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Time Dilation

Possibly the most unintuitive result from Relativity is the idea of time dilation. In its most basic form, it says that when an object speeds up relativistically, time will appear to slow down for the object. Before the Special Theory, this idea had never been considered. The idea that time was an absolute measure ruled the thoughts of scientists until the early twentieth century. However, Einstein proposed the idea that time was relative; it was different for every body in the universe.

To understand why this happens, we can look at the picture to the left. Let's say we are in a train and had a laser shining off of a mirror and reflecting back. The light would travel a certain distance. Now, let's say the train is moving fast compared to the ground. Someone looking into the train from the ground would see the light travel similar to the second diagram; however someone inside the train sees the light travel similar to the first diagram. The distance the light travels in the perspective of someone on the ground is greater than the distance it travels from the perspective of someone in the train. Because we know that light always travels at a constant speed, c, so the light cannot be speeding up or slowing down. So how could the light travel two distances in the same amount of time?

The answer is time dilation. When the train is moving fast, time slows down for it relative to someone on the ground. Likewise, for the person in the train, the ground is moving fast relative to the train. Therefore, with respect to someone on the ground, time is moving slower for the person in the train. This is what allows the light to appear to travel two different distances in the same interval of time, relative to a single reference point, of course.

To calculate how slow time is going for some other inertial frame, we must make the calculation below. In the final equation below, To is the "proper time" or the time that the moving body reads. This is multiplied by gamma, which is defined as one over the square root of one minus the square of the velocity another inertial reference frame is reading the body divided by the speed of light squared.

Note that gamma can never reach zero. As the velocity of the body approaches c, the time dilation grows very big; however, the velocity can never be c! If v were equal to c, gamma would be equal to zero, which is an undefined result.