Chapter 4: Time Dilation
"Moving clocks run slower." This seemingly impossible statement is one of the most verified predictions in all of physics. Time itself flows differently depending on how fast you're moving—a direct consequence of the constancy of the speed of light.
The Light Clock Thought Experiment
To derive time dilation, Einstein imagined a "light clock"—a device that measures time by bouncing a pulse of light between two mirrors.
Setup
- • Two mirrors separated by distance L (in the clock's rest frame)
- • A light pulse bounces back and forth between them
- • One "tick" = time for light to travel from bottom mirror to top and back
- • In the rest frame: \( \Delta \tau = \frac{2L}{c} \) (proper time)
Clock at Rest (Observer A)
Light travels straight up and down. Distance = 2L.
Time for one tick: \( \Delta \tau = \frac{2L}{c} \)
Clock Moving (Observer B)
Light travels diagonally—a longer path! Distance = \( 2\sqrt{L^2 + (v\Delta t/2)^2} \)
But light still travels at c for observer B!
Deriving the Time Dilation Formula
Let Δt be the time for one tick as measured by observer B (who sees the clock moving at velocity v).
Step-by-Step Derivation
Step 1: Path length for moving clock
In time Δt/2, the clock moves horizontally by \( v \cdot \frac{\Delta t}{2} \)
Light travels diagonally: \( d = \sqrt{L^2 + \left(\frac{v\Delta t}{2}\right)^2} \)
Step 2: Light travels at c
For round trip: \( c \Delta t = 2\sqrt{L^2 + \left(\frac{v\Delta t}{2}\right)^2} \)
Step 3: Square both sides
\( c^2 \Delta t^2 = 4L^2 + v^2 \Delta t^2 \)
Step 4: Solve for Δt
\( \Delta t^2 (c^2 - v^2) = 4L^2 \)
\( \Delta t = \frac{2L}{\sqrt{c^2 - v^2}} = \frac{2L}{c\sqrt{1 - v^2/c^2}} \)
Step 5: Express in terms of proper time
Since \( \Delta \tau = 2L/c \), we get:
\( \Delta t = \gamma \Delta \tau \)
where \( \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} \)
The Lorentz Factor γ
\( \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} = \frac{1}{\sqrt{1 - \beta^2}} \)
where β = v/c is the velocity as a fraction of the speed of light
Values at Different Speeds
| Speed (v/c) | γ (Lorentz factor) | Time dilation effect |
|---|---|---|
| 0 (at rest) | 1.000 | No effect |
| 0.1c (30,000 km/s) | 1.005 | 0.5% slower |
| 0.5c (150,000 km/s) | 1.155 | 15.5% slower |
| 0.9c | 2.294 | 2.3× slower |
| 0.99c | 7.089 | 7× slower |
| 0.999c | 22.37 | 22× slower |
| 0.99999c | 223.6 | 224× slower |
At everyday speeds, γ ≈ 1 and time dilation is negligible. At speeds approaching c, the effect becomes extreme. A muon traveling at 0.9999c experiences time 70× slower than us!
Proper Time vs. Coordinate Time
Proper Time (τ)
- • Time measured by a clock traveling with the object
- • The "personal" time of the moving object
- • Always the shortest time between two events on a worldline
- • Invariant—all observers agree on proper time
Coordinate Time (t)
- • Time measured in a particular reference frame
- • Uses synchronized clocks spread throughout space
- • Always longer than proper time (γ ≥ 1)
- • Different observers measure different coordinate times
Key insight: The proper time is what a clock actually measures as it moves along its worldline. When we say "moving clocks run slower," we mean that coordinate time Δt in the lab frame is longer than proper time Δτ measured by the moving clock.
Experimental Verification
Time dilation isn't just theoretical—it's been measured with incredible precision:
1. Cosmic Ray Muons (1941)
Muons created in the upper atmosphere have a half-life of 1.5 microseconds. At near-light speeds, they should decay before reaching Earth. But we detect them! Time dilation extends their lifetime in our frame, allowing them to reach the surface. This was one of the first confirmations of special relativity.
2. Hafele-Keating Experiment (1971)
Cesium atomic clocks were flown around the world on commercial airliners. Compared to ground clocks, the flying clocks showed exactly the time difference predicted by special (and general) relativity: about 59 nanoseconds for the eastward flight. The experiment confirmed both time dilation from velocity and gravitational time dilation from altitude.
3. Particle Accelerators (ongoing)
Unstable particles at CERN travel at 0.99999c with γ ≈ 7000. Their measured lifetimes are thousands of times longer than at rest, exactly as predicted by time dilation.
4. GPS Satellites (everyday application)
GPS satellites orbit at 14,000 km/h. Their clocks run about 7 microseconds/day slower due to their velocity (special relativity) but 45 microseconds/day faster due to weaker gravity (general relativity). Without relativistic corrections, GPS would accumulate errors of ~10 km per day!
Preview: The Twin Paradox
Time dilation leads to a famous puzzle: If twin A travels to a star at high speed while twin B stays on Earth, A will age less than B. But from A's perspective, isn't B the one moving? Shouldn't B age less?
The Resolution
The situation is not symmetric! Twin A must accelerate to leave, decelerate at the star, and accelerate to return. A experiences non-inertial motion, breaking the symmetry. Twin A is definitively younger upon return—this is not a paradox but a real physical effect.
We'll explore this in detail in Part II: Classic Paradoxes.