## Section 2.4: Light Clocks, Time Dilation, and Length Contraction

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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 consists of a box with a light pulse emitter and a light detector at its bottom wall and a mirror on its top wall. Light is emitted from the bottom wall and every time the light pulse returns to the bottom wall the detector triggers a tick of the clock and another light pulse is emitted **(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*_{0}, where *L*_{0}/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*_{0} 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 (green) light clock records. Set β (= *u*/*c*) 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 (green) clock (as recorded by the stationary observer) must tick slower than that of the stationary (red) 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 when viewed in each clock's own reference frame, where each clock is stationary. However, the distance traveled by the moving clock, as seen from the point-of-view of an observer in the stationary clock's reference frame, involves both horizontal and vertical components, and it is only the vertical component of the light pulse's motion that contributes to the clock ticks (as seen from the reference frame of the clock at rest in the animation). We can calculate these distances by using the Pythagorean theorem:

(*c*Δ*t*')^{2} = (*u*Δ*t*')^{2} + *L*_{0}^{2} , (2.1)

where Δ*t*' is the time interval that an observer in the stationary frame sees the light travel time to be. We can simplify this equation to (Δ*t*')^{2} = (βΔ*t*')^{2} + (*L*_{0}/*c*)^{2 }by dividing by the speed of light. By grouping common terms we find that:

(1 − β^{2})Δ*t*' ^{2} = (*L*_{0}/*c*)^{2} = Δ*t*_{0}^{2} , (2.2)

since Δ*t _{0}* =

*L*

_{0}/c for the stationary clock (and for the moving clock as observed in the moving clock's frame of reference). Therefore Δ

*t*' = γΔ

*t*where

_{0}γ = 1/(1 − β^{2})^{0.5 }. (2.3)

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 recorded by an observer in the stationary frame, and not what the moving observer records. The time interval of a stationary clock remains Δ*t*_{0} (whether it is the red clock in the stationary frame or the green clock as recorded in its reference frame).

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In this animation, the light clocks are rotated 90 degrees **(position is given in meters by dragging the cross-haired cursor around the animation)**. Restart. The results we saw for the time dilation still occurs for the rotated 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 moving clock to be contracted. In fact, the clock must appear to be contracted to *L*' = *L*_{0}/γ where *L*_{0} is the length of the clock as seen in its own (stationary) reference frame and and *L*' is the length of the moving clock as seen from the stationary frame.^{3}

^{3}For all of the algebra, see pages 26-29 of K. Krane, *Modern Physics*, 2nd ed., John Wiley and Sons, New York (1996).

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