**
**
Relativity

Length
Contraction

**A**
and **B** have relative velocity v as before but now **B**
holds a rod in a direction parallel to the direction of v.

End 1 of the
rod passes **A** at the instant when the flash of light is
sent towards the mirror.

End 2 of the
rod passes **A** at the instant when the reflected light
returns to the torch.

From **A**’s
point of view the situation is

i) light leaves the torch

............................................................ |

ii) light
returns to the torch |

................................. |

**
****A**
will therefore conclude that the length of the rod
is given by =
2vt_{o}

From **B**’s
point of view the situation is

.................... |

So, using
the same "light beam clock" observer **B**
measures the length of the same rod to be
where

o
= 2vt

and, using the time dilation
formula, we have

1. |
The
length of a rod as measure by an observer who is at
rest relative to it is called the proper length,
o.
(In this example, **B** gives the "proper
length".) |

2. |
Other
inertial observers will measure improper lengths for
the same object |

3. |
Improper
length < proper length, hence the term "length
contraction". (This effect is often called the
Fitzgerald contraction after the Irish mathematician
who predicted it using a different theory.) |

4. |
This
contraction only affects the length of a rod held *parallel
to the direction of the relative motion*. |

You have probably noticed
that the factor (1 - v²/c²)^{-1/2} appears very
often in relativistic equations. To save time, it is usually
abbreviated by the letter
.