= Proper lifetime inferred from a delayed decay flash
{title2=$\tau=t_{\rm receive}\sqrt{(1-\beta)/(1+\beta)}$}
An inertial particle passes a stationary detector, then recedes at <speed> $v=\beta c$ before decaying. The detector's reception time obeys $t_{\rm receive}=t_d+vt_d/c$. Dividing the decay time $t_d$ by its <Lorentz factor> gives the title formula for elapsed <proper time>. A reception event at the detector and a decay event at the particle are not interchangeable endpoints for a time-dilation calculation.
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