# Klein-Gordon equation propagators: intersection with the support of the source

Let $$(M,g)$$ be a globally hyperbolic. Let $$P = \Box - m^2$$ be the Klein-Gordon differential operator. Following Fewster's notes, we may define the retarded/advanced propagators $$E^\pm : C^\infty_0(M)\to C^\infty(M)$$ as follows. First let $$f\in C^\infty_0(M)$$ and consider the problem $$P\phi = f$$

with two conditions imposed on the solution:

1. $$\operatorname{supp} \phi \subset J^\pm (\operatorname{supp}f)$$;
2. $$\operatorname{supp}\phi \cap J^\mp (\operatorname{supp}f)$$ is compact;

One shows that the problem has unique solution and define $$E^\pm(f)$$ to be the solution to the corresponding problem.

I'm trying to gain intuition on this.

First consider the $$E^+$$ case. Condition (1) seems to mean that "to the past of when $$f$$ is turned on the solution vanishes". In that sense, it seems it allows us to say that the solution is created by the source $$f$$.

Now for the $$E^-$$ case. Now condition (1) seems to mean that "to the future of when $$f$$ ceases to exist the solution vanishes". In that sense, it seems that the solution is in fact what creates $$f$$.

Condition (2), on the other hand, I can't see how to interpret.

So, is my intuition on condition (1) correct? What is the intuition for demanding condition (2) when defining $$E^\pm$$?

Let's consider the $$E^+$$ case. The set $$J^+(\text{supp} f)$$ is the region of $$M$$ that can be reached causally starting from the support of $$f$$, either by light rays or by massive particles. So it's a causality condition: it means that $$\phi(x)$$ has to vanish if $$x$$ cannot be reached by a light ray starting from the support of $$f$$.
The second condition is less intuitive. The set $$J^{-}$$ is obtained from the support of $$f$$ by evolving backwards in time. There is bound to be some overlap between $$J^+$$ and $$J^-$$: there are points in $$M$$ that can be reached going in both directions. However, the support of $$\phi$$ should not reach back in time indefinitely: $$\phi$$ is generated by the source $$f$$ which only kicks in at some definite time, say $$t=0$$. So the support of $$\phi$$ can have a little bit of overlap with $$J^{-}(f)$$, but $$\phi$$ should completely vanish for $$t < t_0$$ for some time $$t_0$$.
Above I've introduced a time coordinate $$t$$, and since you're working on some abstract manifold $$M$$ you probably want to avoid that. But it's the gist of the idea.