C4 - proper time

define the proper time $τ$ that all observers agree with, where time is measured in a co-moving frame with the clock
Alice is travelling on a spaceship at $0.8 c$ with a clock, and Bob watches her from earth
Alice measures the proper time $τ$
Bob measures $t$ on his coordinate clock
$τ < t$ , hence there is time dilation
importantly, $τ$ is the time measured by a clock present at both event
observing a clock in its own rest frame:

d x^{'} = d y^{'} = d z^{'} = 0 d s^{2} = c^{2} d t^{' 2} = c^{2} d τ^{2}

d τ^{2} = \frac{d s^{2}}{c^{2}} = d t^{2} (1 - \frac{1}{c^{2}} (v_{x}^{2} + v_{y}^{2} + v_{z}^{2})) = d t^{2} (1 - \frac{v^{2}}{c^{2}}) = \frac{d t^{2}}{γ^{2}} ∴ d τ = \frac{d t}{γ}

\frac{d}{d τ} = γ \frac{d}{d t}