I am taking a course in classical electrodynamics and I am facing this problem
Prove that $A_\mu A^\mu=0$ is a good gauge fixing or not. If it is, there are residual gauge freedom?
I know that, for any $A_\mu$, I have to find a function $\alpha$ that make $(A_\mu + \partial_\mu \alpha )( A^\mu + \partial^\mu \alpha)=0$. I also know that I can not make $(A_\mu + \partial_\mu \alpha )=0$ (as a tensor), so I have tried to expand the product, but I haven't managed to get an explicit expression for $\alpha$. Is this a good approach or should I try something else?