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.

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

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

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
10 s
3.6 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
5 s