Global Hadamard form for the Green Function in Schwarzschild space-time

Understand

The retarded Green function of a wave equation on a 4-dimensional curved background spacetime is a (generalized) function of two spacetime points and diverges when these are connected by a null geodesic.

• The Hadamard form makes explicit the form of this divergence but only when one of the points is in a normal neighbourhood of the other point.
• In this paper we derive a representation for the retarded Green function for a scalar field in Schwarzschild spacetime which makes explicit its {\it complete} singularity structure beyond the normal neighbourhood.
• We interpret this representation as a sum of Hadamard forms, the summation being taken over the number of times the null wavefront has passed through a caustic point: the sum of Hadamard forms applies to the non-smooth contribution to the full Green function, not only the singular contribution.