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).
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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.
- Lorentz factor (no unit, ≥ 1)
- speed of the moving frame (m s⁻¹)
- speed of light in a vacuum (3.0 × 10⁸ m s⁻¹)
A spaceship moves past Earth at v = 0.80c. Find its Lorentz factor γ.
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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.
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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.
- time measured by the observer who sees the clock moving (s)
- proper time — measured on the moving clock itself (s)
- Lorentz factor (no unit)
The spaceship above (v = 0.80c, γ = 1.67) carries a clock that measures a proper time Δt₀ = 2.0 s for an event on board. What time does an Earth observer measure for that same event?
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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.
- length measured by the observer who sees it moving (m)
- proper length — measured in the object's rest frame (m)
- Lorentz factor (no unit)
The spaceship (v = 0.80c, γ = 1.67) has a proper length L₀ = 100 m. What length does an Earth observer measure as it flies past?
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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.
- velocity of the object in the second frame (m s⁻¹)
- velocity of the object in the first frame (m s⁻¹)
- velocity of the second frame relative to the first (m s⁻¹)
- speed of light in a vacuum
As seen from Earth, two spacecraft head straight toward each other, each moving at 0.50c. How fast does one ship measure the other to approach?
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How this is tested — Lorentz transformations are HL only (A.5):
Paper 1A
- A one-step 'find γ' or 'which is the proper time?'
- 'Is the result above or below c?'
Paper 2
- Determine a dilated time, a contracted length, or a relative speed.
- Often set for a muon or a fast spacecraft.
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).
A probe travels past a space station at v = 0.60c. A signal lamp on the probe flashes with a proper period of 5.0 s. Determine (a) the Lorentz factor and (b) the period of the flashes as measured by the station.
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