I'm wondering a bit about the classical Maxwell equations in flat spacetime and their Cauchy problem. For the following, I suppose that the sources are already given and don't react to their own electromagnetic field (this approximation is the simplest case). I consider the following equation to be solved: $$\partial_a F^{ab} = \square \, A^b - \partial^b (\partial_a A^a) = J^b, \tag{1}$$ which gives 4 second order equations for the 4 unknowns $A^a(t, x^i)$, subject to the following initial conditions (at initial time $t_0$): \begin{align} A_a(t_0, x^i) &= \mathfrak{g}_a(x^i), \tag{2} \\[1em] \partial_0 A_a \Big|_{t_0} &= \mathfrak{h}_a(x^i). \tag{3} \end{align} The 8 functions $\mathfrak{g}_a(x^i)$ and $\mathfrak{h}_a(x^i)$ are the initial data. The 4 equations (1) aren't fully independant, because of charge conservation: $\partial_a J^a = 0$, so we only have 3 independant equations for the 4 unknowns. We then fix the gauge by imposing the Lorenz gauge condition: $$\partial_a A^a = 0. \tag{4}$$ Then equation (1) becomes $$\square A^a = J^a. \tag{5}$$ We get 4 independant differential equations for the 4 unknowns $A^a$. All is good.
What is not clear to me is the remaining gauge freedom, since (4) doesn't fix completely the gauge. If $A_a$ is a solution to (4)-(5) with initial data (2)-(3), then so is $$\tilde{A}_a = A_a + \partial_a \theta, \tag{6}$$ where $\theta$ is an arbitrary function satisfying $$\square \theta = 0, \tag{7}$$ and $\partial_a \theta = 0$ at $t = t_0$.
The solution doesn't appears to be unique, unless we could prove that $\theta$ is actually a simple constant. How could we prove that from the equations (4)-(5)? AFAIK, this is the Cauchy problem for these equations.
For a slight increase of time: $t = t_0 + \Delta t$, equation (7) gives $$\square \theta \equiv \partial_0^2 \theta - \nabla^2 \theta \approx \frac{1}{\Delta t} \Big( \partial_0 \theta \Big|_{t_0 + \Delta t} - \partial_0 \theta \Big|_{t_0} \Big) - \nabla^2 \theta \Big|_{t_0} = 0. \tag{8}$$ The second and third terms vanish since $\partial_a \theta = 0$ at $t = t_0$. Thus $\partial_0 \theta \Big|_{t_0 + \Delta t} = 0$, which implies $\theta$ doesn't depend on time. This also implies that it can't depend on space neither, and so $\theta$ is a constant and the Cauchy problem is solved (apparently, since (6) then gives $\tilde{A}_a \equiv A_a$).
I feel uneasy with the previous reasoning, and I suspect that something is wrong with it. I believe something is missing. I need help in fixing the reasoning and confirm that it's all good.
I've found a similar question here:
but its answers aren't satisfying at all. They don't answer the question, actually.
