If we form the infinitesimal version of the Lorentz transformation of
coordinates:

(15.66) | |||

(15.67) | |||

(15.68) | |||

(15.69) |

Point is moving at velocity in frame , which is in turn moving at velocity with respect to the ``rest'' frame . We need to determine (the velocity of in ). We will express the problem, as usual, in coordinates and to the direction of motion, exploiting the obvious azimuthal symmetry of the transformation about the direction.

Note that

(15.70) |

(15.71) |

Similarly, (e.g. -- ) is given by

(15.72) |

or

(15.73) |

Note also that if
and
, then

(15.74) |

(15.75) |

(15.76) | |||

(15.77) |

What about the other limit? If
, then

(15.78) |

We observe that the three spatial components of ``velocity'' do *not* seem
to transform like a four vector. Both the and the
components are mixed by a boost. We can, however, make the velocity into a
four vector that does. We *define*

(15.79) | |||

(15.80) |

where is evaluated using the magnitude of

Now we can ``guess'' that the 4-momentum of a particle will be .
To prepare us for this, observe that

(15.81) |