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Time dilation and length contraction are two of the most puzzling features of special relativity. One of the best ways to visualize these effects is with the aid of a light clock. A light clock consists of a box with a light pulse emitter and detector on its bottom wall and a mirror on its top wall. A light pulse is emitted upward from the bottom wall, reflects off the top wall, and returns to the bottom wall, triggering a tick of the clock, whence another flash is emitted. 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 meter (note that the cursor in the lower left corner of the window can be used to measure position in meters). In addition, note that the light clock reads time in "stationary" meters (i.e. meters traveled in a stationary clock). Why can we do this? In other words, how is the conversion accomplished and what does it depend on? Compute the time interval between successive ticks.
Now consider what a stationary observer sees when viewing a moving clock. Set β to 0.5 and press the set value and play button. The green clock then moves at half the speed of light (ignore the length contraction of the horizontal size of the light clock as it is irrelevant to this discussion). Given Einstein's postulate about the constancy of the speed of light, what can we say about the ticking of the moving clock (as seen by the stationary observer) relative to the ticking of the stationary clock?
We get this result because the speed of light is constant in any reference frame. Equal time intervals are only obtained when the distances traveled by the light pulses are 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. Show that the light travel time Δt' as observed from the stationary frame is given by the following expression:
( c Δt' )2 = ( u Δt' )2 + L2 .
This equation can be written as ( Δt' )2 = ( β Δt' )2 + ( L/c )2 if we divide 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.
It takes more clicks as measured by the stationary clock to measure the time interval of a moving clock. As observed from stationary frames, moving clocks run slower. This phenomenon is the basis of time dilation.
How fast must the moving clock travel to tick at one half the rate of the stationary clock? Use the animation above to check your answer.
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.
Finally, we tilt the light clocks sideways. Restart. The results that we obtained for time dilation must still occur for the sideways clocks: as measured in the stationary frame, the moving clock ticks slower. Note that we must accommodate this property while keeping the relative speed of the frames and the speed of light constant. In the animation above, set β so that the moving clock ticks at half the rate of the stationary clock. Along the axis of motion, how long is the moving clock (as seen from the stationary frame) relative to the length of the stationary clock? Why is this modification necessary? In other words, if the length of the moving clock were unchanged, would we get consistent results?
The required modification of the moving clock is called relativistic length contraction. Objects moving relative to us must be shorter along the axis of motion.
Authored by Mario Belloni, Wolfgang Christian, and Tim Gfroerer.
© 2004 by Mario Belloni and Wolfgang Christian.