* Time dilation* is the difference in time measured by an observer who is moving relative to another observer. Consider two frames of reference, each with an observer and a clock, which consists of a photon bouncing between two mirrors A and B. If the frame of observer X is at rest, while the frame of observer Y is moving away from it, observer Y sees the photon of his clock travelling vertically between the mirrors (diagram I) and measures , where is known as the

*, is the distance between the mirrors and is the speed of light. Observer X, however, sees the same photon moving in a triangular path (as the observer X’s clock moves away from him) and measures (diagram II), where is the time recorded using his own clock.*

**proper time**Substituting , where is the speed of the moving frame relative to the rest frame, and in , we have

where is called the * Lorentz factor*, which is useful in deriving the Thomas half.