To study the relativity of time, we will consider a MRU wagon with velocity v relative to the ground.
On the roof of the wagon a flat mirror is placed and on the floor a lantern is glued at a distance d from the mirror as shown below:
The flashlight emits a pulse of light from the floor that goes to the mirror and back to her. It is important to define two events here:
1st) flashlight emitting the pulse of light;
2º) pulse of light arriving and returning to the lantern.
Let's establish two frameworks to analyze the elapsed time interval between the two events. Are they:
From the point of view of the frame R ', the light follows the path shown in the figure above, propagating at speed c and traveling the distance 2d during the time interval Δt '.
So for R 'we can write:
Observe in the following figure the light's trajectory in relation to the reference frame R.
Now, analyzing from the point of view of the referential R, the light also makes the path in question with speed c, having traveled a distance ç. Δt during the time interval Δt. Remember that R saw the car, with speed v, if it traveled the distance v. Δt.
Starting from the right triangle of the figure above, we can write:
We can substitute this result by obtaining:
Since the expression presented in the denominator of the equation above is less than 1, we can conclude that Δt is greater than Δt '. Like this:
For a referential R that moves relative to where events occur, the time interval Δt between events is greater than the interval Δt 'measured by reference R', which is at rest relative to the event location. This phenomenon is called time dilation.