intervoxel said:
Now I see, JANm, thank you.
The Lorentz condition is a statement of the law of conservation of charge -- It imposes a constraint between A and V. If I select a V invariant in time I get Coulomb gauge:
A' = A + grad(V)
V' = V - 0
You would not want to explicitly set the scalar potential to be time invariant. The primary reason being that it restricts the physical phenomenon that can occur. For example, under the Coulomb gauge we can derive the expression that
\frac{1}{c^2}\nabla\frac{\partial^2 \Phi}{\partial t^2} = \mu_0\mathbf{J}_t
where J_t here is the irrotational current, the curl free portion of the currents. Thus, by fixing a time invariant scalar potential you are forcing the condition that there can be no irrotational currents, among other things I am sure.
The reason for confusion is that I believe you are still trying to satisfy the Lorenz condition, which is the defining property that gives rise to the Lorenz gauge. The Lorenz gauge requires that the Lorenz condition be satisfied,
\nabla\cdot\mathbf{A}+\frac{1}{c^2}\frac{\partial \Phi}{\partial t} = 0
This gives rise to the two decoupled inhomogeneous wave equations for \Phi and A.
The Coulomb gauge, however, satisfies the inhomogeneous wave equation,
\nabla^2\mathbf{A} - \frac{1}{c^2}\frac{\partial^2 \mathbf{A}}{\partial t^2} = -\mu_0\mathbf{J} + \frac{1}{c^2}\nabla \frac{\partial \Phi}{\partial t}
We can see that by decomposing the current into irrotational and solenoidal components we can decouple the above wave equation as previously stated above. While this is very similar to what arises in the Lorenz gauge, it allows a different set of freedoms in the choice of our potentials. You can find a detailed explanation of this in Jackson's text.
