Time Dilation
by Darren Ng

Observer O' is at rest inside the vehicle. A mirror is fixed to the ceiling. A laser is mounted at distance d below the mirror, on the floor (run animation below for event 1).

At some instance, the laser emits a pulse of light toward the mirror (event 1). After reflecting from the mirror, the light pulse arrives back at the laser (event 2).

Observer O' carries a clock C'. Using it, he measured the interval between the 2 events as , calculated as below:

This time is measured by O' who is at rest in the moving vehicle.

Now consider the same set of events observed by a man standing on the land (animation below). To him, the mirror and the laser is moving to the right with a speed v.

By the time the light pulse reaches the mirror, the mirror has moved a distance to the right where is the time for the light pulse to travel from O' to the mirror and back as measured by the man on the land, O with his clock C.

Comparing these two situations, we see the light must travel farther in the second case than in the first case. According to second postulate of relativity, both observers must measure the same c for speed of light.

Because the light pulse travels farther in the second case, the time interval measured by observer on the land must be longer than , the time interval measured by the observer moving inside the vehicle.

Using Pythagorean theorem on the triangle in the second case, we obtain a relationship between and .

Solving for gives,

because , we obtain

where
           

This result says that the time interval measured by the observer moving with respect to the clock is longer than the time interval measured by the observer at rest with respect of the clock, because is always greater than unity.

Thus we can conclude that time between ticks of a moving clock , is observed to be longer than the time between ticks of an identical clock at rest, .

A moving clock runs slower than a clock at rest by a factor of .

This applies to all physical processes including chemical reactions and biological processes. Thus if you happen to move in space at a high constant speed with respect to the stationary Earth, your time runs slower and you age slower.