= Time dilation
{title2=$\Delta t=\gamma\Delta\tau$}
{wiki}
If one clock measures a <proper time> interval $\Delta\tau$ while moving at constant <speed> $v$ in an <inertial frame>, that frame measures $\Delta t=\gamma\Delta\tau$, where $\gamma$ is the <Lorentz factor>. The two <spacetime events> must lie on that clock's <world line>.
Suppose a clock moving at $0.6c$ advances by $4\,\mu\mathrm s$ between two ticks. Set $t=x=t'=x'=0$ at the first tick. The laboratory's synchronized clocks assign $\Delta t=1.25\times4=5\,\mu\mathrm s$. During that interval the clock travels $\Delta x=0.6c\times5\,\mu\mathrm s=3c\,\mu\mathrm s$, approximately $899.38\,\mathrm m$. The <Minkowski interval> checks the result:
$$
s^2=-c^2(5\,\mu\mathrm s)^2+(3c\,\mu\mathrm s)^2=-c^2(4\,\mu\mathrm s)^2.
$$
Both frames therefore recover the same $4\,\mu\mathrm s$ of <proper time>. This calculation is useful whenever a process is timed on a moving apparatus but described using laboratory clocks.
A <light clock> makes the mechanism visible. Two mirrors separated by $h$ perpendicular to the motion exchange a pulse; define one tick as its complete return to the lower mirror. In the clock's <rest frame> the tick takes $\Delta\tau=2h/c$. In the laboratory each half-tick lasts $T$ and the light traverses a diagonal, so the <Pythagorean theorem> gives
$$
(cT)^2=h^2+(vT)^2,\qquad T=\gamma h/c,\qquad \Delta t=2T=\gamma\Delta\tau.
$$
For $h=1\,\mathrm m$ and $v=0.6c$, each leg covers $1.25\,\mathrm m$ while the mirrors move $0.75\,\mathrm m$. The return tick lasts $2.5\,\mathrm m/c$ instead of $2\,\mathrm m/c$: the light has the same <speed of light> in both accounts. https://newt.phys.unsw.edu.au/einsteinlight/jw/module4_time_dilation.htm[Joe Wolfe, UNSW: time dilation and the light clock]{ref}.
\Image[/relativity-light-clock.png]
{title=One return tick of a transverse light clock at rest and moving at 0.6c; the laboratory light path is longer}
Reciprocal <time dilation> is consistent: each <inertial frame> times the other frame's single moving clock using its own synchronized clock network. These comparisons use different simultaneous <spacetime events>. Comparing reunited clocks instead requires their complete <world lines>, as in the <twin paradox>.
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