Physics HL · Chapter 6: Relativity
6.4 Spacetime Diagrams and Causality
Interpret worldlines, light cones, and slanted primed axes to reason about cause, simultaneity, and measured intervals.
Estimated time: 24 minutes
Worldlines and the ct Axis
Spacetime diagrams plot position on the horizontal axis and ct on the vertical axis. A particle at rest traces a vertical worldline. Uniform motion appears as a straight tilted worldline. Because photons travel at c, light rays appear as 45 degree lines when axes are scaled uniformly in x and ct.
A key geometric test follows: no material worldline can tilt beyond the light line. That visual rule encodes the speed limit directly and helps you reject impossible diagram interpretations quickly.
Primed Axes and Non-Euclidean Intuition
When a second inertial frame moves relative to the first, its x-prime and ct-prime axes appear slanted on the same diagram. The chapter stresses that scales on these axes are not read with ordinary Euclidean assumptions. You must project parallel to the relevant axes of that frame.
This is where diagram fluency pays off. Time dilation, length contraction, and simultaneity changes can be read graphically without repeating full algebra every time. The diagram and the Lorentz equations are two views of one structure, not competing methods.
Reading Simultaneity on Minkowski Grids
On a spacetime diagram, simultaneous events in frame S lie on lines parallel to the x-axis. Simultaneous events in S' lie on lines parallel to x'. Because x' is tilted, the same pair of events can be horizontal in one frame and slanted in the other. This visual rule is the geometric form of relativity of simultaneity.
A useful exam habit is to separate three line-families: worldlines (motion histories), S-simultaneity slices (parallel to x), and S'-simultaneity slices (parallel to x'). If you label these explicitly, ordering questions become straightforward and most sign mistakes disappear.
Simulation: High-Beta Interval Stress Test
Push beta close to 1 to connect diagram intuition with large gamma and dramatic disagreement between frame-time assignments.
Relativity Clock + Spacetime Lab
γ
2.9311
v
0.940 c
v (m/s)
2.818e+8
t from τ
17.59 s
Minkowski diagram (x, ct)
This diagram is coordinate-based: the same event pair gets different simultaneity slices in S and S′, while light-cone boundaries preserve causal structure.
Interpretation guidance: at beta = 0.94, gamma is about 2.93. Use the simulation bars to compare one local ship interval against the mapped lab interval, then imagine those as separations along ct' and ct projections on a Minkowski diagram. The same event pair is being projected onto different axis sets; that is the spacetime meaning of 'time dilation' in coordinate language.
Light Cones and Causal Ordering
Events inside a future light cone can be influenced by the cone's origin event through subluminal or light-speed signals. Events outside that cone cannot be causally linked to the origin without superluminal transfer. This gives a clean physical test for whether one event can cause another.
Because causal order is constrained by light cones, relativity of simultaneity does not imply causal contradiction. Different frames can disagree on timing of spacelike-separated events while still agreeing on all cause-and-effect chains for timelike-connected events.