Physics HL · Chapter 6: Relativity
6.2 Time Dilation and Length Contraction
Define proper intervals carefully and use gamma to connect moving and rest-frame measurements.
Estimated time: 30 minutes
Proper Time as a Local Clock Reading
Proper time is the interval between two events that occur at the same place in one frame. In that frame, a single clock can record both events. This local measurement is not 'more true'; it is simply the interval with a special geometric role in Lorentz transformations.
For an observer who sees the clock moving, the interval is dilated by gamma.
The source chapter uses clock-tick scenarios and decay lifetimes to reinforce frame logic. The frame where the process happens at one location measures proper time. Any frame in which that process spans different positions measures a longer interval by factor gamma.
Proper Length and Contraction Along Motion
L0 is proper length measured in the object's rest frame. Contraction applies only parallel to motion.
Length contraction often causes confusion because it requires simultaneous endpoint measurements in the measuring frame. If the rod moves in your frame, you must record both ends at the same time in your frame. Changing simultaneity conditions changes the measured spatial separation.
Evidence: Muons and Precision Timing
The chapter's muon example ties both effects together. In the lab frame, fast muons survive long enough to travel farther than naive non-relativistic estimates allow. In the muon's frame, lifetime stays proper while atmospheric distance contracts. Different stories, same physical outcome, and both stories are linked by Lorentz rules.
A second practical theme is timing systems such as satellite-based navigation. Even with moderate orbital speeds, tiny relativistic timing offsets accumulate into large position errors if ignored. Relativity here is not decorative theory; it is operational calibration.
For timelike-separated events, proper time comes from the invariant interval, so a moving path between the same events gives a smaller local elapsed time.
Simulation: Relativity Clock Lab
Vary beta = v/c and compare proper-time and coordinate-time intervals through the gamma factor.
Relativity Clock + Spacetime Lab
γ
1.7471
v
0.820 c
v (m/s)
2.458e+8
t from τ
20.97 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.
Test Yourself
A ship moves at v = 0.80c. If 15.0 s pass on the ship clock (proper time), enter the lab-frame interval in seconds.
Hint: Use gamma = 1/sqrt(1-0.80^2).