A Roller Coaster Ride through Relativity/Appendix B

Length Contraction

Using the context of the river race, if we see Albert and Beatrice arrive back at exactly the same instant and knowing that Beatrice cannot row faster than Albert, we must conclude that Beatrice has a shorter distance to row.

How much shorter? Suppose that Albert rows a distance l_A and Beatrice rows l_B.

${\text{Time for Albert to finish}}={2l_{A} \over {\sqrt {c^{2}-v^{2}}}}$
${\text{Time for Beatrice to finish}}={l_{B} \over c-v}+{l_{B} \over c+v}={2l_{B}c \over c^{2}-v^{2}}$

Now these must be equal so:

${2l_{A} \over {\sqrt {c^{2}-v^{2}}}}={2l_{B}c \over c^{2}-v^{2}}$

from which we get:

$l_{B}={\sqrt {1-v^{2}/c^{2}}}\,l_{A}$

What this means is that from the point of view of a stationary observer, a rod of proper length l₀ (ie whose length is l₀ when at rest) will have a length l given by

$l={\sqrt {1-v^{2}/c^{2}}}\,l_{0}=l_{0}/\gamma$

when it moves with a velocity v in a direction parallel to its length. (It will always appear to have the same length l₀ perpendicular to the direction of motion.)

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