*Please wait for the animation to completely load.*

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 pulses’ 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} = ( v Δt' )^{2} + L^{2} .

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 - β^{2} )^{ }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 - β^{2} )^{0.5}.

It takes more clicks as measured by the stationary clock to obtain 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.