I've been given the following gauge fixing condition:
$$A_\mu A^\mu = 0 $$
And I've been asked to show if it is a valid gauge fixing condition or not. I believe that it isn't because I've already encountered a term similar to $A_\mu A^\mu$ in the Proca action and I've read that the mass term in that action makes that the Proca action is not Lorentz invariant.
I've tried to show that for a given configuration $(\phi,A)$ in this "gauge" I can't always transform it to a configuration in the Lorentz gauge that I already know that it is well defined.
I haven't been able to show it this far and I've been wondering if there is a mistake in my argument or if the mistake is in my calculations. Is this the right way to prove this? Is this a valid gauge fixing condition? Why?