Combining the principle of relativity with the constancy of the speed of light leads to a conclusion about time that is foreign to classical mechanics:
(Theorem) If an observer is moving at velocity v relative to an inertial frame K, then after Δt time elapsed in K, the observer's own clock will show only a corresponding Δt' = Δt · √1 – v2 / c2 time elapsed.
Thus, in general, the (coordinate) time that elapses between two events depends on the inertial frame in which it is measured. The effect is called time dilation since Δt > Δt' holds whenever v > 0 and Δt' > 0. In other words, the observer moving relative to K always sees as if time was passing faster in K.
Note: time dilation is unnoticeable in our everyday life.
The formula works for polygonal trajectories as well, provided that the speed is the same along all edges.
(Assumption) Any motion (of a point-like entity) can be approximated with arbitrary accuracy by polygonal motion.
This makes the formula valid for any motion of constant speed; even for circular ones as a matter of fact.