Proper lifetime inferred from a delayed decay flash ($\tau =t_{\mathrm{receive}}\sqrt{(1-\beta )/(1+\beta )}$)

An inertial particle passes a stationary detector, then recedes at speed $v=\beta c$ before decaying. The detector's reception time obeys $t_{\mathrm{receive}}=t_d+v{t}_d/c$. 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.