Lorentz transformations (Physics HL)

The big idea: Fly around the world in a fast jet and your watch ends up a fraction of a microsecond behind the clocks that stayed home — at high speed, moving clocks really do run slow. To keep the speed of light c the same for everyone, time and length stretch and shrink with motion; the Lorentz transformations are the rules that link what one observer measures to what a moving observer measures.

Everything below is built from one number: the Lorentz factor γ (gamma).

Two pieces of jargon: Proper time Δt₀ — the time between two events measured by a clock that is present at both events (it is the shortest possible time).

Proper length L₀ — the length of an object measured in the frame where it is at rest (it is the longest possible length).

Every relativistic effect is scaled by the Lorentz factor γ. It depends only on the speed v as a fraction of c. At everyday speeds γ ≈ 1 (relativity hides itself); as v approaches c, γ shoots up toward infinity.

Common slip: Keep speeds as a fraction of c. Writing v = 0.80c means v/c = 0.80, so the term is 0.80², not (0.80 × 3 × 10⁸)². Working in units of c keeps the numbers clean.

Moving clocks tick slowly: A clock that is moving relative to you ticks slower than your own. The proper time Δt₀ (measured on the moving clock) is stretched by γ to give the longer time Δt that you measure.

Lengths squeeze along the motion: An object that moves past you is measured to be shorter along its direction of motion than its proper length. Only the dimension along the motion contracts — width and height are unchanged. The proper length L₀ is divided by γ.

You cannot just add speeds: At low speeds you simply add velocities (0.5 + 0.5 = 1.0). Near c that breaks the cosmic speed limit, so relativity uses a velocity-addition rule. The denominator keeps every result below c — you can never reach the speed of light by adding speeds.

How this is tested — Lorentz transformations are HL only (A.5):

The classic trap: Always find γ first, then decide: multiply by γ for time (Δt = γΔt₀), divide by γ for length (L = L₀/γ). The proper quantity is the one measured in the rest frame.

Three easy marks: (1) Always find γ first. (2) Identify the proper quantity: proper time is on the moving clock, proper length is in the rest frame. (3) Multiply by γ for time (Δt = γΔt₀); divide by γ for length (L = L₀/γ).