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Time Dilation

The object of this page is to show explicitly how it is possible for two observers in inertial frames moving relative to each other at a relativistic speed to each see the other's clocks as running slow and as being unsynchronized, and yet if they both look at the same clock at the same time from the same place (which may be far from the clock), they will agree on what time it shows!

A simple experiment which reveals in a direct way the quantitative relation connecting the time interval between two events as measured from two different inertial frames is the following. Imagine a passenger sitting on a train that moves with uniform velocity V with respect to the ground. The experiment will consist of turning on a flashlight aimed at a mirror directly above on the ceiling and measuring, in the two frames, the time it takes the light to travel up and be reflected back down to its starting point. The situation is illustrated in the figure below:

Lets consider what two observers, one on the train and one on the ground would measure. We set S' to be the frame of the train, and S to be that of the ground:

• In S', the light ray follows a strictly vertical path from A to B, reflects off the mirror, and follows the vertical path from B to C. This is called a "proper time interval" since it is measured by a single clock at one place, the departure and arrival of the light ray occurring at the same place in the train's (S') frame. The proper time interval in this case is simply given by:

  

• In S, the train and passenger move to the right during the interval between emission and detection of the light. He will measure the time interval from the readings on two stationary clocks, one at the position the experiment began (turning-on of flashlight) and a second at the position the experiment ended (arrival of light to flashlight). Hence, he compares the reading of one moving clock (the passenger's watch) to the readings on two stationary clocks. For the S-observer, the light ray follows the oblique path shown in the figure, and the time interval is then given by:

  

Note that we have already seen that the transverse distance from the source/detector to the mirror is the same for each observer. Thus, it is apparent that an observer on the ground (in S) observes the light to travel a greater distance than does the passenger. Because the speed of light is the same in both frames, the ground observer sees more time elapse between the departure and the return of the ray of light than does the passenger. He concludes that the passenger's clock runs slow.

We can easily make this quantitative. Taking the ratio of the two time intervals, one obtains:

or

Beta is bounded to be less than one (V is less than c), and thus gamma is always greater than 1. We see that the proper time interval always is less than the time interval measured in a different inertial frame. This is time dilation - the clocks in S' seem to run slow as compared to the clocks in S.

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