Proper time

Proper time refers to the passage of time as experienced by the thing being observed specifically at its location. For example, an astronaught in orbit will experience time differently than an identical twin remaining on the planet. The time experience for the astronaught at his location such as his rate of aging or his wristwatch's time may be thought of as measures of his proper time. Lets say the astronaught uses coordinates $\left(ct',x',y',z'\right)$, then his proper time may be designated by $\tau$ is given by

$\tau =t'|_{(x',y',z')=(0,0,0)}$

If we observe him with coordinates we designate unprimed and describe his path through spacetime called his world line, with the line element, then that line element gives us the flow of his proper time according to

$ds^{2}=dc\tau ^{2}=g_{\mu \nu }dx^{\mu }dx^{\nu }$

For example, lets say the astronaught moved about in flat spacetime and that we use an inertial frame to observe him so that the line element corresponds to the metric of special relativity which we use to describe his path

$ds^{2}=dc\tau ^{2}=dct^{2}-\left(dx^{2}+dy^{2}+dz^{2}\right)$

factoring out our coordinate time differential yields

$dc\tau ^{2}=\left(1-{\frac {v^{2}}{c^{2}}}\right)dct^{2}$

simplify

$dt={\frac {d\tau }{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$

which is special relativity's time dilation formula.