We seek a relativistic generalization of momentum (a vector quantity) and
energy. We know that in the low speed limit, ,

(15.82) |

(15.83) |

The only possible form for this generalization of these equations consistent
with our requirement that the laws of nature remain invariant are:

(15.84) |

(15.85) |

This immediately yields the limiting forms:

(15.86) |

(15.87) |

There are several possible ways to evaluate the full forms of these functions. Jackson's (based on scattering theory) is tedious and conceals the structure of the result. Furthermore, after telling us that selecting clever initial directions with an eye to simplifying the algebra ``lacks motivation'' he derives a result by selecting particular initial directions. The guy loves algebra, what can I say. Feel free to study his approach. It works.

I, on the other hand, am too lazy to spend most of a period deriving a result that is ``obvious'' in the correct notation. I am therefore going to ``give'' you the result and motivate it, and then verify it trivially be expressing it as a four-vector. This works nearly as well and is not anywhere near as painful.

We begin by considering elastic scattering theory. An elastic collision of
two identical particles must conserve momentum and energy in all inertial
frames. In the center of mass frame (which we will consider to be )

(15.88) |

(15.89) |

Now,

(15.90) |

(15.91) |

A moments quiet reflection (egad, another pun!) should convince you that in
terms of the general transformation:

(15.92) |

(15.93) |

We thus begin with

(15.94) |

(15.95) |

(15.96) |

(15.97) |

Thus

(15.98) |

To get the energy equation, we use the same approach. Recall that a binomial
expansion of is given by

(15.99) |

(15.100) |

(15.101) |

There are several questions to be answered at this point, some experimentally
and some theoretically. We need to measure the rest masses and theoretically
verify that only this transformation correctly preserves the energy momentum
conservation laws in elastic collisions as required. Beyond that, there are
still some uncertainties. For example, there could in principal be an
additional constant energy added to the energy term that was not scaled by
and the laws of physics would still be expressible, since they are
not sensitive to absolute energy scale. We will take advantage of that
freedom in several instances to add or subtract an infinite *theoretical*
constant in order to make the rest mass come out to the observed *experimental* mass m. This is called renormalization.

To obtain the same result a different way, we turn to the notation of
4-vectors. We observe that the common factor of above in both
and also occurs when one makes velocity into a four vector. This
suggests that energy and momentum can similarly be made into four vectors that
transform like the coordinates under a boost. If we try the combination

(15.102) | |||

(15.103) |

we see that it works exactly. It results in an invariant

(15.104) |

The total energy can thus be expressed in terms of the three momentum as

(15.105) |

(15.106) |

This completes our review of ``elementary relativity theory''. We shall now
proceed to develop the theory in a new, *geometric* language which is
suitable to our much more sophisticated needs. To do this, we will need to
begin by generalizing the notion of a four dimensional vector space with a set
of transformations that leave an appropriately defined ``length'' invariant.