Illustration 4: Light Clocks, Time Dilation, and Length Contraction

β = u / c = 

Please wait for the animation to completely load.

One of the best ways to visualize time dilation and length contraction of moving objects is with the construction of a light clock.  A light clock is a clock that clicks every time a light pulse emitted from one location returns to the same location after reflecting off a wall as shown in the animation (position is given in meters by dragging the cross-haired cursor around the animation)Restart.  The total vertical distance traveled by the light for the stationary clock is L, where L/2 is the distance between the walls of the clock.  For simplicity, in this animation the distance between the walls is 0.5 meters and therefore the total vertical distance is L or 1.0 meter.  Given this, the clock reads time in meters.  What does this mean?  Light travels one meter every 3.33 X 10-9 seconds.  Therefore every click measures the time it takes for light to travel 1 meter or 3.33 X 10-9 seconds.

Now consider what a stationary observer relative to a moving light clock sees.  Set β to 0.5 and press the set value and play button.  The green clock moves at half the speed of light (ignore the length contraction of the horizontal size of the light clock as it is irrelevant for this discussion).  Given Einstein's postulate about the constancy of the speed of light, the moving clock (as seen by the stationary observer) must tick slower that of the stationary clock. 

This result occurs for the light clock because the speed of light is constant in any reference frame; therefore the distances traveled by the two light pulses must be the same as viewed in the frame of the stationary clock. However the distance traveled by the moving clock involves both horizontal and vertical components, and it is only the vertical component of the light pulsesí motion that contributes to the clock ticks.  We can calculate these distances by using the Pythagorean theorem:

( c Δt' )2 = ( u Δt' )2 + L2     

where Δt'  is the time interval that an observer in the stationary frame sees the light travel time to be.  We can simplify this equation to ( Δt' )2 = ( β Δt' )2 + ( L/c )2 by dividing by the speed of light.  By grouping common terms we find that: 

( 1 - β2 ) Δt' 2 = ( L/c )2 = Δt 2    

since Δt = L/c for the stationary clock (and for the moving clock as seen in the moving clock's frame of reference).  Therefore Δt' = γ Δt where

γ = 1/ ( 1 - β2 )0.5.

Therefore it takes more clicks as measured by the stationary clock to measure a time interval of a moving clock.  Observed from stationary frames, moving clocks run slower.  This is called time dilation. 

Note that we are talking about what is seen by an observer in the stationary frame and not what the moving observer sees.  The stationary observer sees the time in between clicks of the moving clock to be Δt'.  The time interval of a stationary clock remains Δt (whether it is the red clock in the stationary frame or the green clock as seen in its reference frame). Also note that this has nothing to do with light-travel time.

β = u / c = 

Please wait for the animation to completely load.

In this animation, the light clocks are tilted sideways (position is given in meters by dragging the cross-haired cursor around the animation)Restart.  The results we saw for the time dilation must still occur for the sideways clocks.  Therefore given the fact that in the stationary frame the moving clock still ticks slower, the only way for this to happen is for the distance between the walls of the clock to be contracted.  In fact the clock must be contracted to L' = L / γ where L is the length of the clock as seen in the reference frame of the clock (whether moving or stationary) and L' is the length of the moving clock as seen from the stationary frame.4

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4For all of the algebra, see pages 26-29 of K. Krane, Modern Physics, 2nd ed. (John Wiley and Sons, New York, 1997).

 

 

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