While reading the reference
Eric Poisson and Adam Pound and Ian Vega,The Motion of Point Particles in Curved Spacetime, available here,
there is something that I don't quite understand. Eq.(16.6) is an evolution equation for de Green functional. Then in Eq.(16.7) Poisson et. al. look for a specific solution and they state that the separation of the Green functional is valid only in the convex neighborhood of a field point $x$. I assume that is because the Cauchy problem is valid only in that neighborhood... My question is why? Why is the Cauchy problem related to the imposition that the two points must be connected by a unique geodesic?
