Time dilation

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 t_0=\frac{2L}{c}, where t_0 is known as the proper time, L is the distance between the mirrors and c 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 t=\frac{2D}{c} (diagram II), where t is the time recorded using his own clock.

Substituting D=\sqrt{\left ( \frac{vt}{2}\right )^{2}+L^{2}}, where v is the speed of the moving frame relative to the rest frame, and L=\frac{ct_0}{2} in t=\frac{2D}{c}, we have

t=\gamma t_0\; \; \; \; \; \; \; \; 249

where \gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}} is called the Lorentz factor, which is useful in deriving the Thomas half.


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