The big idea: A space-time diagram (also called a Minkowski diagram) is a graph of where and when. We put ct (the speed of light × time) up the vertical axis and position x along the horizontal axis.
Using ct instead of plain t means both axes are 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. |
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.
Worked example — reading a world line
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.
Solution
- Read the speed straight off the diagram:
- Put in the numbers read off the line:
- Work it out — keep it as a fraction of c:
Final answer
v = 0.5c. A light ray would move 8 units in x for 8 units of ct (a 45° line, v = c); this world line is steeper, so it is slower than light.
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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)
Worked example — calculating the interval
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.
Solution
- Start from the given interval formula:
- Work out the time term (cΔt)²:
- Work out the space term (Δx)²:
- Subtract, then take the square root — keep the unit:
Final answer
Δs = 1200 m — and every inertial observer gets exactly this value.
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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Where it shows up: Space-time diagrams are HL only (A.5):
- Paper 1A — a one-step 'what does a vertical world line mean?', 'what angle is a light ray?', or 'is the interval the same for all observers?'. - Paper 2 — calculate the invariant interval, or describe / interpret a sketched space-time diagram (which line is fastest, why simultaneity tilts).
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.
IB-style question — identifying world lines
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.
Solution
- Vertical means x never changes — line P is an object at rest (v = 0).
- A line at 45° has ct = x, so it is a ray of light (the fastest possible).
- Line R leans at 70° (between vertical and 45°), so it is a real object moving slower than light:
Final answer
P = at rest, R = a moving object, Q = light. Slowest → fastest: P, R, Q.