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Adding Velocities: A Walk on the Train

Michael Fowler, UVa Physics, 12/1/07

The Formula










Walking Across the Train

Imagine now a rather wide train, of width w, and the walker begins the walk across the train, which is now equipped with clocks on both sides, when the clock where he begins reads t = 0. For walking speed uy' (relative to the train, and across the train is the y-direction) when he reaches the clock at the other side it will read w/uy'. How is this seen from the ground? The width of the train w will be the same, there is no Lorentz contraction in the y-direction for motion in the x-direction. The beginning and ending clocks will also be synchronized as seen from the ground, since they are separated in the y-direction but not the x-direction. However, they are clocks moving at relativistic speed, so they will exhibit the familiar time dilation factor. That is, when they read w/uy', a clock on the ground will read

Thus, as observed from the ground, walking directly across the train is slowed down by the time dilation factor, just as is every other activity on the train as seen from the ground.

However, for steady motion on the train in an arbitrary direction, velocity components (ux', uy') the cross-train velocity transforms in a more complicated way, because the train clocks at the beginning and end of the walk are now separated in the x-direction, so if they register an elapsed time of w/uy a ground observer would add a lack of synchronicity term

Thus the time for the walk as observed from the ground

From this we find the general formula for transformation of transverse velocities:

For the special case of walking directly across the train, ux' = 0, we recover the earlier result, that transverse velocity is simply slowed by the time dilation effect.

Testing the Addition of Velocities Formula

Actually, the first test of the addition of velocities formula was carried out in the 1850s! Two French physicists, Fizeau and Foucault, measured the speed of light in water, and found it to be c/n, where n is the refractive index of water, about 1.33. (This was the result predicted by the wave theory of light.)

They then measured the speed of light (relative to the ground) in moving water, by sending light down a long pipe with water flowing through it at speed v. They discovered that the speed relative to the ground was not just v + c/n, but had an extra term, v + c/n - v/n2. Their (incorrect) explanation was that the light was a complicated combination of waves in the water and waves in the aether, and the moving water was only partially dragging the aether along with it, so the light didn’t get the full speed v of the water added to its original speed c/n.

The true explanation of the extra term is much simpler: velocities don’t simply add. To add the velocity v to the velocity c/n, we must use the addition of velocities formula above, which gives the light velocity relative to the ground to be:

(v + c/n)/(1 + v/nc)

Now, v is much smaller than c or c/n, so 1/(1 + v/nc) can be written as (1 - v/nc), giving:

(v + c/n)(1 - v/nc)

Multiplying this out gives v + c/n - v/n2 -v/n×v/c, and the last term is smaller than v by a factor v/c, so is clearly negligible.

Therefore, the 1850 experiment looking for “aether drag” in fact confirms the relativistic addition of velocities formula! Of course, there are many other confirmations. For example, any velocity added to c still gives c. Also, it indicates that the speed of light is a speed limit for all objects, a topic we shall examine more carefully in the next lecture.

Copyright © Michael Fowler, 1996, 2008

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