Chapter 5: Invariant Spacetime Interval
While observers disagree on time intervals and spatial distances, they all agree on the spacetime interval. This invariant quantity is the foundation of relativistic geometry, playing the same role as distance in Euclidean geometry.
The Spacetime Interval
\( \Delta s^2 = c^2\Delta t^2 - \Delta x^2 - \Delta y^2 - \Delta z^2 \)
All inertial observers calculate the same value for Δs²!
This is analogous to the Pythagorean theorem \( d^2 = x^2 + y^2 \) being invariant under rotations. The minus signs make spacetime geometry "hyperbolic" rather than Euclidean.
Classification of Intervals
Timelike (Δs² > 0)
Time separation dominates. Events can be connected by a massive particle. All observers agree on time ordering. \( \Delta s = c\Delta\tau \) (proper time).
Lightlike (Δs² = 0)
Events connected by light. Null separation. Photon worldlines have zero interval.
Spacelike (Δs² < 0)
Space separation dominates. No causal connection. Order of events is frame-dependent.\( \Delta s^2 = -\Delta\sigma^2 \) (proper distance).
Physical Interpretation
Timelike: Proper Time
For timelike intervals, \( \Delta\tau = \Delta s/c \) is the proper time—the time measured by a clock traveling between the events. This is the "wristwatch time" of a traveler.
Spacelike: Proper Distance
For spacelike intervals, \( \Delta\sigma = \sqrt{-\Delta s^2} \) is the proper distance—the spatial separation measured by an observer for whom the events are simultaneous.