Chapter 3: Spacetime Diagrams
Spacetime diagrams (Minkowski diagrams) are powerful tools for visualizing special relativity. They let us see time dilation, length contraction, and the relativity of simultaneity as geometric relationships, making the abstract concrete.
Basic Setup
Axes
- • Vertical axis: ct (time × c, so both axes have length units)
- • Horizontal axis: x (space)
- • Each point = an event in spacetime
Light Lines
- • Light travels at 45° (slope = 1 since x = ct)
- • All observers agree light lines are at 45°
- • Light cone = future and past light rays from origin
Worldlines
- • Stationary object: vertical line (moving through time only)
- • Moving object: tilted line (slope < 1 since v < c)
- • Must always be inside the light cone (no FTL travel)
A Moving Observer's Axes
For an observer moving at velocity v, the ct' and x' axes are tiltedtoward the light line.
Key Features
- • The ct' axis is the worldline of the moving observer
- • The x' axis (simultaneity surface) is tilted the same angle from horizontal
- • Both axes tilt toward the light line as v → c
- • The axes meet the light line at v = c (impossible)
The angle θ from the vertical satisfies tan(θ) = v/c. Lines of constant t' are tilted; events that are simultaneous in S' are NOT simultaneous in S!
Visualizing Relativistic Effects
Time Dilation
Equal proper time intervals on the moving clock correspond to longer coordinate time intervals in the rest frame. The moving clock's tick marks are spread out along the time axis.
Length Contraction
The ends of a moving rod trace out two worldlines. "Length" means the spatial separation at the same time—but "same time" differs between frames! The rod appears shorter in the frame where it moves.
Relativity of Simultaneity
Lines of constant t are horizontal; lines of constant t' are tilted. Events on a horizontal line are simultaneous in S but NOT in S'—and vice versa.
Light Cones and Causality
Future Light Cone
Upper 45° cone. Events here CAN be influenced by the origin. Timelike separated.
Past Light Cone
Lower 45° cone. Events here COULD HAVE influenced the origin. Timelike separated.
Elsewhere
Outside both cones. No causal connection possible. Spacelike separated.