Proper time

Physical motivation and derivation of the negative sign in the time interval

Consider two events that coincide in time and space for one observer. Then they must agree for all the observers. Therefore, if we consider a spaceship travelling from Earth to Alpha Centauri, if the crew observes that their clock is at 0 when they are leaving, and at 3 years when they are arriving, ANY OBSERVER must agree in the fact that, with respect to the time measurement system of the ship, the travel has had a duration of 3 years. This quantity is an invariant, and it is called the proper time (seen in this video).

From here we can conclude why the minus sign in Minkowski metric?