106 Chapter 3
Using an Equation to Understand Length Contraction
(Lorentz Contraction)
Let’s use an equation to see how length contracts.
In this case, let’s assume that a rocket is flying at a constant velocity v (see Figure 3-1).
When the person riding in the rocket measures the positions of the front and back ends
of the rocket, he finds the front end is at position x¢
2
, and the back end is at position x¢
1
.
Therefore, the rocket’s length is .
Now what happens if this situation is observed from outside the rocket, for example,
from a space station as in Figure 3-2?
To calculate the contraction in the length of a rocket as it moves past an observer at
close to the speed of light, let’s consider two points in the rocket’s frame of reference: x¢
1
at
the front of the ship and x¢
2
at the back of the ship. Using the Lorentz transformation, intro-
duced in “Wait a Second—What Happens with the Addition of Velocities?” on page 48, we can
calculate how an observer who watches the ship pass measures the points at the front x
1
and back of the ship x
2
, in his reference frame. The length that the observer on the outside
of the ship measures will be shorter than length that the astronaut measures. This effect,
relativistic length contraction, comes from the contraction of space at speeds close to the
speed of light.
Mr. A in the rocket thinks he’s at rest.
Figure 3-1: A person riding in the rocket measures the positions of the front and
back ends of the rocket.
l x x
0 2 1
=
Figure 3-2: The rocket viewed from a space station
Mr. B in the
space station
Direction of motion
Velocity v
l
0
x´
1
x´
2
RELATIVITY_03.indb 106 3/15/2011 3:27:23 PM

Get The Manga Guide to Relativity now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.