Proper time ($\tau$)

Proper time is the time measured by a clock following a timelike worldline. With metric signature $(-,+,+,+)$ and units $c=1$,along that worldline.

Restoring the speed of light, the same metric signature gives $d\tau =\sqrt{-d{s}^2}/c=dt\sqrt{1-|\mathbf{v}{|}^2/c^2}$ for a future-directed timelike curve. Integrating along a clock's world line gives its elapsed proper time, even when its velocity varies. For uniform motion this becomes the time dilation formula; it is not generally the square root of the interval between endpoints of a curved world line.

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.