Let us consider the experiment described below for the purpose of measuring time by two different observers. Inside a train moving at the velocity v , a pulse of light is emitted at the floor of the train. This pulse travels straight to the ceiling of the train where it bounce back to the source. How long does it take for the pulse to complete the round trip journey for both, an observer inside the train, and an observer standing outside the train next to the tracks? Notice that the height of the train cart is a train dimension that it is perpendicular to the motion of the train. In other words, the velocity of the train is perpendicular to this dimension. It is assumed that dimensions perpendicular to the motion are not affected by the results of special relativity. Let us call this height d.
The two hypotenuses are the same; therefore, over the hypotenuses, the speed of light should be calculated accordingly with the following expression:
In this expression, the denominator correspond to the time measured by the observer standing next to the track, Earth Observer.
Considering one of the right angle triangles, the Pythagorean theorem implies that . In this expression, the two dimensions d and D can be obtained from the times measured by the observer in the train and the observer on the Earth. Substituting those values in the previous expression, we get . From these equation, the time measured by the observer on the Earth can be solved in terms of the time measured by the observer traveling in the train.
and the time, accordingly to the observer on the Earth, is larger than the time measured by the observer on the train. Again, notice that the time measured by the observer on the train is made with a single clock while the time measured by the observer on the Earth is done with a sequence of clocks. Times measured with the equivalent to a single clock are called PROPER TIMES while times measured with a sequence of synchronized clocks are called IMPROPER TIMES.