We are now ready to get serious about electrodynamics. We have developed a
beautiful, geometric system for describing the *coordinates* in terms of
which electrodynamics must be formulated for the speed of light to be an
invariant. We have developed a group of coordinate transformations that
preserves that invariance. Now we must face up to the fact that our original
equations of electrodynamics are not in a ``covariant'' formulation that makes
these constraints and transformation properties manifest. For example, we do
not yet know how the electric and magnetic fields themselves transform under a
LT!

Let us then reformulate our basic equations in 4-tensor form. We will make the equations themselves 4-scalars, 4-vectors, or 4-tensors of higher rank so that we can simply look at them and deduce their transformation properties. In addition, we will simplify the notation when possible.

We begin at the beginning. All we really know about electromagnetic fields is
their (defined) action on a charged particle:

(16.135) |

(16.136) |

Thus we can write

(16.137) |

(16.138) |

Now, if this energy-force 4-vector equation is to be covariant (so its transformed form is still a 4-vector) then the right hand sides must form a 4-vector too. Thus we must be able to express it (as a contraction of co and contra variant tensors) so that this property is ``manifest''. We know (experimentally) that charge is a Lorentz scalar; that is, charge is invariant under LT's. forms a contravariant 4-vector.

From this we can deduce the 4-tensor form for the electromagnetic field!
Since the space parts
form the time component of a
four vector, **E** must be the time-space part of a tensor of rank two.
That is,

(16.139) |

For example, we already have observed that the continuity equation is a
covariant 4-scalar:

(16.140) |

(16.141) |

(16.142) |

Next, consider the wave equations for the potentials in the Lorentz
gauge (note well that Jackson for no obvious reason I can see *still* uses Gaussian units in this part of chapter 11, which is goiing
to make this a pain to convert below - bear with me):

(16.143) |

so that:

(16.144) | |||

(16.145) |

Therefore, if we form the 4-vector potential

(16.146) |

(16.147) |

(16.148) |

Now we can construct the components of **E** and **B** from the covariant
4-vector potential. For example, we know that:

(16.149) |

(16.150) |

(16.151) |

The components of the electric and magnetic fields (all six of them) thus
transform like the components of a **second rank, antisymmetric, traceless
field strength tensor**^{16.7} :

(16.152) |

(16.153) |

(16.154) |

Another important version of this tensor is the **dual field strength
tensor**
. In terms of the totally antisymmetric
tensor of the fourth rank and the normal field strength tensor it is given by:

(16.155) |

Finally, we must write Maxwell's equations in covariant form. The
inhomogeneous equations are (recall)

(16.156) | |||

(16.157) |

The quantity on the right is proportional to the four current. The quantity on the left must therefore contract a 4-derivative with the field strength tensor. You should verify that

(16.158) |

The homogeneous equations

(16.159) | |||

(16.160) |

also form a four vector (of zero's) and must hence be the contraction of a field strength tensor. But which one? Well, the second homogeneous equation requires that and both require that , so it must be the dual:

(16.161) |

(16.162) |

Now that we have written Maxwell's equations (and the consequences of ME) in
four dimensional form (remarking all the while that they are unusually
beautiful and concise in this notation) we are done. Before we go on to
deduce (from these results) how electric and magnetic fields LT, however, we
should complete the question with which we began the discussion, namely, how
does Newton's law become covariant? The answer is (now that we know what the
field strength tensor is)

(16.163) |

As a postscript to our discussion (recalling that sometimes the fields
propagate in some medium and not free space) we note that in this case the
homogeneous equation(s) remain unchanged, but the inhomgeneous equations are
modified (by using
and
instead of
and
). The
inhomogeneous equation is then

(16.164) |

Let us pause for a moment of religious silence and contemplate a great wonder of nature. This is the scientist's version of ``prayer in school''.