The big idea: Mark where you are at each moment of a journey — home at 8 am, the bus stop at 8:10 — and join the dots: you have just drawn a space-time diagram (also called a Minkowski diagram), a graph of where meets when.
Physicists put ct (the speed of light × time) up the vertical axis and position x across the bottom. Using ct keeps both axes in metres, so a flash of light travels at a tidy 45°.
What is a world line?: The path an object traces on the diagram is its world line — a record of its position at every instant. A single point on the diagram is an event (a definite place at a definite time).
| Object | Its world line | Why |
|---|---|---|
| At rest (stationary) | Vertical line | x stays fixed while ct keeps climbing — time passes, position does not. |
| Moving at steady speed | Tilted straight line | x changes as ct climbs; the faster it goes the more it tilts toward the x-axis. |
| A ray of light | Line at 45° | Light covers x = ct, so the line rises one unit of ct for each unit of x. |
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Slope tells you the speed: On a (ct vs x) diagram a world line that is steep (close to vertical) is slow, and a line leaning toward the x-axis is fast.
Nothing material can reach 45° — that is the light line, the cosmic speed limit. Real objects always have world lines steeper than 45°.
Reading speed off the diagram: For a world line that rises Δ(ct) while moving Δx sideways, the object's speed is
v = Δx ÷ Δt = c × (Δx ÷ Δ(ct)).
A light ray has Δx = Δ(ct), giving v = c.
A world line on a space-time diagram rises 8 units of ct while moving 4 units across in x. Find the object's speed as a fraction of c, and compare it with a light ray.
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The quantity everyone agrees on: Different observers disagree about how much time (Δt) and how much space (Δx) separate two events — that is length contraction and time dilation. But there is one combination they all measure the same: the space-time interval Δs.
It is the relativistic version of distance, and it is invariant — the same in every inertial frame.
- space-time interval (m)
- speed of light, 3.00×10⁸ m s⁻¹
- time separation of the two events (s)
- space separation of the two events (m)
Two events are separated by Δt = 5.0 μs in time and Δx = 900 m in space. Using c = 3.00×10⁸ m s⁻¹, find the space-time interval Δs between them.
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Simultaneous is not absolute: Two events that happen at the same time in one frame need not be simultaneous in another. Whether two things happen 'at once' depends on who is looking — there is no universal 'now'.
On the diagram: For a stationary observer, events that are simultaneous lie on a horizontal line (the line of simultaneity, all at the same ct).
For a moving observer the line of simultaneity is tilted up toward the light line by the same angle their world line tilts. So a line that is flat for one observer is sloped for the other — they pick out a different set of 'now' events.
What observers AGREE on
- The space-time interval Δs between two events
- The order of cause-and-effect (timelike) events
- That a light ray travels at 45° (speed c)
What observers DISAGREE on
- How much time Δt separates two events
- How much space Δx separates them
- Whether two events are simultaneous
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How this is tested — space-time diagrams are HL only (A.5):
Paper 1A
- A one-step 'what does a vertical world line mean?' or 'what angle is a light ray?'
- 'Is the interval the same for all observers?' (yes).
Paper 2
- Calculate the invariant interval.
- Describe or interpret a sketched diagram (which line is fastest, why simultaneity tilts).
The classic trap: In (Δs)² = (cΔt)² − (Δx)² the space term is subtracted — don't add it. Light is always at 45° (ct = x), and the interval is the same in every frame, so compute it in the easiest one.
Three easy marks: (1) Light is always 45° because ct = x. (2) The interval is invariant — same in every frame, so compute it in the easiest frame. (3) Watch the sign: in (Δs)² = (cΔt)² − (Δx)² the space term is subtracted.
On a space-time diagram (ct up, x across), three world lines are drawn: line P is vertical, line Q is at 45°, and line R leans at 70° to the x-axis. Identify what each represents and order them from slowest to fastest.
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