Special Relativity: Time Dilation
Time Dilation
When something moves close to the speed of light, time passes slower for it compared to someone watching from outside. It isn't a trick of the clock — time itself really stretches out, and moving objects shrink along their direction of motion by that same exact factor.
The formula
Δt = γ · Δt₀ , γ = 1 / √(1 − v²/c²)
- Δt — dilated time (s): the time interval measured by an observer watching the clock move past
- Δt₀ — proper time (s): the time interval measured by someone moving along with the clock
- v — relative speed (m/s): how fast the clock moves relative to the observer
- c — speed of light (m/s): the universal speed limit, about 3 × 10⁸ meters per second
- γ — Lorentz factor (none): how much time stretches and length shrinks at that speed
Worked example
A spaceship zooms past Earth at 0.8 times the speed of light. An astronaut's heartbeat takes 3 seconds by her own clock. How long does that heartbeat take according to an observer on Earth?
- v = 0.8 c
- Δt₀ = 3 s
- γ = 1 / √(1 − 0.8²) = 5/3
- Δt = γ · Δt₀ = (5/3) · 3 s
Δt = 5 s
Test yourself
The astronaut's heartbeat now takes 6 seconds by her own clock, still at 0.8c. How long does the Earth observer measure it takes?
- 8 s
- Correct answer: 10 s
- 3.6 s
Right! Gamma is still five thirds, so 6 s times five thirds equals 10 s.
Now the ship travels at 0.6 times the speed of light, and the proper time is 4 seconds. What does the Earth observer measure?
- 3.2 s
- 4.6 s
- Correct answer: 5 s
Exactly! At 0.6c, gamma equals 1.25, so 4 s times 1.25 equals 5 s.
Where you see this
GPS satellites carry clocks that run measurably off from ground clocks, and the system corrects for it every day — weaker gravity at altitude makes satellite clocks run fast, slightly outpacing the slowing from their orbital speed. Cosmic-ray muons born in the upper atmosphere should decay long before reaching the ground, yet arrive in droves: their internal clock runs slow at near-light speed.
Common mistakes
The recurring slip is which time grows: the dilated time measured by the outside observer is ALWAYS longer than the proper time on the mover's own clock — you multiply by gamma, never divide. At 0.8c, gamma is 5/3, so a 6-second heartbeat on the ship takes 10 s on Earth; at 0.6c, gamma is 1.25, and 4 s of proper time becomes 5 s. The same factor shrinks lengths along the motion.
How it connects
This opens relativity, modern physics' second pillar alongside the quantum lessons: no photons, no rungs, just the constancy of light speed forcing space and time to bend around it. Its most famous byproduct, E = Δm · c², arrives at the module's end in nuclear binding energy.