TITLE: Hodge decomposition in Minkowski space
QUESTION [16 upvotes]: This question is motivated by the physical description of magnetic monopoles. I will give the motivation, but you can also jump to the last section.
Let us recall Maxwell’s equations: Given a semi-riemannian 4-manifold and a 3-form $j$. We describe the field-strength differential form $F$ as a solution of the equations
$\mathrm{d}F=0$
$\mathrm{d}\star F=j$ (where $\star$ denotes the Hodge star).
If the second de-Rham-cohomology vanishes (for example in Minkowski space), $F$ is exact and we can write it as $F=\mathrm{d}A$, where $A$ denotes a 1-form.
Now let us consider monopoles: We use two 3-forms $j_m$ (magnetic current) and $j_e$ (electric current) and consider the equations
$\mathrm{d}F=j_m$
$\mathrm{d}\star F=j_e$.
Essentially, it is described in this paper, but the author Frédéric Moulin (a physicist) uses coordinates. Now he assumes that (in Minkowski space) $F$ can be decomposed using two potentials — into an exact (in the image of the derivative) and a coexact (in the image of the coderivative) form: $F=\mathrm{d}A-\star\mathrm{d}C$. Is there a mathematical justification for this assumption (maybe it is just very pragmatic)?
The actual question
Given a 2-form $F$ on 4-dimensional Minkowski space (more generally: semi-riemannian manifolds)—are there any known conditions such that $F$ decomposes into an exact and a coexact form: $F=\mathrm{d}A+\star\mathrm{d}C$)?
For compact riemannian manifolds there is the well-known Hodge decomposition: There is always a decomposition into an exact, a coexact and a harmonic form. In the non-compact case you might be able to get rid of the harmonic form by only considering “rapidly decaying” forms (Wikipedia suggests that, but I do not have a good reference, in euclidean space there is the Helmholtz decomposition, and non-trivial (smooth) harmonic 1-forms do not vanish at infinity).
That is why I also ask: Are there “rapidly decaying” harmonic 2-forms in Minkowski space? Any references where I could see what is known about harmonic forms and Hodge theory in the semi-riemannian case are also welcome.

REPLY [11 votes]: Willie's answer is of course correct. But the part that I think is most interesting in the physical context is only briefly mentioned in point 3. of the last paragraph. Let me expand on that.
The relevant condition that make everything work nicely is that the background Lorentzian spacetime be globally hyperbolic. There are two important spaces of forms, $\Omega^p_0$ and $\Omega^p_{SC}$, $p$-forms with compact and spacelike compact supports, respectively. A set $X$ is spacelike compact if it is contained in the causal influence set of a compact set $Y$, $X\subseteq J(Y)$.
On Lorentzian manifolds, the D'Alembertian operator $\square = d\delta + \delta d$ is the analog of the Laplacian operator $\Delta$ in Riemannian signature. Note that I'm using the notation $\delta = (-1)^p{\star d \star}$. It is not elliptic, but it is hyperbolic with very nice properties. Instead of the nice analytical properties of the elliptic Laplacian, we have the following exact sequence (the image of each map is equal to the kernel of the next one):
$$ 0 \to \Omega^p_0 \stackrel{\square}{\to} \Omega^p_0 \stackrel{G}{\to} \Omega^p_{SC} \stackrel{\square}{\to} \Omega^p_{SC} \to 0 , $$
where $G=G^+-G^-$ is the causal Green function, defined as the difference of the retarded ($G^+$) and advanced ($G^-$) Green functions. The retarded and advanced Green functions of the D'Alembertian always exist on Globally hyperbolic manifolds (see, for instance, here or here).
A version of the result that you want is this: if $F\in \Omega^2_{SC}$, $d F = 0$ and $\delta F = 0$, then there exist two forms $A\in \Omega^1_{SC}$ and $B\in \Omega^3_{SC}$, with $\delta A = 0$ and $d B = 0$, such that $F = d A + \delta B$. Moreover, there exist $\alpha \in \Omega^1_0$ and $\beta \in \Omega^3_0$ such that $A=G\alpha$ and $B=G\beta$. This result can be found, for instance, as Prop.2.2 here. I think you can straight forwardly adapt the proof to the case with non-vanishing currents $j_e$ or $j_m$.