Existence of advanced and retarded Green operators

Statement

Every normally hyperbolic operator on a vector bundle over a globally hyperbolic spacetime has unique future and past Green operators $G^+$ and $G^-$.

For each compactly supported source $f$, the section $G^+f$ is the unique solution of $Pu=f$ supported in $J^+(\operatorname{supp}f)$, and $G^-f$ is the unique solution supported in $J^-(\operatorname{supp}f)$. Their difference

solves the homogeneous equation after applying $P$ and is called the causal propagator.