To summarize from the last chapter, two useful Green's functions for the
inhomogeneous wave equation:

(18.1) |

(18.2) |

(18.3) |

(18.4) |

(18.5) |

For the moment, let us ignore Dirac's observations and the radiation
field and focus instead on only the ``normal'' causally connected
retarded potential produced by a single charged particle as it moves in
the absence of external potentials. This potential is ``causal'' in
that the effect (the potential field) follows the cause (the motion of
the charge) in time, where the advanced potential has the effect
preceding the cause, so to speak. Let me emphasize that this is not a
particularly consistent assumption (again, we the theory is manifestly
time symmetric so ``past'' and ``future'' are pretty much arbitrary
namings of two opposed directions), but it yields some very nice
results, as well as some problems. In that case:

(18.6) |

(18.7) |

(18.8) |

(18.9) |

To do the integral, we need the ``manifestly covariant'' form of the retarded
Green's function. Note that:

(18.10) |

(where ). In terms of this, is given by

(18.11) |

(18.12) | |||

(18.13) |

The vector potential at a point gets a contribution only where-when that point lies on the light cone in the future (picked out by the function) of the world line of the charge (picked out be the function). The contribution is proportional to at that (retarded) time. It dies off like , although that is obscured by the form of the function.

To evaluate this (and discover the embedded ), we use the rule (from
way back at the beginning of the book, p. 30 in J1.2)

(18.14) |

(18.15) |

(18.16) |

(18.17) |

From this we see that

(18.18) |

(18.19) |

where

Recall that
. Thus:

(18.20) |

(18.21) |

Similarly

(18.22) |

where again things must be evaluated at retarded times on the particle trajectory. Note well that both of these manifestly have the correct non-relativistic form in the limit .

We can get the fields from the 4-potential in any of these forms. However,
the last few forms we have written are compact, beautiful, intuitive, and have
virtually no handles with which to take vector derivatives. It is simpler to
return to the integral form, where we can let
act on the
and functions.

(18.23) |

(18.24) |

(18.25) |

This is inserted into the expression above and integrated by parts:

(18.26) |

There is no contribution from the function because the derivative of a theta function is a delta function with the same arguments

(18.27) |

We can now do the integrals (which have the same form as the potential
integrals above) and construct the field strength tensor:

(18.28) |

This result is beautifully covariant, but not particularly transparent
for all of that. Yet we will need to find explicit and useful forms for
the fields for later use, even if they are not as pretty. Jackson gives
a ``little'' list of ingredients (J14.12) to plug into this expression
when taking the derivative to get the result, which is obviously quite a
piece of algebra (which we will skip):

(18.29) |

(18.30) |

``Arrrgh, mateys! Shiver me timbers and avast!'', you cry out in dismay.
``This is easier? Nonsense!'' Actually, though, when you think about it (so
think about it) the first term is clearly (in the low velocity, low
acceleration limits) the *usual static field*:

(18.31) |

The second term is proportional to the *acceleration* of the charge;
both **E** and **B** are transverse and the fields drop off like
and hence are ``long range'' but highly directional.

If you like, the first terms are the ``near'' and ``intermediate'' fields and the second is the complete ``far'' field; only the far field is produced by the acceleration of a charge. Only this field contributes to a net radiation of energy and momentum away from the charge.

With that (whew!) behind us we can proceed to discuss some important expressions. First of all, we need to obtain the power radiated by a moving charge.