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Time dilation

Special relativity says that a clock that is moving runs more slowly as seen by an observer at rest. The factor gamma is 1 when the speed is zero, and it grows without limit as the speed approaches the speed of light. The time on board, the proper time, is the shortest. The speed of light is the same for everyone, and nothing with mass can reach it. At ordinary speeds gamma is so close to 1 that we do not notice the effect.

γ=11−v2/c2\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}gamma
Δt=γ Δt0\Delta t = \gamma\,\Delta t_0time dilation

Symbols

vvspeedm/s
ccspeed of lightm/s
Δt0\Delta t_0proper times
Δt\Delta ttime in the rest frames

Example

A spaceship moves at 0.6c, and 4 years pass on board.

γ=1/1−0.62=1.25\gamma = 1/\sqrt{1 - 0.6^2} = 1.25, so on Earth 1.25⋅4=51.25 \cdot 4 = 5 years pass.

Gamma is never less than 1. The length in the direction of motion becomes correspondingly shorter, L=L0/γL = L_0/\gamma.
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