Axial Gauge V Rubakov: Find Residual Gauge Transforms for Maxwell Eqns

[gasar8]

V. Rubakov: Classical Theory of Gauge Fields, Problem 4: Find the residual gauge transformations and the general solution of the Maxwell equations in the axial gauge ($\vec{\textbf{n}} \cdot \vec{\textbf{A}}=0$), where $\vec{\textbf{n}}$ is some fixed unit three-vector, which is constant in spacetime.

I am using Rubakov notation, so gauge transformation is $A'_{\mu}(x)=A_{\mu}(x)+ \partial_{\mu} \alpha(x)$ and $\eta_{\mu \nu}=diag(1,-1,-1,-1).$ If $\vec{\textbf{n}} \cdot \vec{\textbf{A}}=0 \Longrightarrow \vec{\textbf{n}} \cdot \vec{\textbf{A}'}=0$, so:
$$\vec{\textbf{n}}(\vec{\textbf{A}}-\vec{\nabla} \cdot \alpha)=0\\ \vec{\textbf{n}}(\vec\nabla \cdot \alpha)=0.$$
I assume that form this relation, $\vec{\textbf{n}}$ must be orthogonal to $\vec\nabla \cdot \alpha$, but is there any further or deeper explanation? Do I have to choose particular $\vec{\textbf{n}}$, for example $(0,0,0,1)$? In this case:
$$A'_z=0 \\ A_z-\partial_z \alpha = 0 \\ \Longrightarrow \alpha = \int A_z dz+f(x,y,t)$$
I still don't know how to proceed to Maxwell equations.

 

[NateD]

Is there any way to derive them from this relation? The general solution of the Maxwell equations in the axial gauge is given by:A'_{\mu} = \partial_{\mu} \alpha + \epsilon_{\mu \nu \lambda \sigma} \partial^{\nu} \phi^{\lambda \sigma}where \alpha is an arbitrary scalar and \phi_{\mu \nu} is an arbitrary antisymmetric tensor. The residual gauge transformations are of the form:\alpha'=\alpha+\partial_{\mu}\alpha \\\phi'_{\mu \nu}=\phi_{\mu \nu}+\partial_{\mu}\xi_{\nu}-\partial_{\nu}\xi_{\mu}where \xi_{\mu} is an arbitrary vector.