Key Idea: Relativity is about how the same events look to observers moving differently. At everyday speeds, Galilean relativity (just add/subtract velocities) works fine. But near the speed of light, Einstein's special relativity takes over: moving clocks run slow, moving lengths contract, and nothing can reach c.
Galilean
- x′ = x − vt and t′ = t
- u′ = u − v: fine at everyday speeds
- Newton's laws hold in every inertial frame
Einstein
- Same laws and same c for every inertial observer
- Lorentz: x′ = γ(x − vt), t′ = γ(t − vx/c²)
- Δt = γΔt₀, L = L₀/γ; muons are the evidence
Space-time
- ct up, x across; light at 45°
- tan θ = v/c; x′ and ct′ tilt towards the light line
- (Δs)² = (cΔt)² − (Δx)² is invariant
It is HL only (A.5).
Einstein's two postulates
1. The laws of physics are the same in all inertial frames (no experiment finds an absolute rest frame). 2. The speed of light in a vacuum is the same, c, for every inertial observer — no matter how the source or observer moves. Postulate 2 is the strange one: it forces time and space themselves to stretch and shrink so everyone still measures light at c.
The formulas you're given
- Lorentz factor (always ≥ 1)
- proper time — measured by one clock at both events (the shortest)
- proper length — measured in the object's rest frame (the longest)
- relativistic velocity addition (always stays below c)
- the invariant space-time interval — the same for all observers
- Galilean velocity addition: everyday speeds only
- position and time of an event in the moving frame
- angle between a world line and the ct axis
IB-style worked examples (one per micro)
Two trains travel toward each other along a straight track at 25 m s⁻¹ and 30 m s⁻¹ relative to the ground. Determine the speed of one as measured from the other.
Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
Next:
A spaceship moving at 0.50c shines a torch forward. State the speed of the light measured by (a) an astronaut on the ship and (b) an observer on a nearby planet.
Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
Next:
A clock on a spaceship moving at 0.80c records a proper time of 2.0 s for an event. Determine the time an Earth observer measures.
Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
Next:
Two events are separated by Δt = 5.0 μs and Δx = 900 m. Determine the space-time interval Δs. (c = 3.00×10⁸ m s⁻¹.)
Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
Important: 1. Proper time is the SHORTEST (one clock at both events) and proper length is the LONGEST (rest frame). Identify which is which before substituting. 2. Time dilation multiplies by γ (Δt = γΔt₀); length contraction divides by γ (L = L₀/γ). Don't swap them. 3. γ is always ≥ 1. If you get γ < 1, you've made a sign error under the root. 4. Never use simple Galilean addition near c — use the relativistic formula; the answer always stays below c.
Quick self-check
Tap each card to reveal the answer.
What stays the same for every observer? The speed of light, c (postulate 2) — and the space-time interval Δs.
Which clock reads the proper time? The single clock present at both events — it reads the shortest (proper) time.
Lorentz factor at v = 0.866c? γ = 2.0 — the classic 'γ = 2' speed worth memorising.
A 100 m rocket flies past at 0.80c (γ = 1.67). Its measured length? 60 m — L = L₀/γ = 100/1.67.
Two ships approach, each at 0.50c. Their relative speed? 0.80c — u' = (0.5c+0.5c)/(1+0.25) = 0.80c, never 1.0c.
On a space-time diagram, what angle is a light ray? 45° — because ct = x for light; a stationary object is a vertical world line.
Exam tips
Exam tips
- Compute γ first — everything else (time, length) hangs off it.
- Label the proper time / proper length before you substitute; it's the most common slip.
- γ = 2.0 at v = 0.866c is a handy memory check.
- Relativistic answers for speed always stay below c — if you get ≥ c, recheck.
- All inertial observers agree on the space-time interval Δs, even though they disagree on Δt and Δx.