_obs = λ_src√((1 + β)/(1 − β)).
Worked example
A proton is accelerated to a kinetic energy equal to twice its rest energy. Then K = 2mc², so the total energy is E = 3mc² and γ = 3. From γ = 3, 1 − β² = 1/9, giving β = √(8/9) = 0.943. The momentum is p = γmv = γβmc = 3 × 0.943 × 938 MeV/c ≈ 2650 MeV/c, and the total energy is 3 × 938 ≈ 2814 MeV.
Traps
Treating a massive particle as if it could reach c; applying time dilation in the wrong frame - the proper time belongs to the clock present at both events; letting length contraction act perpendicular to the motion; dropping the rest-energy term in E² = (pc)² + (mc²)²; assuming simultaneity is absolute; confusing the relativistic Doppler factor with γ.