Lorentz transformations (Physics HL)

The big idea: To keep the speed of light c the same for everyone, time and length stretch and shrink with motion. Everything below is built from one number: the Lorentz factor γ (gamma).

Clocks on airliners: In 1971 atomic clocks were flown round the world on airliners and compared with identical clocks left on the ground. They came back out of step by tens to hundreds of nanoseconds: part of the difference is gravity, and part is time dilation, because moving clocks run slow.

Lorentz transformations: The Lorentz transformations are the rules that link what one observer measures to what a moving observer measures.

Reading the Lorentz equations: At low speed γ ≈ 1 and vx/c² ≈ 0, so they become the Galilean x′ = x − vt and t′ = t.

The vx/c² term is new: two events at different places that are simultaneous in S (Δt = 0) are not simultaneous in S′.

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.

Where Δt = γΔt₀ comes from: Put a clock at rest in S′, so its two ticks are at the same place there: Δx′ = 0, and the time between them is the proper time Δt₀ = Δt′.

The Lorentz equations run the other way too (swap the frames, v becomes −v): Δt = γ(Δt′ + vΔx′/c²) = γΔt₀.

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.

The evidence: muons

One experiment, two explanations: The Earth observer explains the muons with time dilation (their clocks run slow); the muon explains them with length contraction (the atmosphere is thinner). Both use the same γ and agree on how many muons reach the ground.

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.

Always below c: 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).