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An Illustration of Length Contraction

In this article we will work out an example of length contraction. To facilitate understanding, snapshots of the process are provided, in addition to the numerical calculations.

Imagine a panel on which are mounted five lamps, equally spaced. Each lamp is capable of displaying five different colors. At any given time (according to the clock on the panel), all lamps display the same color. A computer is programmed to display red at time 0.0, orange at time 0.6, green at time 1.2, cyan at time 1.8, and blue at time 2.4. At each of these times, a camera mounted on the panel snaps a picture.

The panel is moving at 60% of light speed relative to an observation platform. The motion is from right to left across the page. There is a camera mounted to the platform, and a clock. The clocks are set so that they both read 0.0 as the panel comes in view of the camera on the platform. At that time, the camera on the platform snaps a picture.

There is a spacetime event at each lamp. We will calculate the position and time coordinates for each event using the Lorentz transformation. The equations for the transformation are shown here. The form has been modified to obtain a more convenient scale for time, as is customary. The time coordinate is the clock time multiplied by c, the speed of light.

Xb = (Xa - V • Ta) / Sqrt(1 - V2)

Tb = (Ta - V • Xa) / Sqrt(1 - V2)

where:

Xa = position in coordinate system A (the platform)

Xb = position in coordinate system B (the panel)

Ta = time on a clock on the platform, multiplied by c

Tb = time on a clock on the panel, multiplied by c

V = velocity of the panel, divided by c; 0.6 in this example

c = speed of light

Sqrt represents the square root function


This is the table of events. The record of the events is in the snapshot taken by the camera mounted on the platform. Accordingly, the time coordinate on the platform is 0.0 for all events.


Event
(lamp)
Time
(platform)
Position
(platform)
Time
(panel)
Position
(panel)
10.00.00.00.0
20.00.80.61.0
30.01.61.22.0
40.02.41.83.0
50.03.22.44.0

Here are the snapshots that were taken by the two cameras. The top five snapshots were taken by the camera on the panel, at the times indicated, according to the clock on the panel. The bottom snapshot was taken by the camera on the platform. The contraction along the line of motion is easily seen. The colors tell the story of the time transformation. The first lamp is red, so we know that we are seeing that lamp at time 0.0 on the panel clock. The orange color tells that we are seeing the second lamp at time 0.6 on the panel clock. And so on to the final lamp, where the blue color tells us that we are seeing that lamp at time 2.4 on the panel clock.


These results are so strange that the immediate reaction must be to question their validity. While it is true that the experiment as described is beyond our capabilities, physicists are satisfied that an abundance of experimental evidence is supportive of the theory. It is a fact that no experiment has ever contradicted the theory. More information on the subject may be found here and here.



Last edited: 2015-04-03 16:31:21

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