Lorentz factor ($\gamma$)

For speed $v<c$, the Lorentz factor is

A rod of rest length $L_0$ moving parallel to its length has simultaneous laboratory length $L_0/\gamma$, where $\gamma$ is the Lorentz factor. Transverse lengths are unchanged by the corresponding Lorentz transformation.

If a clock measures proper-time interval $\Delta \tau$ while moving at constant speed $v$ in an inertial frame, that frame measures $\Delta t=\gamma \Delta \tau$.

The twin paradox compares proper time along the different world lines of a travelling observer and a stay-at-home observer who reunite. Reciprocal inertial time dilation produces an apparent contradiction only if the traveller's change of inertial frame is ignored.