Result

Time dilation and length contraction

(Q1.2.1) Q1.2 · Time Dilation and Length Contraction →

Δ t = γ Δ t_{0}, L = \frac{L _{0}}{γ}, γ = \frac{1}{1 - v ^{2} / c ^{2}}

Statement

Time measured in a frame where the clock moves runs slower by γ; length measured along the direction of motion is shorter by the same factor.

Symbols and units

$Δ t, Δ t_{0}$	dilated, proper time interval	s
$L, L_{0}$	contracted, proper length	m
$γ$	Lorentz factor	—
$v$	relative speed between the frames	m/s
$c$	speed of light	2.998×10⁸ m/s

When it applies

Inertial (non-accelerating) reference frames only.

Don't use when

Only valid between inertial (non-accelerating) reference frames — an accelerating frame requires general relativity or a more careful piecewise treatment.

Derivation

A light clock ticking in a moving frame, viewed from a stationary one, appears to tick more slowly because the light must trace a longer diagonal path; working through that geometry produces the factor γ.

γ = \frac{1}{1 - v ^{2} / c ^{2}}

Worked examples

The speed limit

As v → c:

γ = \frac{1}{1 - v ^{2} / c ^{2}} \to \infty

Reaching c would require infinite γ, hence infinite energy — which is why nothing with mass ever does.

Watch out for

Δt₀ and L₀ are PROPER time and PROPER length — measured in the frame where the clock or object is at rest. Which frame counts as 'moving' depends entirely on which observer is asked; there is no absolute frame of rest. As v → c, γ → ∞, which is why nothing with mass can reach the speed of light.