Chapter 6: Proper Time and Worldlines
The proper time is the most fundamental quantity in special relativity—it's what clocks actually measure. Understanding proper time and worldlines provides deep insight into the geometry of spacetime and connects to the variational principles of physics.
Proper Time τ
\( d\tau = \sqrt{1 - v^2/c^2} \, dt = \frac{dt}{\gamma} \)
Time measured by a clock traveling with the particle
The total proper time along a worldline from A to B:
\( \tau = \int_A^B d\tau = \int_A^B \sqrt{1 - v(t)^2/c^2} \, dt \)
Worldlines
A worldline is the path a particle traces through spacetime—the complete history of its position at every instant.
Types of Worldlines
- • Timelike: Massive particles (v < c)
- • Null/Lightlike: Photons (v = c)
- • Geodesic: Free particles (no forces)
Proper Time as "Arc Length"
Just as arc length measures distance along a curve in space, proper time measures "length" along a worldline in spacetime.
Principle of Maximal Aging
Free particles (no forces) follow worldlines that maximize proper timebetween two events.
This is the relativistic analog of "straight lines are the shortest distance." In spacetime:
- • Straight worldlines have MAXIMUM proper time (unlike Euclidean geometry)
- • This explains the twin paradox: the stay-at-home twin ages more!
- • Geodesics in GR are paths that extremize proper time
Connection to Lagrangian Mechanics
The action for a free relativistic particle is proportional to proper time:
\( S = -mc^2 \int d\tau \)
Particles follow paths that extremize this action
This connects special relativity to the principle of least action, the foundation of all modern physics. The negative sign means maximizing proper time minimizes the action.