The Lorenz gauge, for a variety of reasons, is in my opinion the ``natural'' gauge of electrodynamics. For one thing, it is elegant in four dimensional space-time, and we are gradually working towards the epiphany that we should have formulated all of physics in four dimensional space-time from the beginning, even if we're considering non-relativistic phenomena. Working in it, most problems are relatively tractible if not actually easy. We will therefore consider it first.

Above we derived from MEs and their definitions the two equations of
motion for the potentials
and
:
&phis#phi;+ ()t & = & -
&rho#rho;

+ - 1c^2t & = & -
+ (+ 1c^2&phis#phi;t)

*If* we can guarantee that we can *always* find a gauge
transformation from a given solution to these equations of motion,
, a new one such that new
such that the new
ones satisfy the constraint (the Lorenz gauge condition):
+ 1c^2&phis#phi;t = 0
then the two equations of motion both became the *inhomogeneous wave
equation* for potential waves that propagate at the speed of light into
or out of the charge-current source inhomogeneities. This precisely
corresponds to our intuition of what should be happening, is elegant,
symmetric, and so on. Later we'll see how *beautifully* symmetric
it really is.

We must, however, prove that such a gauge condition actually exists. We
propose:
&phis#phi;& = & &phis#phi;_0 - &Lambda#Lambda;t

& = & _0 + &Lambda#Lambda;
and substitute it into the desired gauge condition:
+ 1c^2&phis#phi;t & = & _0 +
&Lambda#Lambda;+ 1c^2&phis#phi;_0t -
1c^2&Lambda#Lambda;t

& = & 0
or
&Lambda#Lambda;- 1c^2 &Lambda#Lambda;t = _0 +
1c^2&phis#phi;_0t = f(,t)
for some computable inhomogeneous sourcevfunction
.

This equation is solvable for an enormous range of possible s (basically, all well-behaved functions will lead to solutions, with issues associated with their support or possible singularities) so it seems at the very least ``likely'' that such a gauge transformation always exists for reasonable/physical charge-current distributions.

Interestingly, the gauge function
that permits the Lorenz
condition to be satisfied so that
satisfy wave equations is
itself the solution to a wave equation! It is also interesting to note
that there is *additional* gauge freedom within the Lorenz gauge.
For example, if one's original solution
itself satisfied
the Lorenz gauge condition, then a gauge transformation to
where
is *any free scalar wave*:
&phis#phi;& = & &phis#phi;_0 - &Lambda#Lambda;t

& = & _0 + &Lambda#Lambda;

&Lambda#Lambda;- 1c^2 &Lambda#Lambda;t & = & 0
continues to satisfy the Lorenz gauge condition. Not only are we
nearly guaranteed that solutions that satisfy the Lorenz gauge
condition exist, we have discovered an *infinity* of them, connected
by a *restricted gauge transformation*.

In the Lorenz gauge, then, everything is a wave. The scalar and vector potentials, the derived fields, and the scalar gauge fields all satisfy wave equations. The result is independent of coordinates, formulates beautifully in special relativity, and exhibits (as we will see) the causal propagation of the fields or potentials at the speed of light.

The other gauge we must learn is not so pretty. In fact, it is really
pretty ugly! However, it is still *useful* and so we must learn it.
At the very least, it has a few important things to teach us as we work
out the fields in the gauge.